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Grade 5 Data and Graphing Worksheets

Introducing statistics.

Our  grade 5 data and graphing exercises extend previous graphing skills (double bar and line graphs, line plots and circle graphs with fractions) and introduce basic probability and statistics (range, mode, mean, median).

Grade 5 data and graphing worksheet

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• 6th Grade Math
• Data graphing

Grade 6 data and graphing worksheets: Creating and interpreting graphs

Do you want to reinforce your 6 th grader’s data and graphing skills with fun and engaging activities? If so, you’ve come to the right place. In this article, we’ll share why data and graphs are useful for 6 th graders, simple steps to follow when creating and interpreting graphs , and where to find enriching Grade 6 data and graphing worksheets for your students.

How to make and read graphs: Activities and worksheets for 6th Grade

Do your students like puzzles, games, and stories? Do they enjoy finding out new things and sharing them with others? If you answered yes, they might also love these data and graphing activities and worksheets for 6th Grade! In this article, they will learn how to make and read different types of graphs using data from various sources. They will also discover how to use data and graphing skills to have fun and learn more about their favorite topics.

Data and graphing are ways of collecting, organizing, and visually presenting information. They can help us explore the world around us, answer questions, solve problems, and communicate our ideas better.

So, by the end of this article, your 6th-grade students will see that data and graphing are useful, exciting, and creative!

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Start practice on Sixth Grade here

What are data and graphs, and why are they useful for 6th graders.

Data are pieces of information that can be measured or counted. For example, the number of students in your class, the colors of your clothes, the temperature of the day, and the scores of a game are all data. We can collect data by observing, surveying, experimenting, or researching.

Graphs are visual representations of data that show how they are related or change over time. For example, a bar graph can show how many students prefer different types of music, a line graph can show how the temperature changes throughout the year, and a pie chart can show how the scores of a game are distributed among players.

Graphs can help us understand data better by making them easier to compare, analyze, and communicate. For 6th graders, data and graphing are useful skills because they can help them with math, science, social studies, and other subjects. They can also help them develop critical thinking, creativity, and communication skills to prepare them for future challenges.

Types of data: How to classify data into categories, numerical or ordinal

There are three main types of data that we can use to make graphs : categorical, numerical, and ordinal .

Categorical data are data that belong to different groups or categories. For example, the names of animals, the types of fruits, and the colors of cars are categorical data. We can use words or symbols to label categorical data.

Numerical data are data that have numerical values or measurements. For example, the heights of plants, the weights of rocks, and the distances of planets are numerical data. We can use numbers or units to label numerical data.

Ordinal data are data that have an order or rank. For example, the student's grades, the ratings of movies, and the difficulty levels are ordinal data. We can use numbers or words to label ordinal data.

Types of graphs: How to choose the best graph for different types of data

We can use many types of graphs to display different types of data. The most common are bar graphs , line graphs , pie charts , pictographs , histograms , scatter plots , and stem-and-leaf plots . Each type of graph has its advantages and disadvantages depending on the type of data and the purpose of the graph. Here are some general guidelines on how to choose the best graph for different types of data :

• Bar graphs : Use bar graphs to compare categorical or ordinal data across groups or categories. For example, we can use a bar graph to show how many students like different types of sports.
• Line graphs : Use line graphs to show numerical data that change over time or across a continuous variable. For example, we can use a line graph to show how people’s height changes as they grow older.
• Pie charts : Use pie charts to show categorical or ordinal data as parts of a whole or percentages. For example, we can use a pie chart to show how much time we spend on different daily activities.
• Pictographs : Use pictographs to show categorical or ordinal data using pictures or symbols representing a specific number or amount. For example, we can use a pictograph to show how many books we read in a month using book icons.
• Histograms : Use histograms to show numerical data grouped into intervals or bins. For example, a histogram can show how many students have different heights in a class.
• Scatter plots : Scatter plots show numerical data with two variables or dimensions. For example, we can use a scatter plot to show how the weight and height of students are related.
• Stem-and-leaf plots : Use stem-and-leaf plots to show numerical data with two or more digits. For example, we can use a stem-and-leaf plot to show how many points each player scored in a game.

Creating graphs: How to use tables, scales, labels, and titles to make graphs

Before creating a graph , we must organize our data in a table. A table is a way of arranging data in rows and columns. A table can help us see the patterns and relationships in our data more clearly. To make a table, we’ll decide what variables or categories to include and how to label them. For example, if you want to make a bar graph of how many students like different types of music, you need to choose the types of music as your categories and the number of students as your variable. You can label the categories as "Music Type" and the variable as "Number of Students."

After making a table, we’ll choose a scale for our graph. A scale is a way of measuring or dividing the data along an axis. An axis is a line that shows the range or direction of the data. A graph usually has two axes: a horizontal axis (x-axis ) and a vertical axis (y-axis ). To choose a scale, we must consider our data's minimum and maximum values and how to divide them into equal intervals or units. For example, if you want to make a bar graph of how many students like different types of music, and the minimum and maximum values are 0 and 20, you can choose a scale of 1 unit = 1 student for both axes.

After choosing a scale, label your axes with the names and units of your variables or categories. For example, if you want to make a bar graph of how many students like different types of music, you can label the x-axis as "Music Type" and the y-axis as "Number of Students."

After you label your axes, draw your graph using the data from your table and the scale from your axes. For example, if you want to make a bar graph of how many students like different types of music, you need to draw a bar for each type of music that matches the number of students from your table.

After drawing your graph, we must give it a title that describes what it shows. For example, if you want to make a bar graph of how many students like different types of music, you can give it a title like "Students' Music Preferences."

Interpreting graphs: How to read and understand the information in graphs

Once we have made a graph, you will interpret it. Interpreting a graph means reading and understanding the information that it shows. To interpret a graph, we can ask ourselves these questions:

• What is the title of the graph? What does it tell about the topic or purpose of the graph?
• What are the labels and units of the axes? What do they tell about the variables or categories in the graph?
• What are the scales of the axes? What do they tell about the range or intervals of the data in the graph?
• What are the shapes and colors of the bars, lines, pies, or points in the graph? What do they tell about the values or relationships of the data in the graph?
• What are the patterns or trends in the graph? What do they tell about how the data change or vary over time or across groups or categories?
• What are the outliers or errors in the graph? What do they tell about any unusual or incorrect data in the graph?

Comparing graphs: How to compare and contrast different graphs and data sets

Sometimes, we may want to compare two or more graphs or data sets that show similar or different information. Comparing graphs means finding out how they are alike or different. To compare graphs, we’ll look at some aspects:

• The titles : Do they show similar or different topics or purposes?
• The labels and units : Do they show similar or different variables or categories?
• The scales : Do they show similar or different ranges or intervals?
• The shapes and colors : Do they show similar or different values or relationships?
• The patterns or trends : Do they show similar or different changes or variations?
• The outliers or errors : Do they show similar or different unusual or incorrect data?

Analyzing graphs: How to identify patterns, trends, outliers, and errors in graphs

One of the most essential skills for 6th graders is to be able to analyze graphs and data. Analyzing means to examine something closely and carefully to understand what it means. When we analyze graphs, we can look for four things: patterns, trends, outliers, and errors .

Patterns are repeated or regular shapes or arrangements in the data. For example, if we have a graph that shows the number of books read by different students in a month, we might see a pattern of increasing or decreasing numbers. Patterns can help us see how the data is organized and related.

Trends are general directions or tendencies in the data. For example, if a graph shows a city's average temperature over a year, we might see a trend of rising or falling temperatures. Trends can help us see how the data changes over time or across categories.

Outliers are data points that are very different from the rest of the data. For example, if we have a graph showing the height of different plants in a garden, we might see an outlier much taller or shorter than the others. Outliers can help us see if there are any unusual or exceptional cases in the data.

Errors are mistakes or inaccuracies in the data or the graph. For example, if we have a graph showing the population of different countries worldwide, we might see an error with a wrong number or a missing label. Errors can affect how we interpret and trust the data and the graph.

To analyze graphs and data, we need to ask ourselves some questions:

• What is the purpose and audience of the graph?
• What type of data and graph is used?
• What are the scales, labels, and titles of the graph?
• What patterns and trends can we see in the data?
• Are there any outliers or errors in the data or the graph?
• What conclusions or inferences can we make from the graph?

Your 6th-grade students can improve their critical thinking and problem-solving skills by asking and answering the questions above.

Evaluating graphs: How to critique and improve graphs based on their purpose and audience

Another vital skill for 6th graders is to be able to evaluate graphs and data. Evaluating means to judge something based on specific criteria or standards. When we evaluate graphs, we can use four criteria: clarity, accuracy, relevance, and ethics .

Clarity means how easy it is to understand the graph and the data. A clear graph should have the following:

• A suitable type of graph for the type of data
• A clear title that describes what the graph shows
• Labels for the axes, categories, and units
• Scales that are appropriate and consistent
• Colors, shapes, and symbols that are distinct and meaningful

Accuracy means how correct and precise the graph and the data are. An accurate graph should have:

• Data that is collected from reliable sources and methods
• Data that is represented correctly and consistently
• No errors or mistakes in the data or the graph
• No distortion or manipulation of the data or the graph

Relevance means how vital and valuable the graph and the data are for the purpose and audience. A relevant graph should have:

• Data that is related to the topic or question
• Data that is sufficient and appropriate for the purpose
• Data that is current and up-to-date
• Data that is interesting and engaging for the audience

Ethics means how fair and respectful the graph and the data are for the people and situations involved. An ethical graph should have the following:

• Data that is collected and used with consent and honesty
• Data that is respectful of privacy and confidentiality
• Data that is unbiased and objective
• Data that is presented with honesty and integrity

To evaluate graphs and data, we need to ask ourselves some questions:

• How clear is the graph and the data?
• How accurate are the graph and the data?
• How relevant is the graph and the data?
• How ethical is the graph and the data?

Your 6 th graders can improve their communication and presentation skills by asking and answering the above questions.

What are the main takeaways and applications of data and graphing skills?

Although Data and graphing skills are useful for 6th graders, they are also essential for everyone who wants to learn more about themselves and their world. By learning how to create, interpret, compare, analyze, and evaluate graphs , we can:

• Organize information in a visual way
• Understand information in a deeper way
• Communicate information in an effective way
• Make decisions based on information

Data and graphing skills can help us in many areas of life, such as:

• Science : We can use graphs to explore natural phenomena such as weather, climate, plants, animals, etc.
• Math : We can use graphs to explore mathematical concepts such as numbers, operations, fractions, decimals, etc.
• Social Studies : We can use graphs to explore social issues such as population, culture, history, geography, etc.
• Art : We can use graphs to express our creativity and emotions, such as colors, shapes, patterns, etc.

Data and graphing skills can also help us develop essential habits of mind such as:

• Curiosity : We can use graphs to ask questions and seek answers
• Creativity : We can use graphs to generate ideas and solutions
• Critical Thinking : We can use graphs to analyze and evaluate information
• Collaboration : We can use graphs to share and discuss information with others

Bonus: Where to find exciting resources to improve 6th graders' graphing skills?

If you want your 6 th graders to learn more about data and graphing or practice their skills, there are many exciting online resources that you can use. Here are some of them:

• Khan Academy : This website has videos, exercises, and quizzes on many topics, including data and graphing. Grade 6 students can learn how to create and interpret different graphs, such as bar graphs, line graphs, pie charts, histograms, etc. They can also learn to analyze data using mean, median, mode, range, and other measures. https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-data-statistics
• Math is Fun : This website has interactive activities and games that help 6 th graders learn and practice data and graphing skills. They can create graphs using online tools or play games that test their knowledge of graphs and data. They can also find definitions and examples of data and graphing terms. https://www.mathsisfun.com/data/index.html
• IXL : This website has online practice questions on data and graphing for 6th grade. Students can choose from different topics, such as organizing data, creating graphs, interpreting graphs, comparing data sets, and more. They can get instant feedback and explanations for their answers. https://www.ixl.com/math/grade-6

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Data and graphing are essential skills we will use in many areas of life. They help us organize, display, understand, and communicate information in a visual way. They also help us make sense of the world around us and discover new things.

In this article, your 6 th Grade students have learned about the basics of data and graphing, such as how to classify data into different types, how to choose the best graph for their data, how to create and interpret graphs, how to compare and analyze data sets, and how to evaluate graphs based on their purpose and audience.

You also learned where to find exciting resources to improve your 6 th graders’ data and graphing skills online. They can use these websites to watch videos, exercise, play games, or create graphs.

We hope you and your students enjoyed this article and learned something new. Data and graphing can be fun and exciting if you know how to use them. Remind your students to practice their skills regularly and apply them to real-life situations.

Happy graphing!

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PICTOGRAPHS

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PICTOGRAPH WORKSHEETS

Pictograph worksheets or picture graph worksheets in this section teach children to represent data using pictures or symbols. This graphical representation is much easier as the numerical data collected is represented using pictures. Worksheets without key for beginners, with key for grade 2 through grade 6, reading, interpretting, and drawing pictographs are all given for practice. Also, worksheets to represent data using Tally marks and Pictographs and matching data to the given pictographs are here for best understanding of both these topics. You can download all these free pictograph worksheets for abundant practice. 

Let us learn to represent the information on a picture graph

1. The first step is to collect data.    Example :  Collect this data from your class. Suggest your friends 5 colors and ask them their favorite color. Lets consider 5 colors. Red,  Blue, Green, Yellow and Orange.

2. Make a small table to record this information.  With the given information, you can easily make a pictograph. Consider the  key as 2. Select a simple picture for the pictograph.

3.  8 students like Red color. According to the key 2 students can be represented using a  picture. How many 2’s are in 8. Its 8/2 = 4. Use 4 pictures. 7 students like Blue color. Now divide 7 by 2. 7/2 = 3.5 or 3 1/2. So we will use 3 full  pictures which represent 6 students and 1 half picture representing 1 student.  Similarly you can represent pictures for Green, Yellow, and Orange.

Observations

•  The most preferred color- Yellow
•  The least preferred color- Green
•  The total number of students surveyed= 8+ 7+ 4+ 12+ 10 = 41

Here we have prepared worksheets according to the Common core Standards. Worksheets can be used from grade 1 through grade 6. 

RELATED DATA HANDLING TOPICS   :    Tally marks

  PICTOGRAPHS / PICTURE GRAPH WORKSHEETS PDF

Reading picture graph - basic.

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Picture graph - with Key - Easy

Picture graph - with Key - Medium

Picture graph - with Key - Hard

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Drawing Pictograph - Hard

Drawing Tally marks & Pictograph

Drawing Tally marks & Pictograph (with pictures)

Matching Pictograph

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Representing Data

Here we will learn about representing data, including how to create and interpret the different tables, charts, diagrams and graphs we can use to represent data.

There are also representing data worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is representing data?

Representing data allows us to display and interpret collected data.

Different types of data need to be represented in different formats so we need to know how to construct different types of tables, charts, diagrams or graphs.

Here are some of the charts and graphs you may come across.

Tally charts

A tally chart is a simple way to organise data.

Tally charts can be used for different types of data.

For example,

20 students were asked about their favourite sport. Here are their answers.

Here are these results in a tally chart.

Step-by-step guide: Tally charts

A pictograph (or pictogram) is a chart which uses a symbol to represent a certain frequency. This is then used in a chart.

For example, here is a pictogram to show the number of parcels an office sends in a week.

A full circle represents 4 parcels. So, a semi-circle represents 2 parcels and so on.

Step-by-step guide: Pictograph

Stem and leaf diagram

A stem and leaf diagram organises numerical data into groups, but also retains the value of each piece of data. We can then analyse the data in more depth and find the mode, or the median, and so on.

Here is a stem and leaf diagram for the heights of children.

We can see that the smallest child is 137 \ cm and the tallest child is 165 \ cm. This means we can calculate the range of the data by subtracting the lowest value from the highest value.

Range =165-137=18 \ \text{cm}

Step-by-step guide: Stem and leaf diagram

x and y axis

Many statistical graphs need a pair of axes drawn to start the graph. The x -axis is the name of the horizontal axis and the y -axis is the name of the vertical axis. Care needs to be taken with the scale on the axes. For statistics the x -axis and the y -axis are given different labels.

Here is a bar chart to show the gender of people working in a company. For this bar chart the y axis becomes frequency (f) and the x -axis has the label “gender”.

Step-by-step guide: x and y axis

A bar chart represents data by using vertical (or horizontal) bars of equal width. They are used for categorical or discrete numerical data.

Bar charts can become more complicated and extend to comparative bar charts or compound bar charts (or stacked bar charts).

For example, 

Here is a comparative bar chart showing the favourite pets of boys and girls in a class.

Step-by-step guide: Bar charts

Scatter graphs

Scatter graphs are used to represent bivariate data (two variables). This is when data occurs in pairs of values. The data is plotted on a pair of axes as coordinates.

Scatter graphs can be used to comment about correlation, the relationship between the two variables.

A line of best fit can be drawn through the data and used to help estimate a missing value.

Here is a scatter graph showing the Maths and English scores of a group of students. There is a positive correlation. A line of best fit has been drawn and is being used to estimate a missing English score.

Step-by-step guide: Scatter graphs

Two way tables

Two way tables are a type of table used to organise more complex data where there are two aspects which need to be looked at.

For example, here is a two way table to represent whether people in an office drink tea, or coffee, or neither. Whether the person is male or female is also seen in the table.

Step-by-step guide: Two way tables

A pie chart is a chart with a visual representation of all items of data within a data set using a circle. The sectors of a pie chart are proportional to the different items in the data set. Pie charts are used to represent categorical data.

For example, here is a pie chart to show the favourite type of music.

Step-by-step guide: Pie charts

This is sometimes called a time series graph. Data is plotted on a pair of axes and the points connected by line segments.

For example, a line graph showing sales over a week.

Step-by-step guide: Line graph

Vertical line diagram

A vertical line diagram is very similar to a bar chart. Vertical lines are used instead of bars.

For example, a vertical line diagram to show the shoe sizes of pupils in a dance school.

Time series graph

A time series graph is a line graph which shows data over a given time period. They can be used to show a trend in the data and are useful for making predictions about the future.

The horizontal axis is always used to show the time. The vertical axis represents the variable being recorded against time.

For example, a time series graph showing average temperature over two years. The years are split up into quarters.

Step-by-step guide: Times series graph

Frequency polygon

A frequency polygon is a graph that shows the frequencies of grouped data. The points are plotted using the midpoints of the class intervals against the frequencies and then the points are joined up with straight lines.

Here is the frequency table showing the results of a History test.

Here is the frequency polygon showing the results of the History test.

Step-by-step guide: Frequency polygon

Frequency diagram

Frequency diagrams are usually bar charts, vertical line diagrams or frequency polygons with frequency displayed on the vertical axis.

Step-by-step guide: Frequency diagram

Frequency graphs

Frequency graphs include the charts mentioned in frequency diagrams. But, it also extends to including more complex graphs such as cumulative frequency graphs and histograms.

Step-by-step guide: Frequency graphs

A histogram is used to represent continuous numerical data. They look like bar charts, but they are different. In a bar chart, the heights of the bars represent the frequencies, whereas in a histogram the area of the bars represent the frequencies.

The vertical axis is labelled with frequency density.

The widths of the bars are usually different.

For example, here is a histogram representing the heights of samplings.

The frequency of the first bar is 20 \times 1=20. This means that 20 saplings were between 0 and 20 \ cm tall.

Step-by-step guide: Histograms

Cumulative frequency

Cumulative frequency is the running total of frequencies in a frequency distribution. It can be used to draw a cumulative frequency graph which is useful for representing or analysing the distribution of a large grouped data set.

Cumulative frequency is closely linked to box plots too.

Cumulative frequency can also be used to find estimates for the median value and quartiles of a data set.

For example, here is a cumulative frequency graph for the mass of apples produced by 200 apple trees.

The estimate of the median is 36kg.

Step-by-step guide: Cumulative frequency

A box plot is a diagram showing the following information for a set of data,

• Lowest value (or smallest value)
• Lower quartile
• Upper quartile
• Highest value (or largest value)

They are useful as they show the overall distribution of the data. They can be used to compare one or more sets of data.

For example, here is a box plot showing the amount of money shoppers spend in a local supermarket.

The median is £15.

Step-by-step guide: Box plots

How to represent data

In order to represent data:

Tip 1 Check the type of table, chart, diagram or graph.

Tip 2 Check labels, keys and titles.

Tip 3 Check the scale.

Tip 4 Be as accurate as you can.

Explain how to represent data

Representing data worksheet

Get your free representing data worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Representing data examples

Example 1: tally charts.

Complete the frequencies for the tally chart below.

1 Check the type of table, chart, diagram or graph.

This is a tally chart.

4 Be as accurate as you can.

Be careful counting the tally marks.

This means 5,

Example 2: stem and leaf diagram

Members of a running club were asked their age. Their ages are in this stem and leaf diagram. Write down the mode of their ages.

Check the type of table, chart, diagram or graph.

This is a stem and leaf diagram.

Check labels, keys and titles.

There is a key, which is needed to help us write down the correct age for the mode.

The mode of the ages of the people in the running club is 27 years.

Example 3: scatter graph

Here is a scatter graph. Clive was ill and missed paper 1. He scored 60 on paper 2.

Use the scatter graph and line of best fit to estimate the score on paper 1.

This is a scatter graph. It shows the relationship with bivariate data. The pairs of data are the scores on Paper 1 and Paper 2.

The horizontal axis is for Paper 1 scores and the vertical axis is for Paper 2 scores.

Check the scale.

The scale on both axes are different.

Be as accurate as you can.

Draw a straight line across from 60 on Paper 2. Then go straight down and read off the value for Paper 1.

The estimated score on Paper 1 for Clive is 30 marks.

Example 4: pie chart

24 pupils were asked which subject was their favourite subject. Here is a pie chart to show the results. Work out how many pupils said Science was their favourite subject.

This is a pie chart, so the size of the sector represents the proportion of pupils who chose the different subjects as their favourite.

We need the Science section.

The Science section is a right-angle, so a quarter of the pupils.

So 6 pupils chose Science.

Example 5: two way tables

Complete the two way table.

This is a two way table. We need to look at the column totals and the row totals. We look at a row (or a column) and find 1 missing value to fill in. We could start with the bottom row.

Example 6: frequency polygon

A vet weighs all the dogs she sees in a week. Here are the results.

Draw a frequency polygon to show the results.

We need to draw a frequency polygon. We need to use the midpoints of the groups, 5, 15, 25 and so on.

Frequency is on the vertical axis and mass is on the horizontal axis.

There are two axes, both with different scales, so be careful plotting the points.

Plot points with a sharp pencil and crosses to be accurate. Join the points up to complete the frequency polygon.

Common misconceptions

• Use a pencil, a ruler and a protractor

When drawing graphs, diagrams and charts use a sharp pencil and a ruler so that you can be as accurate as possible. For pie charts, use a protractor to measure the angles accurately.

• Lines of best fit are straight

In GCSE maths the lines of best fit on scatter graphs are a straight line.

• Check the scale

Check the scale and how it is broken up into smaller sections. Scales can come in a variety of styles.

Here is 0 to 100 split up into 10 equal sections, 5 equal sections and 4 equal sections.

Practice representing data questions

1.  Which of these tally charts is NOT correct?

Remember that tally marks are grouped together in 5’s. Check that the frequencies match the number of tally marks.

2. What is the mode?

The value 157 \ cm occurs twice. Therefore the mode is 157.

3. Which type of correlation does this scatter graph show?

Negative correlation

Positive correlation

Inverse correlation

No correlation

There is a downward trend in the plots. This shows there is negative correlation.

4. Which pie chart represents the data in this frequency table?

The total of the frequencies is 40. The frequency of A is 10, which is a quarter of 40. So, section A needs to be a quarter of the pie chart. Similarly section C needs to be a quarter too.

The frequency of B is 20, which is half of 40. Section B needs to be half of the pie chart.

5. Complete the two way table.

Check the rows and columns for a missing value. We could start with finding the missing value for Class X and Spanish,

Then continue to fill in the missing values.

We can also check the column totals and the row totals to check if they are correct.

6. Which of these is the correct frequency polygon for the frequency table below?

The points should be plotted using the midpoints of the groups: 10, 30, 50, 70 and 90.

They should be plotted using the correct frequencies: 1, 9, 8, 3 and 2.

The points need joining up, but NOT the last and the first points.

Representing data GCSE questions

1. Sam has drawn this bar chart. Give two criticisms of the bar chart.

For a valid criticism of the scale. For example, the scale on the vertical axis is missing the number 3.

For a valid criticism about the widths of the bars. For example, the bar for lemonade is too wide.

2. Ali conducts a survey on how long students spend on their maths homework in a week. Here are the results.

Draw a frequency polygon for the information in the table.

For plotting 3 or 4 points correctly.

For all points plotted correctly and joining up the points correctly.

3. Shelia spends £1440 each month running her household. Her household costs can be broken up into food, bills and rent.

(a) Work out how much money Sheila spends on food a month.

(b) Work out how much more money Sheila spends on rent than bills each month.

For using the right angle as a quarter. For example, 1440 \div 4.

For finding the angle for rent. For example, 360-90-120=150.

For finding the difference in angles between rent and bills. For example, 150-120=30.

For finding the proportion of the total costs. For example, \frac{30}{360} \times 1440.

Alternative Method:

For finding the proportion for bills, \frac{120}{360} \times 1440=480 \ OR rent, \frac{150}{360} \times 1440=600.

For finding the difference in money between rent and bills, 600-480.

Learning checklist

You have now learned how to:

• Construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data
• Interpret and construct tables and line graphs for time series data
• Use and interpret scatter graphs of bivariate data; recognise correlation and draw estimated lines of best fit
• Construct and interpret diagrams for grouped discrete data and continuous data, i.e. histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use

The next lessons are

• Frequency table
• Mean, median, mode
• Types of sampling method

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Praxis Core Math

Course: praxis core math   >   unit 1, data representations | lesson.

• Data representations | Worked example
• Center and spread | Lesson
• Center and spread | Worked example
• Random sampling | Lesson
• Random sampling | Worked example
• Scatterplots | Lesson
• Scatterplots | Worked example
• Interpreting linear models | Lesson
• Interpreting linear models | Worked example
• Correlation and Causation | Lesson
• Correlation and causation | Worked example
• Probability | Lesson
• Probability | Worked example

What are data representations?

• How much of the data falls within a specified category or range of values?
• What is a typical value of the data?
• How much spread is in the data?
• Is there a trend in the data over time?
• Is there a relationship between two variables?

What skills are tested?

• Matching a data set to its graphical representation
• Matching a graphical representation to a description
• Using data representations to solve problems

How are qualitative data displayed?

• A vertical bar chart lists the categories of the qualitative variable along a horizontal axis and uses the heights of the bars on the vertical axis to show the values of the quantitative variable. A horizontal bar chart lists the categories along the vertical axis and uses the lengths of the bars on the horizontal axis to show the values of the quantitative variable. This display draws attention to how the categories rank according to the amount of data within each. Example The heights of the bars show the number of students who want to study each language. Using the bar chart, we can conclude that the greatest number of students want to study Mandarin and the least number of students want to study Latin.
• A pictograph is like a horizontal bar chart but uses pictures instead of the lengths of bars to represent the values of the quantitative variable. Each picture represents a certain quantity, and each category can have multiple pictures. Pictographs are visually interesting, but require us to use the legend to convert the number of pictures to quantitative values. Example Each represents 40 ‍   students. The number of pictures shows the number of students who want to study each language. Using the pictograph, we can conclude that twice as many students want to study French as want to study Latin.
• A circle graph (or pie chart) is a circle that is divided into as many sections as there are categories of the qualitative variable. The area of each section represents, for each category, the value of the quantitative data as a fraction of the sum of values. The fractions sum to 1 ‍   . Sometimes the section labels include both the category and the associated value or percent value for that category. Example The area of each section represents the fraction of students who want to study that language. Using the circle graph, we can conclude that just under 1 2 ‍   the students want to study Mandarin and about 1 3 ‍   want to study Spanish.

How are quantitative data displayed?

• Dotplots use one dot for each data point. The dots are plotted above their corresponding values on a number line. The number of dots above each specific value represents the count of that value. Dotplots show the value of each data point and are practical for small data sets. Example Each dot represents the typical travel time to school for one student. Using the dotplot, we can conclude that the most common travel time is 10 ‍   minutes. We can also see that the values for travel time range from 5 ‍   to 35 ‍   minutes.
• Histograms divide the horizontal axis into equal-sized intervals and use the heights of the bars to show the count or percent of data within each interval. By convention, each interval includes the lower boundary but not the upper one. Histograms show only totals for the intervals, not specific data points. Example The height of each bar represents the number of students having a typical travel time within the corresponding interval. Using the histogram, we can conclude that the most common travel time is between 10 ‍   and 15 ‍   minutes and that all typical travel times are between 5 ‍   and 40 ‍   minutes.

How are trends over time displayed?

How are relationships between variables displayed.

• (Choice A)   A
• (Choice B)   B
• (Choice C)   C
• (Choice D)   D
• (Choice E)   E
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
• a proper fraction, like 1 / 2 ‍   or 6 / 10 ‍  
• an improper fraction, like 10 / 7 ‍   or 14 / 8 ‍  

Things to remember

• When matching data to a representation, check that the values are graphed accurately for all categories.
• When reporting data counts or fractions, be clear whether a question asks about data within a single category or a comparison between categories.
• When finding the number or fraction of the data meeting a criteria, watch for key words such as or , and , less than , and more than .

• Math Article

Graphical Representation

Graphical Representation is a way of analysing numerical data. It exhibits the relation between data, ideas, information and concepts in a diagram. It is easy to understand and it is one of the most important learning strategies. It always depends on the type of information in a particular domain. There are different types of graphical representation. Some of them are as follows:

• Line Graphs – Line graph or the linear graph is used to display the continuous data and it is useful for predicting future events over time.
• Bar Graphs – Bar Graph is used to display the category of data and it compares the data using solid bars to represent the quantities.
• Histograms – The graph that uses bars to represent the frequency of numerical data that are organised into intervals. Since all the intervals are equal and continuous, all the bars have the same width.
• Line Plot – It shows the frequency of data on a given number line. ‘ x ‘ is placed above a number line each time when that data occurs again.
• Frequency Table – The table shows the number of pieces of data that falls within the given interval.
• Circle Graph – Also known as the pie chart that shows the relationships of the parts of the whole. The circle is considered with 100% and the categories occupied is represented with that specific percentage like 15%, 56%, etc.
• Stem and Leaf Plot – In the stem and leaf plot, the data are organised from least value to the greatest value. The digits of the least place values from the leaves and the next place value digit forms the stems.
• Box and Whisker Plot – The plot diagram summarises the data by dividing into four parts. Box and whisker show the range (spread) and the middle ( median) of the data.

General Rules for Graphical Representation of Data

There are certain rules to effectively present the information in the graphical representation. They are:

• Suitable Title: Make sure that the appropriate title is given to the graph which indicates the subject of the presentation.
• Measurement Unit: Mention the measurement unit in the graph.
• Proper Scale: To represent the data in an accurate manner, choose a proper scale.
• Index: Index the appropriate colours, shades, lines, design in the graphs for better understanding.
• Data Sources: Include the source of information wherever it is necessary at the bottom of the graph.
• Keep it Simple: Construct a graph in an easy way that everyone can understand.
• Neat: Choose the correct size, fonts, colours etc in such a way that the graph should be a visual aid for the presentation of information.

Graphical Representation in Maths

In Mathematics, a graph is defined as a chart with statistical data, which are represented in the form of curves or lines drawn across the coordinate point plotted on its surface. It helps to study the relationship between two variables where it helps to measure the change in the variable amount with respect to another variable within a given interval of time. It helps to study the series distribution and frequency distribution for a given problem.  There are two types of graphs to visually depict the information. They are:

• Time Series Graphs – Example: Line Graph
• Frequency Distribution Graphs – Example: Frequency Polygon Graph

Principles of Graphical Representation

Algebraic principles are applied to all types of graphical representation of data. In graphs, it is represented using two lines called coordinate axes. The horizontal axis is denoted as the x-axis and the vertical axis is denoted as the y-axis. The point at which two lines intersect is called an origin ‘O’. Consider x-axis, the distance from the origin to the right side will take a positive value and the distance from the origin to the left side will take a negative value. Similarly, for the y-axis, the points above the origin will take a positive value, and the points below the origin will a negative value.

Generally, the frequency distribution is represented in four methods, namely

• Smoothed frequency graph
• Pie diagram
• Cumulative or ogive frequency graph
• Frequency Polygon

Merits of Using Graphs

Some of the merits of using graphs are as follows:

• The graph is easily understood by everyone without any prior knowledge.
• It saves time
• It allows us to relate and compare the data for different time periods
• It is used in statistics to determine the mean, median and mode for different data, as well as in the interpolation and the extrapolation of data.

Example for Frequency polygonGraph

Here are the steps to follow to find the frequency distribution of a frequency polygon and it is represented in a graphical way.

• Obtain the frequency distribution and find the midpoints of each class interval.
• Represent the midpoints along x-axis and frequencies along the y-axis.
• Plot the points corresponding to the frequency at each midpoint.
• Join these points, using lines in order.
• To complete the polygon, join the point at each end immediately to the lower or higher class marks on the x-axis.

Draw the frequency polygon for the following data

Mark the class interval along x-axis and frequencies along the y-axis.

Let assume that class interval 0-10 with frequency zero and 90-100 with frequency zero.

Now calculate the midpoint of the class interval.

Using the midpoint and the frequency value from the above table, plot the points A (5, 0), B (15, 4), C (25, 6), D (35, 8), E (45, 10), F (55, 12), G (65, 14), H (75, 7), I (85, 5) and J (95, 0).

To obtain the frequency polygon ABCDEFGHIJ, draw the line segments AB, BC, CD, DE, EF, FG, GH, HI, IJ, and connect all the points.

Frequently Asked Questions

What are the different types of graphical representation.

Some of the various types of graphical representation include:

• Line Graphs
• Frequency Table
• Circle Graph, etc.

Read More:  Types of Graphs

What are the Advantages of Graphical Method?

Some of the advantages of graphical representation are:

• It makes data more easily understandable.
• It saves time.
• It makes the comparison of data more efficient.

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Very useful for understand the basic concepts in simple and easy way. Its very useful to all students whether they are school students or college sudents

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Question/answer, how does the mastery of the graphical representation skill affect a student's performance at an early age.

Mastery of graphical representation at an early age significantly enhances a student's performance by improving their ability to interpret, analyze, and communicate complex information visually. This skill fosters critical thinking, problem-solving abilities, and mathematical understanding, which are foundational for academic success across various subjects. It also boosts confidence in handling data and aids in the development of spatial reasoning skills.

Why is the Graphical representation skill important for Preschool students?

Graphical representation skills are crucial for preschool students as they lay the foundation for critical thinking, problem-solving, and communication. These skills enable young learners to visually express ideas, understand abstract concepts through concrete representations, and develop early literacy and numeracy skills. This fosters cognitive development, creativity, and the ability to interpret and convey information effectively from a young age.

How to train the Graphical representation skill in Preschool students learning about Reading Fiction?

To train preschool students in graphical representation while reading fiction, introduce them to story mapping. Use simple visual aids like storyboards or charts to map out key elements: characters, setting, problem, and solution. Encourage them to draw or use stickers to represent these elements, helping them visualize and understand the story structure and enhancing their comprehension and retention.

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Introduction to Graphs

Table of Contents

15 December 2020                 

Read time: 6 minutes

Introduction

What are graphs?

What are the different types of data?

What are the different types of graphical representations?

The graph is nothing but an organized representation of data. It helps us to understand the data. Data are the numerical information collected through observation.

The word data came from the Latin word Datum which means “something given”

After a research question is developed, data is being collected continuously through observation. Then it is organized, summarized, classified, and then represented graphically.

Differences between Data and information: Data is the raw fact without any add on but the information is the meaning derived from data.

Introduction to Graphs-PDF

The graph is nothing but an organized representation of data. It helps us to understand the data. Data are the numerical information collected through observation. Here is a downloadable PDF to explore more.

• Line and Bar Graphs Application
• Graphs in Mathematics & Statistics

What are the different Types of Data?

There are two types of Data :

Quantitative

The data which are statistical or numerical are known as Quantitive data. Quantitive data is generated through. Quantitative data is also known as Structured data. Experiments, Tests, Surveys, Market Report.

Quantitive data is again divided into Continuous data and Discrete data.

Continuous Data

Continuous data is the data which can have any value. That means Continuous data can give infinite outcomes so it should be grouped before representing on a graph.

• The speed of a vehicle as it passes a checkpoint
• The mass of a cooking apple
• The time taken by a volunteer to perform a task

Discrete Data

Discrete data can have certain values. That means only a finite number can be categorized as discrete data.

• Numbers of cars sold at a dealership during a given month
• Number of houses in certain block
• Number of fish caught on a fishing trip
• Number of complaints received at the office of airline on a given day
• Number of customers who visit at bank during any given hour
• Number of heads obtained in three tosses of a coin

Differences between Discrete and Continuous data

• Numerical data could be either discrete or continuous
• Continuous data can take any numerical value (within a range); For example, weight, height, etc.
• There can be an infinite number of possible values in continuous data
• Discrete data can take only certain values by finite ‘jumps’, i.e., it ‘jumps’ from one value to another but does not take any intermediate value between them (For example, number of students in the class)

Qualitative

Data that deals with description or quality instead of numbers are known as Quantitative data. Qualitative data is also known as unstructured data. Because this type of data is loosely compact and can’t be analyzed conventionally.

Different Types of Graphical Representations

There are many types of graph we can use to represent data. They are as follows,

A bar graph or chart is a way to represent data by rectangular column or bar. The heights or length of the bar is proportional to the values.

A line graph is a type of graph where the information or data is plotted as some dots which are known as markers and then they are added to each other by a straight line.

The line graph is normally used to represent the data that changes over time.

A histogram graph is a graph where the information is represented along with the height of the rectangular bar. Though it does look like a bar graph, there is a fundamental difference between them. With the histogram, each column represents a range of quantitative data when a bar graph represents categorical variables.

The other name of the pie chart is a circle graph. It is a circular chart where numerical information represents as slices or in fractional form or percentage where the whole circle is 100%.

• Stem and leaf plot

The stem and leaf plot is a way to represents quantitative data according to frequency ranges or frequency distribution.

In the stem and leaf plot, each data is split into stem and leaf, which is 32 will be split into 3 stems and 2 leaves.

Frequency table: Frequency means the number of occurrences of an event. A frequency distribution table is a graph or chart which shows the frequency of events. It is denoted as ‘f’ .

Pictograph or Pictogram is the earliest way to represents data in a pictorial form or by using symbols or images. And each image represents a particular number of things.

According to the above-mentioned Pictograph, the number of Appels sold on Monday is 6x2=12.

• Scatter diagrams

Scatter diagram or scatter plot is a way of graphical representation by using cartesian coordinates of two variables. The plot shows the relationship between two variables. Below there is a data table as well as a Scattergram as per the given data.

What is the meaning of Graphical representation?

Graphical representation is a way to represent and analyze quantitive data. A graph is a kind of a chart where data are plotted as variables across the coordinate. It became easy to analyze the extent of change of one variable based on the change of other variables.

Principles of graphical representation

The principles of graphical representation are algebraic. In a graph, there are two lines known as Axis or Coordinate axis. These are the X-axis and Y-axis. The horizontal axis is the X-axis and the vertical axis is the Y-axis. They are perpendicular to each other and intersect at O or point of Origin.

On the right side of the Origin, the Xaxis has a positive value and on the left side, it has a negative value. In the same way, the upper side of the Origin Y-axis has a positive value where the down one is with a negative value.

When X-axis and y-axis intersected each other at the origin it divides the plane into four parts which are called Quadrant I, Quadrant II, Quadrant III, Quadrant IV.

The location on the coordinate plane is known as the ordered pair and it is written as (x,y). That means the first value will be on the x-axis and the second one is on the y-axis. When we will plot any coordinate, we always have to start counting from the origin and have to move along the x-axis, if it is positive then to the right side, and if it is negative then to the left side. Then from the x-axis, we have to plot the y’s value, which means we have to move up for positive value or down if the value is negative along with the y-axis.

In the following graph, 1st ordered pair (2,3) where both the values of x and y are positive and it is on quadrant I. 2nd ordered pair (-3,1), here the value of x is negative and value of y is positive and it is in quadrant II. 3rd ordered pair (-1.5, -2.5), here the value of x as well as y both are Negative and in quadrant III.

Methods of representing a frequency distribution

There are four methods to represent a frequency distribution graphically. These are,

• Smoothed Frequency graph
• Cumulative frequency graph or Ogive.
• Pie diagram.

Advantages and Disadvantages of Graphical representation of data

• It improves the way of analyzing and learning as the graphical representation makes the data easy to understand.
• It can be used in almost all fields from mathematics to physics to psychology and so on.
• It is easy to understand for its visual impacts.
• It shows the whole and huge data in an instance.

The main disadvantage of graphical representation of data is that it takes a lot of effort as well as resources to find the most appropriate data and then represents it graphically.

You may also like:

• Graphing a Quadratic Function
• Empirical Relationship Between Mean, Median, and Mode

Not only in mathematics but almost in every field the graph is a very important way to store, analyze, and represents information. After any research work or after any survey the next step is to organize the observation or information and plotting them on a graph paper or plane. The visual representation of information makes the understanding of crucial components or trends easier.

A huge amount of data can be store or analyze in a small space.

The graphical representation of data helps to decide by following the trend.

A complete Idea: Graphical representation constitutes a clear and comprehensive idea in the minds of the audience. Reading a large number (say hundreds) of pages may not help to make a decision. Anyone can get a clear idea just by looking into the graph or design.

Graphs are a very conceptual topic, so it is essential to get a complete understanding of the concept. Graphs are great visual aids and help explain numerous things better, they are important in everyday life. Get better at graphs with us, sign up for a free trial . 

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Cuemath, a student-friendly mathematics and coding platform, conducts regular Online Classes for academics and skill-development, and their Mental Math App, on both iOS and Android , is a one-stop solution for kids to develop multiple skills. Understand the Cuemath Fee structure and sign up for a free trial.

Frequently Asked Questions (FAQs)

What is data.

Data are characteristics or information, usually numerical, that are collected through observation.

How do you differentiate between data and information?

Data is the raw fact without any add on but the information is the meaning derived from data.

What are the types of data?

There are two types of Data:

What are the ways to represent data?

Tables, charts and graphs are all ways of representing data , and they can be used for two broad purposes. The first is to support the collection, organisation and analysis of data as part of the process of a scientific study.

- Tables, charts and graphs are all ways of representing data, and they can be used for two broad purposes. The first is to support the collection, organisation and analysis of data as part of the process of a scientific study.

What are the different types of graphs?

Different types of graphs include:

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2.4: Using Excel to Create Graphs

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It can be difficult to be perfectly precise when creating graphs by hand. Also, it can be very time consuming if presented with a large amount of data. Technology can be used instead to create professional and precise graphs, and many pieces of technology also do much of the calculation, enabling the use of much larger amounts of data.

In this section, you will learn about several of the most common types of graphs and how to create them in Excel.

The first step will always be to summarize the data into a frequency distribution or a grouped frequency distribution (as appropriate). You can do this by hand, or in Excel sorting the data then using the \(= \text{COUNT}()\) function, or to count the items automatically, use Pivot tables or the \(= \text{COUNTIF}()\) function.

Different kinds of graphs are best used for different kinds of data. Bar Graphs and Pie Charts are used for categorical variables. Whereas Histograms and Frequency Polygons are more generally used for numerical variables.

Bar graphs are regularly used with categorical data to compare the frequency of different categories. However, they can also be used with numerical data as well.

Example \(\PageIndex{1}\)

Twenty people were asked their favorite colors and their responses are shown below. Construct a bar graph in Excel that shows the frequency of each color.

Enter the data into a single column in Excel, each color should be in its own cell.

Create a frequency distribution. Each color will be a category. Then count up how many times that color appears. If you want, you can automatically do this in Excel using a pivot table or using the \(= \text{COUNTIF}()\) function. Another great option is to sort data entered already in Excel so it is easier to count.

Enter the frequency table into Excel. At this point, the top of your spreadsheet may look something like this.

Then highlight the frequency distribution, and go to Insert, then in the Charts section click to Insert a Bar or Column Chart.

Once that button has been selected, this is the option menu that pops up.

The top three options under 2-D Bar are the most common to use. The top left one, Clustered Column, will give a basic bar chart or a grouped bar chart if you're using more than one variable. The top middle one, Stacked Column, is used only if you're using more than one variable, it shows the bars in a stacked format. the top right one, 100% Stacked Column, is used only if you're using more than one variable, it shows stacked bars with sections given as a percent to compare parts of a whole.

Occasionally, the 3-D Bar options are used for effect. You should typically use the vertical column options instead of the horizontal options.

Choose the Clustered Column for this example and out pops our graph.

Many options can be edited such as the title, the color of the bars, axis descriptions, etc. The default title (if you don't highlight the headings in your frequency distribution) is "Chart Title" which should always be changed to something more meaningful.

Pareto Graphs

Pareto graphs are a special kind of bar chart that order the bars largest to smallest.

Example \(\PageIndex{2}\)

Use the same data from the last example to create a Pareto Graph in Excel.

Just like in the previous example, you need to start with a frequency distribution. But this time, you need to sort the frequency distribution by frequency.

Make sure to highlight the entire frequency distribution, not just the frequency column so that the correct frequencies stay with the correct colors.

On the Home ribbon, select Sort & Filter, then Custom Sort.

Then create a bar chart using the new sorted frequency distribution, following the same steps from the previous example.

Always be sure to have a more descriptive title than "Chart title".

Pie charts display data in a circle, with "pie" slices for each category. For Pie charts to work, the data must total 100% so the entire pie is filled. The relative size of the slices display the percent of each category out of the whole.

Example \(\PageIndex{3}\)

Use the same data from the first example to create a Pie Chart in Excel.

Begin by creating a frequency distribution of the data in Excel, just like in Example 1.

Then highlight the frequency distribution, and go to Insert, then in the Charts section click to Insert Pie or Doughnut Charts.

The top left option, Pie, will give a basic pie chart.

The Pie of Pie or Bar of Pie options could be used to highlight a particular section of the whole pie chart. This is used when there is a cluster of very small pie slices. Occasionally, the 3-D options or the Doughnut are used for effect.

Choose the Pie for this example and out pops our graph.

Pie charts are limited because they don't show differences between similarly sized slices. For example, in the above chart, it's difficult to see the difference between the sizes of blue, pink, purple, and red since they're close in size. Excel does give the option to label the size of the slices, which is a good option if this issue occurs.

Histograms are used for numerical data to compare the frequency of different values or classes of values. Histograms show a graphical view of the overall shape of a distribution. Histograms are very similar to bar charts, but should be displayed without space between the bars. Bar charts are mostly used for categorical data, whereas histograms are only used for numerical data.

Example \(\PageIndex{4}\)

Twenty Elementary Statistics students were asked their heights and their responses are shown below in inches. Construct a histogram in Excel that shows the frequency distribution of the heights.

Start the same way as creating a bar chart.

Enter the data into Excel, each number in its own cell. Create a grouped frequency distribution. We need to use a grouped frequency distribution (where there are ranges of numbers instead of individual numbers) because there are too many individual numbers. The idea is to summarize the data, not show the exact same data in graphical form. Decide on the classes, ensure the class sizes are the same for all the classes, then count up how many times that number occurs. To make it easier, sort the data, then use the \(=\text{COUNT}()\) function.

After creating the frequency distribution in Excel, your spreadsheet may look something like this.

Highlight the frequency distribution in Excel, go to Insert, then in the Charts section click Insert a Bar or Col umn Chart, then choose the Clustered Column. This creates a bar chart of this data.

This is not a histogram. To make it a histogram right click on the graph, then choose Format Chart Area from the options.

Click the down arrow next to Chart Options, then choose Series "Frequency". It's called Frequency since that's what I named the column containing the frequencies. Yours may be called Series "...", where the dots could be something else.

Click on the button that looks like a bar chart, then change the Gap Width to 0%.

This gives us a correct histogram.

Excel does offer an option to automatically create a histogram, which is not recommended. If you use the automatic option, Excel overrides the classes you choose. It also displays the classes in interval notation, which is not the best option for readability. Here is the graph created using the automatic histogram function.

The resulting graph is not as informative and will not be considered correct.

There are adjustments that can be made in Excel to fix it, but I've found it much easier to create histograms by starting with a bar chart first then changing the gap width rather than attempting to fix an automatic histogram.

Frequency Polygons

Frequency polygons show the overall shape of a distribution, similarly to histograms. However, they are especially useful for comparing different variables on the same graph. These can also be called Line Graphs, or if you are looking at a relationship over time, then these are used as a Time Series graph.

Example \(\PageIndex{5}\)

Makila and Jannai are in a club that fundraises money by selling cookies. The sell lasts five weeks. Below is shown how many boxes they each sold each week. Create a line graph depicting this data.

Start by entering the data into Excel. After entering the data, it may look something like this.

Highlight all the data entered into Excel, go to Insert, then click the Insert Line or Area Chart option.

Choose the top left option for a Line chart.

This will create a Line chart, showing both people's sales by week.

This chart has a seperate line for each person and a meaningful title. It also has a key that shows which color line goes with which person.

From the chart, Makila made more sales overall, though less in week 3 than Jannai. Makila's sales went up and down, but Jannai's went steadily up each week.

Using Excel to create graphs can give us precise information in an easy to read format. It can be a lot faster and more precise than creating graphs by hand, and it allows the possibility of using large amounts of data.

Data will always need to be organized into a frequency distribution or a grouped frequency distribution first. Graphs should not be created based on raw data.

5.1 Vector Addition and Subtraction: Graphical Methods

Section learning objectives.

By the end of this section, you will be able to do the following:

• Describe the graphical method of vector addition and subtraction
• Use the graphical method of vector addition and subtraction to solve physics problems

Teacher Support

The learning objectives in this section will help your students master the following standards:

• (E) develop and interpret free-body force diagrams.

Section Key Terms

The graphical method of vector addition and subtraction.

Recall that a vector is a quantity that has magnitude and direction. For example, displacement, velocity, acceleration, and force are all vectors. In one-dimensional or straight-line motion, the direction of a vector can be given simply by a plus or minus sign. Motion that is forward, to the right, or upward is usually considered to be positive (+); and motion that is backward, to the left, or downward is usually considered to be negative (−).

In two dimensions, a vector describes motion in two perpendicular directions, such as vertical and horizontal. For vertical and horizontal motion, each vector is made up of vertical and horizontal components. In a one-dimensional problem, one of the components simply has a value of zero. For two-dimensional vectors, we work with vectors by using a frame of reference such as a coordinate system. Just as with one-dimensional vectors, we graphically represent vectors with an arrow having a length proportional to the vector’s magnitude and pointing in the direction that the vector points.

[BL] [OL] Review vectors and free body diagrams. Recall how velocity, displacement and acceleration vectors are represented.

Figure 5.2 shows a graphical representation of a vector; the total displacement for a person walking in a city. The person first walks nine blocks east and then five blocks north. Her total displacement does not match her path to her final destination. The displacement simply connects her starting point with her ending point using a straight line, which is the shortest distance. We use the notation that a boldface symbol, such as D , stands for a vector. Its magnitude is represented by the symbol in italics, D , and its direction is given by an angle represented by the symbol θ . θ . Note that her displacement would be the same if she had begun by first walking five blocks north and then walking nine blocks east.

Tips For Success

In this text, we represent a vector with a boldface variable. For example, we represent a force with the vector F , which has both magnitude and direction. The magnitude of the vector is represented by the variable in italics, F , and the direction of the variable is given by the angle θ . θ .

The head-to-tail method is a graphical way to add vectors. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the pointed end of the arrow. The following steps describe how to use the head-to-tail method for graphical vector addition .

• If there are more than two vectors, continue to add the vectors head-to-tail as described in step 2. In this example, we have only two vectors, so we have finished placing arrows tip to tail.
• To find the magnitude of the resultant, measure its length with a ruler. When we deal with vectors analytically in the next section, the magnitude will be calculated by using the Pythagorean theorem.
• To find the direction of the resultant, use a protractor to measure the angle it makes with the reference direction (in this case, the x -axis). When we deal with vectors analytically in the next section, the direction will be calculated by using trigonometry to find the angle.

[AL] Ask two students to demonstrate pushing a table from two different directions. Ask students what they feel the direction of resultant motion will be. How would they represent this graphically? Recall that a vector’s magnitude is represented by the length of the arrow. Demonstrate the head-to-tail method of adding vectors, using the example given in the chapter. Ask students to practice this method of addition using a scale and a protractor.

[BL] [OL] [AL] Ask students if anything changes by moving the vector from one place to another on a graph. How about the order of addition? Would that make a difference? Introduce negative of a vector and vector subtraction.

Watch Physics

Visualizing vector addition examples.

This video shows four graphical representations of vector addition and matches them to the correct vector addition formula.

• Yes, if we add the same two vectors in a different order it will still give the same resultant vector.
• No, the resultant vector will change if we add the same vectors in a different order.

Vector subtraction is done in the same way as vector addition with one small change. We add the first vector to the negative of the vector that needs to be subtracted. A negative vector has the same magnitude as the original vector, but points in the opposite direction (as shown in Figure 5.6 ). Subtracting the vector B from the vector A , which is written as A − B , is the same as A + (− B ). Since it does not matter in what order vectors are added, A − B is also equal to (− B ) + A . This is true for scalars as well as vectors. For example, 5 – 2 = 5 + (−2) = (−2) + 5.

Global angles are calculated in the counterclockwise direction. The clockwise direction is considered negative. For example, an angle of 30 ∘ 30 ∘ south of west is the same as the global angle 210 ∘ , 210 ∘ , which can also be expressed as −150 ∘ −150 ∘ from the positive x -axis.

Using the Graphical Method of Vector Addition and Subtraction to Solve Physics Problems

Now that we have the skills to work with vectors in two dimensions, we can apply vector addition to graphically determine the resultant vector , which represents the total force. Consider an example of force involving two ice skaters pushing a third as seen in Figure 5.7 .

In problems where variables such as force are already known, the forces can be represented by making the length of the vectors proportional to the magnitudes of the forces. For this, you need to create a scale. For example, each centimeter of vector length could represent 50 N worth of force. Once you have the initial vectors drawn to scale, you can then use the head-to-tail method to draw the resultant vector. The length of the resultant can then be measured and converted back to the original units using the scale you created.

You can tell by looking at the vectors in the free-body diagram in Figure 5.7 that the two skaters are pushing on the third skater with equal-magnitude forces, since the length of their force vectors are the same. Note, however, that the forces are not equal because they act in different directions. If, for example, each force had a magnitude of 400 N, then we would find the magnitude of the total external force acting on the third skater by finding the magnitude of the resultant vector. Since the forces act at a right angle to one another, we can use the Pythagorean theorem. For a triangle with sides a, b, and c, the Pythagorean theorem tells us that

Applying this theorem to the triangle made by F 1 , F 2 , and F tot in Figure 5.7 , we get

Note that, if the vectors were not at a right angle to each other ( 90 ∘ ( 90 ∘ to one another), we would not be able to use the Pythagorean theorem to find the magnitude of the resultant vector. Another scenario where adding two-dimensional vectors is necessary is for velocity, where the direction may not be purely east-west or north-south, but some combination of these two directions. In the next section, we cover how to solve this type of problem analytically. For now let’s consider the problem graphically.

Worked Example

Adding vectors graphically by using the head-to-tail method: a woman takes a walk.

Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25 m in a direction 49 ∘ 49 ∘ north of east. Then, she walks 23 m heading 15 ∘ 15 ∘ north of east. Finally, she turns and walks 32 m in a direction 68 ∘ 68 ∘ south of east.

Graphically represent each displacement vector with an arrow, labeling the first A , the second B , and the third C . Make the lengths proportional to the distance of the given displacement and orient the arrows as specified relative to an east-west line. Use the head-to-tail method outlined above to determine the magnitude and direction of the resultant displacement, which we’ll call R .

(1) Draw the three displacement vectors, creating a convenient scale (such as 1 cm of vector length on paper equals 1 m in the problem), as shown in Figure 5.8 .

(2) Place the vectors head to tail, making sure not to change their magnitude or direction, as shown in Figure 5.9 .

(3) Draw the resultant vector R from the tail of the first vector to the head of the last vector, as shown in Figure 5.10 .

(4) Use a ruler to measure the magnitude of R , remembering to convert back to the units of meters using the scale. Use a protractor to measure the direction of R . While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since R is south of the eastward pointing axis (the x -axis), we flip the protractor upside down and measure the angle between the eastward axis and the vector, as illustrated in Figure 5.11 .

In this case, the total displacement R has a magnitude of 50 m and points 7 ∘ 7 ∘ south of east. Using its magnitude and direction, this vector can be expressed as

The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that it does not matter in what order the vectors are added. Changing the order does not change the resultant. For example, we could add the vectors as shown in Figure 5.12 , and we would still get the same solution.

[BL] [OL] [AL] Ask three students to enact the situation shown in Figure 5.8 . Recall how these forces can be represented in a free-body diagram. Giving values to these vectors, show how these can be added graphically.

Subtracting Vectors Graphically: A Woman Sailing a Boat

A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction 66.0 ∘ 66.0 ∘ north of east from her current location, and then travel 30.0 m in a direction 112 ∘ 112 ∘ north of east (or 22.0 ∘ 22.0 ∘ west of north). If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up? The two legs of the woman’s trip are illustrated in Figure 5.13 .

We can represent the first leg of the trip with a vector A , and the second leg of the trip that she was supposed to take with a vector B . Since the woman mistakenly travels in the opposite direction for the second leg of the journey, the vector for second leg of the trip she actually takes is − B . Therefore, she will end up at a location A + (− B ), or A − B . Note that − B has the same magnitude as B (30.0 m), but is in the opposite direction, 68 ∘ ( 180 ∘ − 112 ∘ ) 68 ∘ ( 180 ∘ − 112 ∘ ) south of east, as illustrated in Figure 5.14 .

We use graphical vector addition to find where the woman arrives A + (− B ).

(1) To determine the location at which the woman arrives by accident, draw vectors A and − B .

(2) Place the vectors head to tail.

(3) Draw the resultant vector R .

(4) Use a ruler and protractor to measure the magnitude and direction of R .

These steps are demonstrated in Figure 5.15 .

In this case

Because subtraction of a vector is the same as addition of the same vector with the opposite direction, the graphical method for subtracting vectors works the same as for adding vectors.

Adding Velocities: A Boat on a River

A boat attempts to travel straight across a river at a speed of 3.8 m/s. The river current flows at a speed v river of 6.1 m/s to the right. What is the total velocity and direction of the boat? You can represent each meter per second of velocity as one centimeter of vector length in your drawing.

We start by choosing a coordinate system with its x-axis parallel to the velocity of the river. Because the boat is directed straight toward the other shore, its velocity is perpendicular to the velocity of the river. We draw the two vectors, v boat and v river , as shown in Figure 5.16 .

Using the head-to-tail method, we draw the resulting total velocity vector from the tail of v boat to the head of v river .

By using a ruler, we find that the length of the resultant vector is 7.2 cm, which means that the magnitude of the total velocity is

By using a protractor to measure the angle, we find θ = 32.0 ∘ . θ = 32.0 ∘ .

If the velocity of the boat and river were equal, then the direction of the total velocity would have been 45°. However, since the velocity of the river is greater than that of the boat, the direction is less than 45° with respect to the shore, or x axis.

Teacher Demonstration

Plot the way from the classroom to the cafeteria (or any two places in the school on the same level). Ask students to come up with approximate distances. Ask them to do a vector analysis of the path. What is the total distance travelled? What is the displacement?

Practice Problems

Virtual physics, vector addition.

In this simulation , you will experiment with adding vectors graphically. Click and drag the red vectors from the Grab One basket onto the graph in the middle of the screen. These red vectors can be rotated, stretched, or repositioned by clicking and dragging with your mouse. Check the Show Sum box to display the resultant vector (in green), which is the sum of all of the red vectors placed on the graph. To remove a red vector, drag it to the trash or click the Clear All button if you wish to start over. Notice that, if you click on any of the vectors, the | R | | R | is its magnitude, θ θ is its direction with respect to the positive x -axis, R x is its horizontal component, and R y is its vertical component. You can check the resultant by lining up the vectors so that the head of the first vector touches the tail of the second. Continue until all of the vectors are aligned together head-to-tail. You will see that the resultant magnitude and angle is the same as the arrow drawn from the tail of the first vector to the head of the last vector. Rearrange the vectors in any order head-to-tail and compare. The resultant will always be the same.

Grasp Check

True or False—The more long, red vectors you put on the graph, rotated in any direction, the greater the magnitude of the resultant green vector.

Check Your Understanding

• backward and to the left
• backward and to the right
• forward and to the right
• forward and to the left

True or False—A person walks 2 blocks east and 5 blocks north. Another person walks 5 blocks north and then two blocks east. The displacement of the first person will be more than the displacement of the second person.

Use the Check Your Understanding questions to assess whether students achieve the learning objectives for this section. If students are struggling with a specific objective, the Check Your Understanding will help identify which objective is causing the problem and direct students to the relevant content.

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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute Texas Education Agency (TEA). The original material is available at: https://www.texasgateway.org/book/tea-physics . Changes were made to the original material, including updates to art, structure, and other content updates.

Access for free at https://openstax.org/books/physics/pages/1-introduction

• Authors: Paul Peter Urone, Roger Hinrichs
• Publisher/website: OpenStax
• Book title: Physics
• Publication date: Mar 26, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/physics/pages/1-introduction
• Section URL: https://openstax.org/books/physics/pages/5-1-vector-addition-and-subtraction-graphical-methods

© Jan 19, 2024 Texas Education Agency (TEA). The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

• School Guide
• Class 9 Syllabus
• Maths Notes Class 9
• Science Notes Class 9
• History Notes Class 9
• Geography Notes Class 9
• Political Science Notes Class 9
• NCERT Soln. Class 9 Maths
• RD Sharma Soln. Class 9
• Math Formulas Class 9
• CBSE Class 9 Maths Revision Notes

Chapter 1: Number System

• Number System in Maths
• Natural Numbers | Definition, Examples, Properties
• Whole Numbers | Definition, Properties and Examples
• Rational Number: Definition, Examples, Worksheet
• Irrational Numbers- Definition, Identification, Examples, Symbol, Properties
• Real Numbers
• Decimal Expansion of Real Numbers
• Decimal Expansions of Rational Numbers
• Representation of Rational Numbers on the Number Line | Class 8 Maths
• Represent √3 on the number line
• Operations on Real Numbers
• Rationalization of Denominators
• Laws of Exponents for Real Numbers

Chapter 2: Polynomials

• Polynomials in One Variable - Polynomials | Class 9 Maths
• Polynomial Formula
• Types of Polynomials
• Zeros of Polynomial
• Factorization of Polynomial
• Remainder Theorem
• Factor Theorem
• Algebraic Identities

Chapter 3: Coordinate Geometry

• Coordinate Geometry
• Cartesian Coordinate System in Maths
• Cartesian Plane

Chapter 4: Linear equations in two variables

• Linear Equations in One Variable
• Linear Equation in Two Variables
• Graph of Linear Equations in Two Variables
• Graphical Methods of Solving Pair of Linear Equations in Two Variables
• Equations of Lines Parallel to the x-axis and y-axis

Chapter 5: Introduction to Euclid's Geometry

• Euclidean Geometry
• Equivalent Version of Euclid’s Fifth Postulate

Chapter 6: Lines and Angles

• Lines and Angles
• Types of Angles
• Pairs of Angles - Lines & Angles
• Transversal Lines
• Angle Sum Property of a Triangle

Chapter 7: Triangles

• Triangles in Geometry
• Congruence of Triangles |SSS, SAS, ASA, and RHS Rules
• Theorem - Angle opposite to equal sides of an isosceles triangle are equal | Class 9 Maths
• Triangle Inequality Theorem, Proof & Applications

Chapter 8: Quadrilateral

• Angle Sum Property of a Quadrilateral
• Quadrilateral - Definition, Properties, Types, Formulas, Examples
• Introduction to Parallelogram: Properties, Types, and Theorem
• Rhombus: Definition, Properties, Formula, Examples
• Kite - Quadrilaterals
• Properties of Parallelograms
• Mid Point Theorem

Chapter 9: Areas of Parallelograms and Triangles

• Area of Triangle | Formula and Examples
• Area of Parallelogram
• Figures on the Same Base and between the Same Parallels

Chapter 10: Circles

• Circles in Maths
• Radius of Circle
• Tangent to a Circle
• What is the longest chord of a Circle?
• Circumference of Circle - Definition, Perimeter Formula, and Examples
• Angle subtended by an arc at the centre of a circle
• What is Cyclic Quadrilateral
• Theorem - The sum of opposite angles of a cyclic quadrilateral is 180° | Class 9 Maths

Chapter 11: Construction

• Basic Constructions - Angle Bisector, Perpendicular Bisector, Angle of 60°
• Construction of Triangles

Chapter 12: Heron's Formula

• Area of Equilateral Triangle
• Area of Isosceles Triangle
• Heron's Formula
• Applications of Heron's Formula
• Area of Quadrilateral
• Area of Polygons

Chapter 13: Surface Areas and Volumes

• Surface Area of Cuboid
• Volume of Cuboid | Formula and Examples
• Surface Area of Cube
• Volume of a Cube
• Surface Area of Cylinder (CSA and TSA) |Formula, Derivation, Examples
• Volume of Cylinder
• Surface Area of Cone
• Volume of Cone | Formula, Derivation and Examples
• Surface Area of Sphere | CSA, TSA, Formula and Derivation
• Volume of a Sphere
• Surface Area of a Hemisphere
• Volume of Hemisphere

Chapter 14: Statistics

• Collection and Presentation of Data

Graphical Representation of Data

• Bar graphs and Histograms
• Central Tendency
• Mean, Median and Mode

Chapter 15: Probability

• Experimental Probability
• Empirical Probability
• CBSE Class 9 Maths Formulas
• NCERT Solutions for Class 9 Maths
• RD Sharma Class 9 Solutions

In today’s world of the internet and connectivity, there is a lot of data available and some or the other method is needed for looking at large data, the patterns, and trends in it. There is an entire branch in mathematics dedicated to dealing with collecting, analyzing, interpreting, and presenting the numerical data in visual form in such a way that it becomes easy to understand and the data becomes easy to compare as well, the branch is known as Statistics . The branch is widely spread and has a plethora of real-life applications such as Business Analytics, demography, astrostatistics, and so on. There are two ways of representing data, 

• Pictorial Representation through graphs.

They say, “A picture is worth the thousand words”.  It’s always better to represent data in graphical format. Even in Practical Evidence and Surveys, scientists have found that the restoration and understanding of any information is better when it is available in form of visuals as Human beings process data better in visual form than any other form. Does it increase the ability 2 times or 3 times? The answer is it increases the Power of understanding 60,000 times for a normal Human being, the fact is amusing and true at the same time. Let’s look at some of them in detail. 

Types of Graphical Representations

Comparison between different items is best shown with graphs, it becomes easier to compare the crux out of the data pertaining to different items. Let’s look at all the different types of graphical representations briefly: 

Line Graphs

A line graph is used to show how the value of particular variable changes with time. We plot this graph by connecting the points at different values of the variable. It can be useful for analyzing the trends in the data predicting further trends. 

A bar graph is a type of graphical representation of the data in which bars of uniform width are drawn with equal spacing between them on one axis (x-axis usually), depicting the variable. The values of the variables are represented by the height of the bars. 

Histograms 

This is similar to bar graphs, but it is based frequency of numerical values rather than their actual values. The data is organized into intervals and the bars represent the frequency of the values in that range. That is, it counts how many values of the data lie in a particular range. 

Line Plot 

It is a plot that displays data as points and checkmarks above a number line, showing the frequency of the point. 

Stem and Leaf Plot 

This is a type of plot in which each value is split into a “leaf”(in most cases, it is the last digit) and “stem”(the other remaining digits). For example: the number 42 is split into leaf (2) and stem (4).  

Box and Whisker Plot 

These plots divide the data into four parts to show their summary. They are more concerned about the spread, average, and median of the data. 

It is a type of graph which represents the data in form of a circular graph. The circle is divided such that each portion represents a proportion of the whole. 

Graphical Representations used in Maths

Graphs in maths are used to study the relationships between two or more variables that are changing. Statistical data can be summarized in a better way using graphs. There are basically two lines of thoughts of making graphs in maths: 

• Value-Based or Time Series Graphs

Frequency Based

Value-based or time series graphs .

These graphs allow us to study the change of a variable with respect to another variable within a given interval of time. The variables can be anything. Time Series graphs study the change of variable with time. They study the trends, periodic behavior, and patterns in the series. We are more concerned with the values of the variables here rather than the frequency of those values. 

Example: Line Graph

These kinds of graphs are more concerned with the distribution of data. How many values lie between a particular range of the variables, and which range has the maximum frequency of the values. They are used to judge a spread and average and sometimes median of a variable under study. 

Example: Frequency Polygon, Histograms.

Principles of Graphical Representations

All types of graphical representations require some rule/principles which are to be followed. These are some algebraic principles. When we plot a graph, there is an origin, and we have our two axes. These two axes divide the plane into four parts called quadrants. The horizontal one is usually called the x-axis and the other one is called the y-axis. The origin is the point where these two axes intersect. The thing we need to keep in mind about the values of the variable on the x-axis is that positive values need to be on the right side of the origin and negative values should be on the left side of the origin. Similarly, for the variable on the y-axis, we need to make sure that the positive values of this variable should be above the x-axis and negative values of this variable must be below the y-axis. 

Advantages and Disadvantages of using Graphical System

Advantages: 

• It gives us a summary of the data which is easier to look at and analyze.
• It saves time.
• We can compare and study more than one variable at a time.

Disadvantage: 

It usually takes only one aspect of the data and ignores the other. For example, A bar graph does not represent the mean, median, and other statistics of the data. 

General Rules for Graphical Representation of Data

We should keep in mind some things while plotting and designing these graphs. The goal should be a better and clear picture of the data. Following things should be kept in mind while plotting the above graphs: 

• Whenever possible, the data source must be mentioned for the viewer.
• Always choose the proper colors and font sizes. They should be chosen to keep in mind that the graphs should look neat.
• The measurement Unit should be mentioned in the top right corner of the graph.
• The proper scale should be chosen while making the graph, it should be chosen such that the graph looks accurate.
• Last but not the least, a suitable title should be chosen.

Frequency Polygon

A frequency polygon is a graph that is constructed by joining the midpoint of the intervals. The height of the interval or the bin represents the frequency of the values that lie in that interval. 

Sample Problems

Question 1: What are different types of frequency-based plots? 

Answer: 

Types of frequency based plots:  Histogram Frequency Polygon Box Plots

Question 2: A company with an advertising budget of Rs 10,00,00,000 has planned the following expenditure in the different advertising channels such as TV Advertisement, Radio, Facebook, Instagram, and Printed media. The table represents the money spent on different channels. 

Draw a bar graph for the following data. 

Solution: 

Steps:  Put each of the channels on the x-axis The height of the bars is decided by the value of each channel.

Question 3: Draw a line plot for the following data 

Steps:  Put each of the x-axis row value on the x-axis joint the value corresponding to the each value of the x-axis.

Question 4: Make a frequency plot of the following data: 

Steps:  Draw the class intervals on the x-axis and frequencies on the y-axis. Calculate the mid point of each class interval. Class Interval Mid Point Frequency 0-3 1.5 3 3-6 4.5 4 6-9 7.5 2 9-12 10.5 6 Now join the mid points of the intervals and their corresponding frequencies on the graph.  This graph shows both the histogram and frequency polygon for the given distribution.

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