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Mean Median Mode Range Worksheets and Help

Welcome to the Math Salamanders Mean Median Mode Range Worksheets. Here you will find a wide range of free printable Worksheets, which will help your child learn how to find the mean, median, mode and range of a set of data points.

These worksheets are aimed at students in 5th and 6th grade.

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Mean Median Mode Range Worksheets

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What is the Mean?

The mean is the average of a set of numbers.

It is found by adding up the set of numbers and then dividing the total by the number of data points in the set.

How to find the mean

Step 1) Add up all the numbers in the set.

Step 2) Divide the total by the total number of data points in the set.

Example 1) Find the mean of 5, 7, 8 and 4

Step 1) Add up the numbers to give a total of 5+7+8+4=24

Step 2) Divide the total by the number of data points. 24 ÷ 4 = 6

Answer: the mean is 6.

Example 2) Find the mean of 8, 2, 5, 7 and 13

Step 1) Add up the numbers to give a total of 8+2+5+7+13=35

Step 2) Divide by the number of data points. 35 ÷ 5 = 7

Answer: the mean is 7.

What is the Median?

The median is the midpoint (or middle value) of a set of numbers.

It is found by ordering the set of numbers and then finding the middle value in the set.

How to find the median

Step 1) Order the numbers in the set from smallest to largest.

Step 2) Find the middle number.

- If there is an odd number of values in the set, then the median is the middle value.

- If there is an even number of values in the set, then the median is the average of the two middle values.

Example 1) Find the median of 5, 7, 8, 2 and 4

Step 1) Put the numbers in order: 2, 4, 5, 7, 8

Step 2) There is an odd number of values in the set so the median is the middle value which is 5.

Answer: the median is 5.

Example 2) Find the median of 23, 27, 16, 31

Step 1) Put the numbers in order: 16, 23, 27, 31

Step 2) There is an even number of values in the set, so the median is the average of the middle two values.

(23+27) ÷ 2 = 25

Answer: the mean is 25

Example 3) Find the median of 7, -4, 9, -7, -2, 5

Step 1) Order the numbers: -7, -4, -2, 5, 7, 9

To get the average, simply add the two values together and divide by 2:

(-2 + 5) ÷ 2 = 1.5

Answer: the mean is 1.5

What is the Mode?

The mode is the most common (or the data point that appears most often) in a set of data.

It can be found by putting the data into an ordered list and seeing which data point occurs most often.

How to find the mode

Step 1) Put the data into an ordered list.

Step 2) Check that you have got the same number of data points.

Step 3) The mode is the data point which is the most common.

Finding the Mode Examples

Example 1) Find the mode of 3, 6, 4, 3, 2, 4, 7, 8, 6, 3, 9

This gives us: 2, 3, 3, 3, 4, 4, 6, 6, 7, 8, 9

Step 2) Check the number of data points in both lists is the same.

Both lists have 11 data points.

Step 3) The mode is the number which occurs most often.

Answer: the mode is 3.

Example 2) Find the mode of 0.6, 0.3, 0.4, 0.2, 0.4, 0.7, 0.6, 0.1, 0.4, 0.9

This gives us: 0.1, 0.2, 0.3, 0.4, 0.4, 0.4, 0.6, 0.6, 0.7, 0.9

Both lists have 10 data points.

Answer: the mode is 0.4.

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What is the Range?

The range is the gap between the smallest and largest data point.

It is found by putting the data into an ordered list and find the difference between the largest and smallest amount.

How to find the range

Step 3) The range is the difference between the largest and smallest data point.

To find the range simply subtract the smallest number from the largest number.

Finding the Range Examples

Example 1) Find the range of 14, 21, 9, 32, 27, 15, 12, 30

This gives us: 9, 12, 14, 15, 21, 27, 30, 32

Both lists have 8 data points.

Step 3) The range is the difference or gap between the largest and smallest numbers.

Answer: the range is 32-9=23.

Example 2) Find the range of 6, 2, -7, 2, -5, 11, 3, -4, 0, 9

This gives us: -7, -5, -4, 0, 2, 2, 3, 6, 9, 11

Answer: the range is 11-(-7)=18.

These printable mean median mode range worksheets have been carefully graded to ensure a progression in the level of difficulty.

Sheets 1, 2 and 3 are designed for 5th graders involve ordering and calculating using positive integers and decimals.

Sheets 4, 5 and 6 are designed for 6th graders and involve ordering and calculating with positive and negative numbers and decimals.

The first sheet involve finding the mean, median, mode and range of some positive whole numbers.

The 2nd sheet involves the use of decimals to 1dp.

The 3rd sheet is similar to the 2nd sheet but has many more data points.

The 4th sheet involves decimals and negative numbers.

The 5th and 6th sheets are similar to the 4th sheets but with increased number of data points.

• Mean Median Mode and Range Sheet 1
• PDF version
• Median Mean Mode and Range Sheet 2
• Median Mean Mode and Range Sheet 3
• Median Mean Mode and Range Sheet 4
• Median Mean Mode and Range Sheet 5
• Median Mean Mode and Range Sheet 6

Mean Median Mode Range Problems

These printable mean median mode range problem sheets will help your child to use and apply their skills to solve problems.

The first problem sheet is more suitable for 5th grade and the second sheet is aimed at 6th graders.

• Median Mean Mode and Range Problems 1
• Median Mean Mode and Range Problems 2

Mean Median Mode Range Walkthrough Video

This short video walkthrough shows the problems from our Median Mean Mode and Range Problems Sheet 2 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, check out the video!

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

• Mean Worksheets

The sheets in this section will help you to find the mean of a range of numbers, including negative numbers and decimals.

There are a range of sheets involving finding the mean, and also finding a missing data point when the mean is given.

• Median Worksheets

The sheets in this section will help you to find the median of a range of numbers, including negative numbers and decimals.

On some of the easier sheets, only odd numbers of data points have been used.

On the harder sheets, both odd and even numbers of data points have been included.

• Mode and Range Worksheets

The sheets in this section will help you to find the mode and range of a set of numbers, including negative numbers and decimals.

There are easier sheets involving fewer data points, and harder ones with more data points.

The sheets in this section will help you to solve problems involving bar graphs and picture graphs.

There are a range of sheet involving reading and interpreting graphs as well as drawing your own graphs.

• Box Plot Worksheets

Here are our selection of box plot worksheets to help you practice creating and interpreting box plots.

Mean, Median, Mode and Range Online Quiz

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Central Tendency Worksheets: Mean, Median, Mode and Range

Mean, median, mode and range worksheets contain printable practice pages to determine the mean, median, mode, range, lower quartile and upper quartile for the given set of data. The pdf exercises are curated for students of grade 3 through grade 8. Interesting word problems are included in each section. Sample some of these worksheets for free!

Finding Average

Average or mean worksheets have plentiful exercises to find the average of numbers, numbers with practical units and decimals.

(49 Worksheets)

Finding Range

Identify the maximum and minimum values to find the range of the given data. Word problems are included for practice.

• Download the set

Finding Median

In these worksheets, 3rd grade and 4th grade children identify the median (middle value) of the represented data. Two word problems included.

Finding Quartiles

Determine the first (lower) quartile, second (median) quartile and the third (upper) quartile of the given data. One word problem included in each pdf worksheet for 5th grade and 6th grade students.

Mean, Median, Mode and Range: Level 1

These printable central tendency worksheets contain a mixed review of mean, median, mode and range concepts. Around 8 data are used in level 1.

Mean, Median, Mode and Range: Level 2

Find the mean, median, mode and range of each set of data. Each sheet has six problems with around 15 data.

Word Problems: Level 1

This exclusive section has five word problems to find the mean, median, mode and range of the given data.

Word Problems: Level 2

A variety of informative data are included as word problems in these central tendency worksheet pdfs requires 7th grade and 8th grade students to determine the values of mean, median, mode and range.

Related Worksheets

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» Mean Absolute Deviation

» Stem and Leaf Plot

» Box and Whisker Plot

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Statistics and probability

Course: statistics and probability   >   unit 3.

• Statistics intro: Mean, median, & mode
• Mean, median, & mode example

Mean, median, and mode

Calculating the mean.

• Calculating the median
• Choosing the "best" measure of center

Mean, median, and mode review

Practice problems.

• 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 ‍  

Finding the median

• Arrange the data points from smallest to largest.
• If the number of data points is odd, the median is the middle data point in the list.
• If the number of data points is even, the median is the average of the two middle data points in the list.

Finding the mode

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Mean, Median, Mode and Range

These lessons help GMAT students review mean, median, mode and range.

Related Pages Mean Median Mode Central Tendency More Statistics Lessons

The following diagrams show how to find the mean, median, mode and range. Scroll down the page for examples and solutions.

Statistics: The Average

Introduction to descriptive statistics and central tendency. Ways to measure the average of a set: median, mean, mode

Mean, Median, and Mode

Learn about Mean, Median, and Mode Mean, median and mode are measures of central tendency. Mean: add together and divide by how many there are. Median: the middle number when the data is listed in order from smallest to largest. If there are 2 middle number, add them and divide by 2. Mode: the most frequent number.

Example: Find the mean, median and mode of this set of data. 35,8,12,50,55,22,34,22,34

Mean, Median and Mode

This lesson shows how to calculate Mean, Median and Mode and some tricks to help you remember the differences of these methods of finding the Center.

The mean is the average. To calculate the mean, you add add the values and divide by the number of values.

• If you had 20 pencils in one box and 30 pencils in a second box, what is the average number of pencils in your boxes?
• Your grades out of 20 are all listed below. What is your average grade out of 20? 15, 16, 20, 12, 18, 20, 17

The median is the center. List the numbers in order and pick the center one. If there are an even number of numbers find the mean of the two middle numbers.

The mode is the number that shows up the most. There can be more than one mode. If the values all appear the same number of times, there is no mode.

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Mean, Median, Mode, and Range

Mean, median, and mode are three kinds of "averages". There are many "averages" in statistics, but these are, I think, the three most common, and are certainly the three you are most likely to encounter in your pre-statistics courses, if the topic comes up at all.

Why do we need different kinds of averages?

The average that we're used to is found by adding all the values in a data set, and then dividing the sum by the number of values in that data set; but this average might be misleading.

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Median, Mode, and Range

A typical example would be the case where nearly every person in a given population lives on about two dollars a day, but there is a small elite with incomes in the millions. The numerical average can mislead by suggesting that the average (in this case, we mean "typical") person earns a few tens of thousands per year.

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But this does not accurately reflect what we mean when we're trying to discuss the "average" income. This is why the average income is almost always expressed by a different sort of average.

What are the three averages?

The three averages are:

• The "mean" is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers.
• The "median" is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order from smallest to largest, so you may have to rewrite your list before you can find the median.
• The "mode" is the value that occurs most often. If no number in the list is repeated, then there is no mode for the list.

The "range" of a list a numbers is just the difference between the largest and smallest values. It expresses "spread", being how far the values are distributed (or how concentrated they are).

• Find the mean, median, mode, and range for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13

The mean is the usual average, so I'll add and then divide:

(13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15

Note that the mean, in this case, isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers; you should not be surprised when it isn't.

The median is the middle value, so first I'll have to rewrite the list in numerical order:

13, 13, 13, 13, 14, 14, 16, 18, 21

There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5 th number:

So the median is 14 .

The mode is the number that is repeated more often than any other, so 13 , I see from my listing above, is the mode.

The largest value in the list is 21 , and the smallest is 13 , so the range is 21 − 13 = 8 .

mean: 15 median: 14 mode: 13 range: 8

Note: The formula for the place to find the median is " ( [the number of data points] + 1) ÷ 2 ", but you don't have to use this formula. You can just count in from both ends of the list until you meet in the middle, if you prefer, especially if your list is short. Either way will work.

The mean is the usual average:

(1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5

The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are an even number of numbers.

Because of this, the median of the list will be the mean (that is, the usual average) of the middle two values within the list. The middle two numbers are 2 and 4 , so:

(2 + 4) ÷ 2 = 6 ÷ 2 = 3

So the median of this list is 3 , a value that isn't in the list at all.

The mode is the number that is repeated most often, but all the numbers in this list appear only once, so there is no mode.

The largest value in the list is 7 , the smallest is 1 , and their difference is 6 , so the range is 6 .

mean: 3.5 median: 3 mode: none range: 6

The values in the list above were all whole numbers, but the mean of the list was a decimal value. Getting a decimal value for the mean (or for the median, if you have an even number of data points) is perfectly okay; don't round your answers to try to match the format of the other numbers.

8, 9, 10, 10, 10, 11, 11, 11, 12, 13

The mean is the usual average, so I'll add up and then divide:

(8 + 9 + 10 + 10 + 10 + 11 + 11 + 11 + 12 + 13) ÷ 10 = 105 ÷ 10 = 10.5

The median is the middle value. In a list of ten values, that will be the (10 + 1) ÷ 2 = 5.5 -th value; the formula is reminding me, with that "point-five", that I'll need to average the fifth and sixth numbers to find the median. The fifth and sixth numbers are the last 10 and the first 11 , so:

(10 + 11) ÷ 2 = 21 ÷ 2 = 10.5

The mode is the number repeated most often. This list has two values that are repeated three times; namely, 10 and 11 , each repeated three times.

The largest value is 13 and the smallest is 8 , so the range is 13 − 8 = 5 .

mean: 10.5 median: 10.5 modes: 10 and 11 range: 5

As you can see, it is possible for two of the averages (the mean and the median, in this case) to have the same value. But this is not usual, and you should not expect it.

Note: Depending on your text or your instructor, the above data set may be viewed as having no mode rather than having two modes, because no single solitary number was repeated more often than any other. I've seen books that go either way on this; there doesn't seem to be a consensus on the "right" definition of "mode" in the above case. So if you're not certain how you should answer the "mode" part of the above example, ask your instructor before the next test.

How can I keep straight which average is which?

About the only hard part of finding the mean, median, and mode is keeping straight which "average" is which. Use this list:

• mean: regular meaning of "average"
• median: middle value
• mode: most often

(In the above, I've used the term "average" rather casually. The technical definition of what we commonly refer to as the "average" is technically called "the arithmetic mean": adding up the values and then dividing by the number of values. Since you're probably more familiar with the concept of "average" than with "measure of central tendency", I used the more comfortable term.)

• A student has gotten the following grades on his tests: 87, 95, 76, and 88 . He wants an 85 or better overall for the course. Assuming all tests are evenly weighted, what is the minimum grade he must get on the last test in order to achieve that overall average?

The minimum grade for the last test is what I need to find. To find the average of all his grades (the known ones, plus the unknown one), I have to add up all the grades, and then divide by the number of grades. Since I don't have a score for the last test yet, I'll use a variable to stand for this unknown value: " x ". Then, setting the expression for the average equal to the desired average, the computation is:

(87 + 95 + 76 + 88 + x ) ÷ 5 = 85

Multiplying through by 5 and simplifying, I get:

87 + 95 + 76 + 88 + x = 425

346 + x = 425

This is the score that he needs on the last test in order to get the overall grade that he is wanting. (If I doubt myself, I can always plug this value into the formula for the average, and confirm that I get 85 as the result.)

He needs to get at least a 79 on the last test.

You can use the Mathway widget below to practice finding the median. Try the entered exercise, or type in your own exercise. Or try entering any list of numbers, and then selecting the option — mean, median, mode, etc — from what the widget offers you. Then click the button to compare your answer to Mathway's.

Please accept "preferences" cookies in order to enable this widget.

(Click here to be taken directly to the Mathway site, if you'd like to check out their software or get further info.)

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Mean, Median, Mode and Range

The mean, the median and the mode are three different measures of average which we can use. The range is a measure of how spread out data is.

The mode is the value that appears most often.

The median is the middle number (when the numbers are in order).

To calculate the mean add up all the numbers and then divide by how many numbers there are.

The range is the difference between the biggest number and the smallest number.

Example 1: Here are a list of 7 numbers: 7   4   7   9   5   1   2 Find: a) The mode b) The median c) The mean d) The range

a) The mode is the most common number The only number that appears more than once is 7 The mode is 7

b) The median is the middle number (but only when the numbers are in order) We have to put the numbers in order: 1   2   4   5   7   7   9 We can now find the middle number:

The median is 5

c) To find the mean we need to add up all the numbers then divide by how many numbers there are:

1 + 2 + 4 + 5 + 7 + 7 + 9 ⁄ 7 = 35 ⁄ 7 = 5

The mean is 5

d) To find the range we take the smallest number away from the biggest number 9 - 1 = 8 The range is 8

Example 2: Here are a list of 8 numbers: 6   6   13   5   11   11   11   9 Find: a) The mode b) The median c) The mean d) The range

a) The mode is the most common number 11 appears three times, that is more than any other number The mode is 11

b) To find the median (the middle number) we need to put the numbers in order: 5   6   6   9   11  11  11  13

We can see that there is not one middle number, we have 9 and 11 in the middle. When this happens the median is half way between the two middle numbers, the middle of 9 and 11 is 10. We can work this out by adding 9 and 11 and then dividing by 2.

9 + 11 ⁄ 2 = 10 The median is 10

5 + 6 + 6 + 9 + 11 + 11 + 11 + 13 ⁄ 8 = 72 ⁄ 8 = 9

The mean is 9

d) To find the range we take the smallest number away from the biggest number 13 - 5 = 8 The range is 8

Calculate the mode, the median, the mean and the range:

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Mean, Median, Mode, and Range Worksheets

Worksheets for calculating the mean (average), median, mode, and range.

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Mean Median Mode Questions

Mean, Median and Mode are three measures of central tendency, and these are the important topics in statistics. In this article, you will get mean, median, and mode questions and answers, along with some practice questions. These questions and answers will help you to practise for the board exams. Also, these questions cover various types of problems asked on mean, median and mode for Classes 9 and 10.

What is the mean formula?

As we know, the mean is the average of the given data set. This can be calculated using the formula given below.

Mean for ungrouped data = Sum of observations/Number of observations

Click here to get more information about mean .

What is the formula for the median?

The median is the middlemost data value of an ordered data set. This can be estimated using the formulas given below.

When n is odd:

Median = (n + 1)/2th observation

When n is even:

Median = [(n/2)th observation + {(n/2)+1}th observation]/2

Learn more about median here.

What is the mode?

Mode is the most frequently occurring value in the data set.

Get more information about mode here.

Mean Median Mode Questions and Answers

1. Find the mean of the first 10 odd integers.

First 10 odd integers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19

Mean = Sum of the first 10 odd integers/Number of such integers

= (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19)/10

Therefore, the mean of the first 10 odd integers is 10.

2. What is the median of the following data set?

32, 6, 21, 10, 8, 11, 12, 36, 17, 16, 15, 18, 40, 24, 21, 23, 24, 24, 29, 16, 32, 31, 10, 30, 35, 32, 18, 39, 12, 20

The ascending order of the given data set is:

6, 8, 10, 10, 11, 12, 12, 15, 16, 16, 17, 18, 18, 20, 21, 21, 23, 24, 24, 24, 29, 30, 31, 32, 32, 32, 35, 36, 39, 40

Number of values in the data set = n = 30

n/2 = 30/2 = 15

15th data value = 21

(n/2) +1 = 16

16th data value = 21

= (15th data value + 16th data value)/2

= (21 + 21)/2

3. Identify the mode for the following data set:

21, 19, 62, 21, 66, 28, 66, 48, 79, 59, 28, 62, 63, 63, 48, 66, 59, 66, 94, 79, 19 94

Let us write the given data set in ascending order as follows:

19, 19, 21, 21, 28, 28, 48, 48, 59, 59, 62, 62, 63, 63, 66, 66, 66, 66, 79, 79, 94, 94

Here, we can observe that the number 66 occurred the maximum number of times.

Thus, the mode of the given data set is 66.

4. Consider the following frequency distribution. Calculate the mean weight of students.

The given distribution has discontinuous class intervals, so we need to make them continuous.

Here, ∑f i = 40 and ∑f i d i = 35

By Assumed mean method,

Mean = a + (∑f i d i /∑f i )

= 43 + (35/40)

= 43 + 0.875

Therefore, the mean weight of the students is 43.875 kg.

5. Find the mean for the following distribution.

Mean = ∑f i x i /∑f i

6. Calculate the median marks of students from the following distribution.

N/2 = 90/2 = 45

Cumulative frequency greater and nearer to 45 is 47, which lies in the interval 40 – 50

Median class is 40 – 50.

Lower limit of the median class = l = 40

Class size = h = 10

Frequency of the median class = f = 20

Cumulative frequency of the class preceding the median class = cf = 27

As we know,

Median = 40 + [(45 – 27)/20] × 10

= 40 + (18/2)

Hence, the median marks of the students = 49.

7. The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches.

Find the mode of the above distribution.

From the given,

Modal class = 4000 – 5000

Lower limit of the modal class = l = 4000

Class width (h) = 1000

Frequency of the modal class = f 1 = 18,

Frequency of the class preceding the modal class = f 0 = 4

Frequency of the class succeeding the modal class = f 2 = 9

Mode = l + [(f 1 – f 0 )/ (2f 1 – f 0 – f 2 )] × h

Mode = 4000 + [(18 – 4)/(36 – 4 – 9)] × 1000

= 4000 + (14000/23)

= 4000 + 608.695

= 4608.7 (approximately)

Thus, the mode of the given data is 4608.7 runs

8. If the median of a distribution given below is 28.5, then find the value of x and y.

N/2 = 60/2 = 30

Median of the given data = 28.5

Median class is 20 – 30 with a cumulative frequency = 25 + x.

Lower limit of median class = l = 20

Cumulative frequency of the class preceding the median class = cf = 5 + x

28.5 = 20 + [(30 − 5 − x)/ 20] × 10

28.5 – 20 = (25 – x)/2

8.5 × 2 = 25 – x

17 = 25 – x

x = 25 – 17

Therefore, x = 8

60 = 5 + 20 + 15 + 5 + x + y

60 = 5 + 20 + 15 + 5 + 8 + y

y = 60 – 53

Therefore, the value of x = 8 and y = 7.

9. For a moderately skewed distribution, the mean and median are respectively 26.8 and 27.9. What is the mode of the distribution?

Mean = 26.8

Median = 27.9

Using the relationship between mean, median and mode,

Mode = 3 Median – 2 Mean

= 3 × 27.9 – 2 × 26.8

= 83.7 – 53.6

Therefore, the mode of the distribution is 30.1.

10. If the mean of the given frequency distribution is 35, then find the missing frequency y. Also, calculate the median and mode for the distribution.

Given that the mean of the distribution is 35.

So, (430 + 45y)/(14 + y) = 35

430 + 45y = (14 + y)35

430 + 45y = 490 + 35y

45y – 35y = 490 – 430

Thus, the missing frequency is 6.

Now, we can calculate the mode as follows:

Mode = 30 + [(7 – 4)/(14 – 4 – 6)] × 10

= 30 + (30/4)

Now, using the formula Mode = 3Median – 2 Mean, we can get the value of median.

37.5 = 3 Median – 2 (35)

3 Median = 37.5 + 70

Median = 107.7/3

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Practice Questions on Mean Median Mode

1. Find the mean, median and mode of the following data:

23, 18, 24, 23, 31, 37, 28, 30, 25, 40, 35, 35, 27, 25

2. The mileage (km per litre) of 50 cars of the same model was tested by a manufacturer, and details are tabulated as given below:

Find the mean mileage.

The manufacturer claimed that the mileage of the model was 16 km/litre. Do you agree with this claim?

3. Find the median of the following frequency distribution.

4. Calculate the mode for the following distribution.

5. Suppose the mean and median of distribution are 10.14 and 8, respectively. Find the mode of the distribution using the empirical relationship between mean, median and mode.

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Mean, Median, Mode and Range -- Sorted Sets (Sets of 5 from 10 to 99) (A)

Welcome to The Mean, Median, Mode and Range -- Sorted Sets (Sets of 5 from 10 to 99) (A) Math Worksheet from the Statistics Worksheets Page at Math-Drills.com. This math worksheet was created or last revised on 2013-02-16 and has been viewed 48 times this week and 1,834 times this month. It may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math.

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Mean, Median, Mode, Range Calculator

Please provide numbers separated by comma to calculate.

Related Statistics Calculator | Standard Deviation Calculator | Sample Size Calculator

The word mean, which is a homonym for multiple other words in the English language, is similarly ambiguous even in the area of mathematics. Depending on the context, whether mathematical or statistical, what is meant by the "mean" changes. In its simplest mathematical definition regarding data sets, the mean used is the arithmetic mean, also referred to as mathematical expectation, or average. In this form, the mean refers to an intermediate value between a discrete set of numbers, namely, the sum of all values in the data set, divided by the total number of values. The equation for calculating the arithmetic mean is virtually identical to that for calculating the statistical concepts of population and sample mean, with slight variations in the variables used:

The mean is often denoted as x̄ , pronounced "x bar," and even in other uses when the variable is not x , the bar notation is a common indicator of some form of the mean. In the specific case of the population mean, rather than using the variable x̄ , the Greek symbol mu, or μ , is used. Similarly, or rather confusingly, the sample mean in statistics is often indicated with a capital X̄ . Given the data set 10, 2, 38, 23, 38, 23, 21, applying the summation above yields:

As previously mentioned, this is one of the simplest definitions of the mean, and some others include the weighted arithmetic mean (which only differs in that certain values in the data set contribute more value than others), and geometric mean . Proper understanding of given situations and contexts can often provide a person with the tools necessary to determine what statistically relevant method to use. In general, mean, median, mode and range should ideally all be computed and analyzed for a given sample or data set since they elucidate different aspects of the given data, and if considered alone, can lead to misrepresentations of the data, as will be demonstrated in the following sections.

The statistical concept of the median is a value that divides a data sample, population, or probability distribution into two halves. Finding the median essentially involves finding the value in a data sample that has a physical location between the rest of the numbers. Note that when calculating the median of a finite list of numbers, the order of the data samples is important. Conventionally, the values are listed in ascending order, but there is no real reason that listing the values in descending order would provide different results. In the case where the total number of values in a data sample is odd, the median is simply the number in the middle of the list of all values. When the data sample contains an even number of values, the median is the mean of the two middle values. While this can be confusing, simply remember that even though the median sometimes involves the computation of a mean, when this case arises, it will involve only the two middle values, while a mean involves all the values in the data sample. In the odd cases where there are only two data samples or there is an even number of samples where all the values are the same, the mean and median will be the same. Given the same data set as before, the median would be acquired in the following manner:

2,10,21, 23 ,23,38,38

After listing the data in ascending order, and determining that there are an odd number of values, it is clear that 23 is the median given this case. If there were another value added to the data set:

2,10,21, 23 , 23 ,38,38,1027892

Since there are an even number of values, the median will be the average of the two middle numbers, in this case, 23 and 23, the mean of which is 23. Note that in this particular data set, the addition of an outlier (a value well outside the expected range of values), the value 1,027,892, has no real effect on the data set. If, however, the mean is computed for this data set, the result is 128,505.875. This value is clearly not a good representation of the seven other values in the data set that are far smaller and closer in value than the average and the outlier. This is the main advantage of using the median in describing statistical data when compared to the mean. While both, as well as other statistical values, should be calculated when describing data, if only one can be used, the median can provide a better estimate of a typical value in a given data set when there are extremely large variations between values.

In statistics, the mode is the value in a data set that has the highest number of recurrences. It is possible for a data set to be multimodal, meaning that it has more than one mode. For example:

2,10,21,23,23,38,38

Both 23 and 38 appear twice each, making them both a mode for the data set above.

Similar to mean and median, the mode is used as a way to express information about random variables and populations. Unlike mean and median, however, the mode is a concept that can be applied to non-numerical values such as the brand of tortilla chips most commonly purchased from a grocery store. For example, when comparing the brands Tostitos, Mission, and XOCHiTL, if it is found that in the sale of tortilla chips, XOCHiTL is the mode and sells in a 3:2:1 ratio compared to Tostitos and Mission brand tortilla chips respectively, the ratio could be used to determine how many bags of each brand to stock. In the case where 24 bags of tortilla chips sell during a given period, the store would stock 12 bags of XOCHiTL chips, 8 of Tostitos, and 4 of Mission if using the mode. If, however, the store simply used an average and sold 8 bags of each, it could potentially lose 4 sales if a customer desired only XOCHiTL chips and not any other brand. As is evident from this example, it is important to take all manners of statistical values into account when attempting to draw conclusions about any data sample.

The range of a data set in statistics is the difference between the largest and the smallest values. While range does have different meanings within different areas of statistics and mathematics, this is its most basic definition, and is what is used by the provided calculator. Using the same example:

2,10,21,23,23,38,38 38 - 2 = 36

The range in this example is 36. Similar to the mean, range can be significantly affected by extremely large or small values. Using the same example as previously:

The range, in this case, would be 1,027,890 compared to 36 in the previous case. As such, it is important to extensively analyze data sets to ensure that outliers are accounted for.

Word Problems: When to Use Mean, Median, and Mode - Expii

Word problems: when to use mean, median, and mode, explanations (4).

(Video) Word Problems: When to Use Mean, Median, and Mode

by Anusha Rahman

This video by Anusha Rahman goes over three examples of when to use mean, median, and mode in word problems.

Example 1: Jessie’s math teacher said that their final class grade is based on the average of all exam grades. Jessie has gotten a 93,87,71, and 97 on her math exams. What measure of central tendency will she use to calculate the average? What is the average?

We know that when we see the word average , we're talking about the mean ! So, we will be using mean to solve this problem.

Mean=93+87+71+974 =3484 =87

So, we know that Jessie's average score on her math exams was 87!

Example 2: Meghan wants to get an accurate picture of household salary in her neighborhood. The salaries for people in her neighborhood are: 87,000;94,000;103,000;97,000; and 55,000. What measure of central tendency will most accurate describe the data?

At first glance, we can see that there is an outlier at 55,000. At outlier is going to mess with the mean, and mode (the frequency) is not going to give us a lot of information. So, we will use the median, or the middle value.

Step 1: Put the salaries in numerical order, from least to greatest.

55,000,87,000,94,000,97,000,103,000

Step 2: Cross of the numbers evenly from each side until you're left with a middle number.

In this case, the median is 94,000.

Example 3: At her birthday party, Charlotte asks everyone to go around and say their favorite Taylor Swift song. In order, people respond with:

All Too Well, Shake it Off, Black Space, New Romantics, All Too Well, New Romantics, All Too Well, and Mr Perfectly Fine

Charlotte wants to know which song is the most popular. What measure of central tendency should she use?

Because the question is asking for the most popular song, we know we are looking for the mode ! We also know that the mean and median cannot be used for nominal or qualitative data.

Our first step is to make a frequency table with the data.

SongFrequencyAll Too Well3Shake It Off1Blank Space1New Romantics2Mr. Perfectly Fine1

From this frequency table, we can see that All Too Well has the highest frequency, so that is the mode of the data set, and it is the most popular song!

When solving word problems with central tendency, it's important to understand the pros and cons of each type . Let's break them down.

PRO: asymmetrical data (with an outlier)

CON: doesn't work with qualitative or nominal data

PRO: symmetrical data (no outlier)

PRO: big data sets

CON: highly affected by outliers

PRO: working with qualitative or nominal data

PRO: very easy to look at and read graphs/charts

CON: very limited in what it tells you

Related Lessons

(video) understanding and calculating measures of central tendency.

by Nurse Killam

This video by Nurse Killam explains when to use mean, median, and mode.

She starts with the definition of a normal distribution . All of the measures of central tendency work best with a normal distribution.

The mean is the average of a data set. We find the mean by adding all of the values, and dividing the sum by the number of values. The mean gets more accurate with more values.

Mean Benefits

• Best measure for symmetrical distributions
• Influenced by all data
• Most reliable
• Good for interval and ratio data

Mean Limitations

• Works best with no outliers

The median is the middle value of a set. We find this by ordering the data, then finding the literal middle number. Even sized sets will have two numbers that share the middle. The median of these sets is the average of those two numbers.

Median Benefits

• Good for asymmetrical data
• Works for ordinal, interval, and ratio data

Median Limitations

• Does not account for extreme scores
• Not algebraically defined
• Not appropriate for nominal data

The mode is the value that occurs the most in a set. We find this by ordering the set and finding the number that repeats the most.

Mode Benefits

• The only appropriate measure for nominal data
• Can also be used for ordinal, interval, and ratio data

Mode Limitations

• Cannot be used if all scores are different
• There can be several modes
• Cannot be used for further calculations

Using the Measures of Central Tendency: Media, Median, and Mode

When you're faced with word problems where you're asking to look into the distribution of a dataset, it can feel intimidating to figure out whether to use mean, median, or mode. The good news is that because each measure is different, they each have their own advantages and disadvantages!

Image source: by Anusha Rahman

Let's go through some example problems.

Which measure of central tendency is the most effective way to analyze the following dataset? 4,7,9,10,11,18,2,20,15,350,14

When to Use Mean, Median, and Mode

Three extremely important measures of central tendency (part of statistics ) are mean , median , and mode . You can find the mean, median, and mode of most data sets. Sometimes, it is more advantageous to only focus on one or two of these measures of central tendency, depending on your data set. Let's find out when we should use the mean, median, or mode to accurately describe a data set.

The mean is the average value in a data set. We can calculate the mean using the following formula:

mean=sum of all valuestotal number of values

Advantages:

The mean is the best measure for symmetrical data distributions. It is affected by all of the data points, and the more data points you know, the more accurate the mean will be. The mean is helpful when working with interval and ratio data.

Limitations:

Outliers can really throw off the mean. For example, say we have the data set {1,4,5,6,6,7,8,8,9}. The average of this data set is 1+4+5+6+6+7+8+8+99=549=6. But, what if we had an outlier in the set? What if our new set was {1,4,5,6,6,7,8,8,9,4946}? When our new mean would be 1+4+5+6+6+7+8+8+9+494610=500010=500. See how much an outlier can throw off the mean?

The median is the value separating the first half of the data set with the second half. To find the median of a data set, first put the data set in order. Then, if there are an odd number of values, cross an equal amount of values off from the beginning of the list and the end of the list. The remaining data point in the middle of your list is the median! If there are an even number of values in your set, the median will be the average of the two middle values.

The median is helpful when describing asymmetrical data , and it also works for ordinal, ratio, or interval data. It is easy to find, especially if your data is in order. Also, it is not usually too affected by outliers.

Limitations

However, the fact that the median is not usually affected by outliers means that we can't use it to account for these extreme values. Also, the median is not helpful when we are working with nominal or qualitative data. Think about it ... how can you order a list of colors and find the middle color?

The mode of a data set is the value that occurs most frequently. We can find the mode by putting the data in order, and then identifying which value(s) repeat the most. Frequency tables , histograms , and frequency distributions can be helpful in identifying the mode.

Finding the mode can be especially helpful when we are working with qualitative data or nominal data , but it can also be used when we have ordinal, ratio, or interval data. It's easy to spot on a graph, frequency table, or even an ordered set of data. It is not usually affected by outliers.

There can be many modes, so finding the mode may not say much about how the data is distributed. On the other hand, if all the data points only occur once, there is no real mode. The mode is unstable, and can be easily influenced by certain changes in the data. Adding just one new data point can change the mode!

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Problem Solving Averages and Range

Subject: Mathematics

Age range: 11-14

Resource type: Worksheet/Activity

Last updated

29 January 2018

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