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What Is Weighted Average?

Weighting a stock portfolio.

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Weighted Average: Definition and How It Is Calculated and Used

A weighted average is a calculation that takes into account the varying degrees of importance of the numbers in a data set. A weighted average can be more accurate than a simple average in which all numbers in a data set are assigned an identical weight.

Key Takeaways

• The weighted average takes into account the relative importance or frequency of some factors in a data set.
• A weighted average is sometimes more accurate than a simple average.
• In a weighted average, each data point value is multiplied by the assigned weight, which is then summed and divided by the number of data points.
• A weighted average can improve the data’s accuracy.
• Stock investors use a weighted average to track the cost basis of shares bought at varying times.

Weighted Average

Paige McLaughlin / Investopedia

What Is the Purpose of a Weighted Average?

In calculating a simple average, or arithmetic mean , all numbers are treated equally and assigned equal weight. But a weighted average assigns weights that determine in advance the relative importance of each data point. In calculating a weighted average, each number in the data set is multiplied by a predetermined weight before the final calculation is made.

A weighted average is most often computed to equalize the frequency of the values in a data set. For example, a survey may gather enough responses from every age group to be considered statistically valid, but the 18 to 34 age group may have fewer respondents than all others relative to their share of the population . The survey team may weigh the results of the 18 to 34 age group so that their views are represented proportionately.

However, values in a data set may be weighted for other reasons than the frequency of occurrence. For example, if students in a dance class are graded on skill, attendance, and manners, the grade for skill may be given greater weight than the other factors.

Each data point value in a weighted average is multiplied by the assigned weight, which is then summed and divided by the number of data points. The final average number reflects the relative importance of each observation and is thus more descriptive than a simple average. It also has the effect of smoothing out the data and enhancing its accuracy.

Investors usually build a position in a stock over a period of several years. That makes it tough to keep track of the cost basis on those shares and their relative changes in value. The investor can calculate a weighted average of the share price paid for the shares. To do so, multiply the number of shares acquired at each price by that price, add those values, then divide the total value by the total number of shares.

A weighted average is arrived at by determining in advance the relative importance of each data point.

For example, say an investor acquires 100 shares of a company in year one at $10, and 50 shares of the same stock in year two at $40. To get a weighted average of the price paid, the investor multiplies 100 shares by $10 for year one and 50 shares by $40 for year two, then adds the results to get a total of $3,000. Then the total amount paid for the shares, $3,000 in this case, is divided by the number of shares acquired over both years, 150, to get the weighted average price paid of $20.

This average is now weighted with respect to the number of shares acquired at each price, not just the absolute price.

The weighted average is sometimes also called the weighted mean.

Advantages and Disadvantages of Weighted Average

Pros of weighted average.

Weighted average provides a more accurate representation of data when different values within a dataset hold varying degrees of importance. By assigning weights to each value based on their significance, weighted averages ensure that more weight is given to data points that have a greater impact on the overall result. This allows for a more nuanced analysis and decision-making process.

Next, weighted averages are particularly useful for handling skewed distributions or outliers within a dataset. Instead of being overly influenced by extreme values, weighted averages take into account the relative importance of each data point. This means you can "manipulate" your data set so it's more relevant, especially when you don't want to consider extreme values.

Thirdly, weighted averages offer flexibility in their application across various fields and disciplines. Whether in finance, statistics, engineering, or manufacturing , weighted averages can be customized to suit specific needs and objectives. For instance, like we discussed above, weighted averages are commonly used to calculate portfolio returns where the weights represent the allocation of assets. Weighted averages can also be used in the manufacturing process to determine the right combination of goods to use.

Cons of Weighted Average

One downside of a weighted average is the potential for subjectivity in determining the weights assigned to each data point. Deciding on the appropriate weights can be challenging, and it often involves subjective judgment where you don't actually know the weight to attribute. This subjectivity can introduce bias into the analysis and undermine the reliability of the weighted average.

Weighted averages may be sensitive to changes in the underlying data or weighting scheme. Small variations in the weights or input values can lead to significant fluctuations in the calculated average, making the results less stable and harder to interpret. This sensitivity can be particularly problematic in scenarios where the weights are based on uncertain or volatile factors which may include human emotion (i.e. are you confident you'll feel the same about the appropriate weights over time?).

Last, the interpretation of weighted averages can be more complex compared to simple arithmetic means. Though weighted averages provide a single summary statistic, they may make it tough to understand the full scope of the relationship across data points. Therefore, it's essential to carefully assess how the weights are assigned and the values are clearly communicated to those who interpret the results.

Accurate representation via weighted significance, aiding nuanced decision-making.

Handles outliers, mitigating extreme value influence for relevance.

Flexible across fields, tailor needs, or objectives.

Subjectivity in determining weights introduces bias and undermines reliability.

Sensitivity to changes in data or weighting scheme affects stability.

Adds complexity compared to arithmetic mean, potentially obscuring analysis

Examples of Weighted Averages

Weighted averages show up in many areas of finance besides the purchase price of shares, including portfolio returns , inventory accounting, and valuation. When a fund that holds multiple securities is up 10% on the year, that 10% represents a weighted average of returns for the fund with respect to the value of each position in the fund.

For inventory accounting, the weighted average value of inventory accounts for fluctuations in commodity prices, for example, while LIFO (last in, first out) or FIFO (first in, first out) methods give more importance to time than value.

When evaluating companies to discern whether their shares are correctly priced, investors use the weighted average cost of capital (WACC) to discount a company’s cash flows. WACC is weighted based on the market value of debt and equity in a company’s capital structure.

Weighted Average vs. Arithmetic vs. Geometric

Weighted averages provide a tailored solution for scenarios where certain data points hold more significance than others. However, there are other forms of calculating averages, some of which were mentioned earlier. The two main alternatives are the arithmetic average and geometric average.

Arithmetic means, or simple averages, are the simplest form of averaging and are widely used for their ease of calculation and interpretation. They assume that all data points are of equal importance and are suitable for symmetrical distributions without significant outliers. Arithmetic means will often be easier to calculate since you divide the sum of the total by the number of instances. However, it is much less nuanced and does not allow for much flexibility.

Another common type of central tendency measure is the geometric mean . The geometric mean offers a specialized solution for scenarios involving exponential growth or decline. By taking the nth root of the product of n values, geometric means give equal weight to the relative percentage changes between values. This makes them particularly useful in finance for calculating compound interest rates or in epidemiology for analyzing disease spread rates.

A weighted average is a statistical measure that assigns different weights to individual data points based on their relative significance, resulting in a more accurate representation of the overall data set. It is calculated by multiplying each data point by its corresponding weight, summing the products, and dividing by the sum of the weights.

Is Weighted Average Better?

Whether a weighted average is better depends on the specific context and the objectives of your analysis. Weighted averages are better when different data points have varying degrees of importance, allowing you to have a more nuanced representation of the data. However, they may introduce subjectivity in determining weights and can be sensitive to changes in the weighting scheme

How Does a Weighted Average Differ From a Simple Average?

A weighted average accounts for the relative contribution, or weight, of the things being averaged, while a simple average does not. Therefore, it gives more value to those items in the average that occur relatively more.

What Are Some Examples of Weighted Averages Used in Finance?

Many weighted averages are found in finance, including the volume-weighted average price (VWAP) , the weighted average cost of capital, and exponential moving averages (EMAs) used in charting. Construction of portfolio weights and the LIFO and FIFO inventory methods also make use of weighted averages.

How Do You Calculate a Weighted Average?

You can compute a weighted average by multiplying its relative proportion or percentage by its value in sequence and adding those sums together. Thus, if a portfolio is made up of 55% stocks, 40% bonds, and 5% cash, those weights would be multiplied by their annual performance to get a weighted average return. So if stocks, bonds, and cash returned 10%, 5%, and 2%, respectively, the weighted average return would be (55 × 10%) + (40 × 5%) + (5 × 2%) = 7.6%.

Statistical measures can be a very important way to help you in your investment journey. You can use weighted averages to help determine the average price of shares as well as the returns of your portfolio. It is generally more accurate than a simple average. You can calculate the weighted average by multiplying each number in the data set by its weight, then adding up each of the results together.

Tax Foundation. “ Inventory Valuation in Europe .”

My Accounting Course. “ Weighted Average Cost of Capital (WACC) Guide .”

CDC. " Measures of Spread ."

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Weighted Arithmetic Mean

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The weighted arithmetic mean is a  measure of central tendency of a set of quantitative observations when not all the observations have the same importance.

We must assign a weight to each observation depending on its importance relative to other observations.

The weighted arithmetic mean equals the sum of observations multiplied by their weights divided by the sum of their weights.

The weighted arithmetic mean was introduced by Cotes, Roger in 1712. His work was published in 1722, six years after his death.

MATHEMATICAL ASPECTS

Let \( { x_1,x_2,\ldots,x_n } \) be a set of n quantities or n observations relative to a  quantitative variable   X to which we assign the weights \( { w_1,w_2,\ldots,w_n } \) .

The weighted arithmetic mean equals:

DOMAINS AND LIMITATIONS

The weighted arithmetic mean is now used in economics, especially in consumer and producer price indices, etc.

Suppose that...

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Cotes, R.: Aestimatio Errorum in Mixta Mathesi, per variationes partium Trianguli plani et sphaerici. In: Smith, R. (ed.) Opera Miscellania, Cambridge (1722)

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(2008). Weighted Arithmetic Mean. In: The Concise Encyclopedia of Statistics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-32833-1_421

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• Weighted Mean and Average: Statistics Made Easy

Welcome to Warren Institute! In this article, we will delve into the fascinating world of Statistics and explore how to find the weighted mean and weighted average. Understanding these concepts is crucial for accurately interpreting data and making informed decisions. We will break down the step-by-step process, providing clear explanations and practical examples along the way. Whether you're a student or a professional in the field, mastering the weighted mean and weighted average will undoubtedly enhance your analytical skills. So, let's embark on this journey together and unlock the power of statistics!

Understanding the Concept of Weighted Mean and Weighted Average

Calculating the weighted mean and weighted average, practical examples of weighted mean and weighted average, advantages and limitations of weighted mean and weighted average, what is the formula for calculating the weighted mean in statistics, how can i find the weighted average when given different weights for each data point, what are some real-life applications of using weighted means and weighted averages in statistics, can you provide an example problem that demonstrates how to find the weighted mean in statistics, is the weighted mean always a better measure of central tendency than the simple arithmetic mean.

The first step in finding the weighted mean and weighted average in statistics is to understand the concept behind these calculations. Weighted mean is a statistical measure that takes into account the importance or weight of each data point when calculating the average. It is calculated by multiplying each data point by its respective weight, summing up these products, and dividing by the sum of the weights.

Weighted average , on the other hand, is a generalization of the weighted mean that can be applied to various situations where different data points have different weights or importance. It is calculated by multiplying each data point by its respective weight, summing up these products, and dividing by the sum of the weights.

To calculate the weighted mean and weighted average, follow these steps:

• Assign weights : Assign weights to each data point based on their relative importance or significance.
• Multiply and sum : Multiply each data point by its respective weight, then sum up these products.
• Sum the weights : Sum up the weights assigned to each data point.
• Divide : Divide the sum of the products by the sum of the weights to obtain the weighted mean or weighted average.

Weighted mean and weighted average are commonly used in various fields, including mathematics education. Here are a few practical examples:

• Calculating final grades: Assigning different weights to different assignments, exams, and projects allows for a more accurate representation of a student's overall performance.
• Market research: When analyzing survey data, weighting the responses based on demographic factors can provide more representative results.
• Financial analysis: In portfolio management, assigning weights to different assets helps in determining the overall performance of the portfolio.

Advantages:

• Reflects the relative importance or significance of each data point.
• Allows for more accurate calculations when dealing with data sets that have varying weights.

Limitations:

• Requires knowledge of the appropriate weights to assign to each data point.
• May not be suitable for all types of data sets or situations.
• Can be sensitive to outliers or extreme values.

frequently asked questions

The formula for calculating the weighted mean in statistics is sum of (value * weight) divided by sum of weights .

To find the weighted average when given different weights for each data point, you can use the following formula:

Weighted Average = (Sum of (Data Point * Weight)) / (Sum of Weights)

Multiply each data point by its corresponding weight, and then add up these products. Then, divide this sum by the total sum of weights. This will give you the weighted average.

Some real-life applications of using weighted means and weighted averages in statistics are:

• Grade calculation: Weighted averages are commonly used in educational settings to calculate final grades. Different assignments or exams may have different weights, and the weighted average is used to determine the overall grade.
• Market research: In market research, weighted averages are used to analyze survey data. Different responses may be assigned different weights based on their importance or representativeness, allowing for a more accurate representation of the population.
• Financial analysis: Weighted averages play a crucial role in financial analysis. For example, stock market indices like the S&P 500 use weighted averages to track the performance of a group of stocks. The weights are typically based on the market capitalization of each stock.
• Polling and elections: Weighted averages are used in polling to account for different demographics and ensure representative results. In elections, different voting regions may have different weights based on their population size, leading to a weighted average of votes.

Sure! Here's an example problem that demonstrates how to find the weighted mean in statistics:

Problem: A teacher wants to calculate the average test score for their class, but wants to give more weight to the final exam. The final exam is worth 40% of the total grade, while the midterm exam is worth 30% and the homework assignments are worth 30%. The scores are as follows:

Final Exam: 90 Midterm Exam: 85 Homework Assignments: 95, 90, 85

Solution: To find the weighted mean, we multiply each value by its corresponding weight and then sum them up.

Weighted Mean = (Final Exam * 0.4) + (Midterm Exam * 0.3) + (Homework Assignments * 0.3)

= (90 * 0.4) + (85 * 0.3) + ((95+90+85)/3 * 0.3)

= 36 + 25.5 + 90

Therefore, the weighted mean test score for the class is 151.5.

No, the weighted mean is not always a better measure of central tendency than the simple arithmetic mean in Mathematics education. The choice between the two depends on the specific context and the nature of the data being analyzed.

In conclusion, understanding how to find the weighted mean and weighted average in statistics is crucial for a comprehensive understanding of Mathematics education. By incorporating weights that reflect the importance or significance of each data point, we can obtain a more accurate representation of the data set. The weighted mean allows us to calculate a single value that takes into account the varying weights, while the weighted average provides a measure of central tendency that considers both the values and their associated weights. These concepts are particularly useful in real-life scenarios where certain data points hold more significance than others. By utilizing the techniques discussed in this article, educators and students alike can enhance their statistical analysis skills and make more informed decisions based on weighted data. So, let's embrace the power of weighted mean and weighted average in Statistics to unlock deeper insights in Mathematics Education.

If you want to know other articles similar to Weighted Mean and Average: Statistics Made Easy you can visit the category General Education .

Michaell Miller

Michael Miller is a passionate blog writer and advanced mathematics teacher with a deep understanding of mathematical physics. With years of teaching experience, Michael combines his love of mathematics with an exceptional ability to communicate complex concepts in an accessible way. His blog posts offer a unique and enriching perspective on mathematical and physical topics, making learning fascinating and understandable for all.

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Weighted Mean

Weighted mean is a type of average that helps in contributing equally to the final mean when some data points are weighted more than the others. It is most commonly used in statistics when the data is associated with the population. If the data is weighted the same across the entire set, then the weighted mean is equal to the arithmetic mean . Let us learn more about the weighted mean, the formula, and solve a few examples to understand the concept better.

What is Weighted Mean?

The weighted mean is a type of mean that is calculated by multiplying the weight associated with a particular event or outcome with its associated quantitative outcome and then summing all the products together. In other words, when some values weigh more than the others that's when the weighted mean is calculated.

Weighted Mean Definition

The weighted mean is defined as the summation of the product of weights and quantities, divided by the summation of weights. The concept of the weighted mean is quite often used in accounts, to give different weights based on time or based on priority.

How to Calculate the Weighted Mean?

While finding the average for an equally weighted set of values, we use the simple process of the arithmetic mean. Where all the values are added and divided by the total number of items in the set. However, the weighted mean is calculated when one of the values has more weight than the others. It can be calculated by using these two simple steps:

• Multiply the numbers in the set by the weights.
• Add the results.

But in certain values, the given data set is more important than the others. A weight (w) n is attached to each of the values (x) n . The general formula to find the weighted mean is given as,

Weighted mean = Σ(w) n (x̄) n /Σ(w) n

• x̄ = the mean value of the set of given data.
• w = corresponding weight for each observation.

The simple steps used to calculate the weight mean through the formula is:

• Step 1: Add all the weighted values together.
• Step 2: Multiply the weighted values and the quantities in the data set.
• Step 3: Add the values together obtained in step 2.
• Step 4: Divide the result by the number obtained in step 1.

Weighted Mean Formula

The weighted mean formula helps to find the mean of the quantities by assigning weights to the quantities. Based on the level of importance of the quantities, weights are assigned to the quantities. The below formula for weighted mean includes variables \(x_1\), \(x_2\), \(x_3\)...\(x_n\), and their weights \(w_1\), \(w_2\), \(w_3\)...\(w_n\) respectively. Here this is similar to the average and the weighted mean represents the summary value of all the available quantities. The weighted mean has the same units as that of the individual quantities.

\[ \bar x = \frac{w_1x_1 + w_2x_2 + ......+ w_nx_n}{w_1 + w_2 + ... + w_n} \]

\[\bar x = \frac{\sum w_nx_n}{\sum w_n} \]

Let us look at an example to understand this better.

Example: Find weighted mean for following data set w = {2, 5, 6, 8, 9}, x = {4, 3, 7, 5, 6}

Given data sets w = {2, 5, 6, 8, 9}, x = {4, 3, 7, 5, 6} and N = 5

Weighted mean = ∑(weights × quantities) / ∑(weights)

= (w 1 x 1 + w 2 x 2 + w 3 x 3 + w 4 x 4 + w 5 x 5 ) / (w 1 + w 2 + w 3 + w 4 + w 5 )

= (2 × 4 + 5 × 3 + 6 × 7 + 8 × 5 + 9 × 6) / (2 + 5 + 6 + 8 + 9)

= ( 8 + 15 + 42 + 40 + 54) / 30

Therefore, the weighted mean is 5.3.

Weighted Mean Vs Arithmetic Mean

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• Categorical Data
• Range in Statistics
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• Mean of Grouped Data

Weighted Mean Examples

Example 1: A teacher provides the following weightage of 20% for class attendance, 30% for project work, 40% for tests, and 10% for home assignments. A student scores 80/100 for class attendance, 4/5 in project work, 35/50 in tests, and 8/10 in home assignments. Find the final score of the student.

\(\begin{align} \bar x &= \frac{w_1x_1 + w_2x_2 + ......+ w_nx_n}{w_1 + w_2 + ... + w_n} \\&=\frac{20\%.\frac{80}{100} + 30\%.\frac{4}{5} + 40\%.\frac{35}{50} + 10\%.\frac{8}{10}}{20\% + 30\% + 40\% + 10\%} \\&=\frac{20\%.0.8 + 30\%.0.8 + 40\%.0.7 + 10\%.0.8}{100\%} \\&=\frac{0.2 \times 0.8 + 0.3 \times 0.8 + 0.4 \times 0.7 + 0.1 \times 0.8}{1} \\&=0.16 + 0.24 + 0.28 + 0.08 \\&=0.76 \end{align} \) Therefore, the final score of the student is 0.76.

Example 2: For a job application 0.8 weightage is given to academic qualification, 0.7 is given to personality, 0.4 is given to the location. The prospective candidate scores 4.5/5 for academic qualification, 3/5 for personality, and 2.8/5 for location. Find the final score received by the candidate.

\(\begin{align} \bar x &= \frac{w_1x_1 + w_2x_2 + ......+ w_nx_n}{w_1 + w_2 + ... + w_n} \\&=\frac{0.8 \times \frac{4.5}{5}+ 0.7 \times \frac{3}{5} + 0.4 \times \frac{2.8}{5}}{0.8 + 0.7 + 0.4} \\&=\frac{0.8 \times 0.9 + 0.7 \times 0.6 + 0.4 \times 0.56}{0.19} \\&=\frac{0.72 + 0.42 + 0.224}{0.19} \\&=\frac{1.364}{0.19} \\&= 7.18 \end{align} \) Hence, the final score of the candidate is 7.18.

Example 3: Ben is a fruit merchant who sells various types of fruits in Chicago. Some fruits are of higher quality and are sold at a higher price. He wants you to calculate the weighted mean from the following data:

\(\begin{align} \bar x &= \frac{w_1x_1 + w_2x_2 + ......+ w_nx_n}{w_1 + w_2 + ... + w_n}

= 100 × 80 + 50 × 70 + 20 × 60 + 15 × 50 / 100 + 50 + 20 + 15

= 8000 + 3500 + 1200 + 750 / 185

= 13450/185

Therefore, the weighted mean is 75.7

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FAQs on Weighted Mean

What is the meaning of weighted mean.

The weighted mean is a mean that is calculated by multiplying the weight associated with a particular event or outcome with its associated quantitative outcome and then summing all the products together.

How Do You Calculate Weighted Mean?

Weighted mean can be calculated in two ways.

• When the weights add to 1 then multiply the weights with the value and add them.
• When the weights add to more than 1 then we use the weighted mean formula.

What is the Weighted Mean Formula?

The weighted mean formula is:

• \(x_1\), \(x_2\), \(x_3\)...\(x_n\) are the variables.
• \(w_1\), \(w_2\), \(w_3\)...\(w_n\) are the weights.

When Should I Use Weighted Mean?

When some values weigh more than the other values i.e. count more, that's when weighted mean is used.

Weighted Average Calculator

• Let's start from the beginning: what is a weighted average?

How to calculate a weighted average

Weighted average formula, weighted vs. unweighted gpa for high school, different averages: arithmetic, geometric, harmonic.

To understand how a weighted average calculator works, you must first understand what a weighted average is. Weighted average has nothing to do with weight conversion, but people sometimes confuse these two concepts. The typical average, or mean, is when all values are added and divided by the total number of values. We can compute this using our average calculator , by hand, or by using a hand-held calculator since all the values have equal weights.

But what happens when values have different weights , which means that they're not equally important? Below you will see how to calculate the weighted mean using the weighted average formula. Also, you'll find examples where the weighted average method may be used - like e.g. calculation of the GPA, average grade, or your final grade.

Prefer watching over reading? Learn all you need in 90 seconds with this video we made for you :

Let's start from the beginning: what is a weighted average?

Weighted average (weighted arithmetic mean) is a concept similar to standard arithmetic mean (called simply the average ), but in the weighted average, not all elements contribute equally to the final result. We can say that some values are more important than others, so they are multiplied by a coefficient called the weight .

For example, during your studies, you may encounter a situation where the grade from an exam is two times more important than the grade from a quiz - and that's exactly what we call the weighted average method. To define it in a more mathematical way, we can write the weighted average formula as:

where x 1 x_1 x 1 ​ , x 2 x_2 x 2 ​ ,…, x n x_n x n ​ are our numbers, and w 1 w_1 w 1 ​ , w 2 w_2 w 2 ​ ,…, w n w_n w n ​ are our weights - the importance of the numbers in averaging.

So, having an A from an exam and a C from a quiz, you'd get a B as a standard average, but assuming that the exam is two times more important, you should get a B+ .

🙋 If you're running a business, you may be interested in checking our WACC calculator , which concerns the Weighted Average Cost of Capital.

One type of average which is typically weighted is a grade point average. As the calculation of GPA may sometimes be tricky, we've created two dedicated tools: the high school GPA and the college GPA calculator — have you checked them yet?

Let's find out how to calculate a weighted average - the easiest way is to look at the simple example:

Suppose a student has two four-credit classes, a three-credit class, and a two-credit class. Assume that the grades of the courses are as follows:

• A for a four-credit class;
• B for the other four credit class;
• A for the three credit class; and
• C+ for the two credit class.

Then, we need to translate the letter grades into numerical values. Most schools in the US use a so-called 4.0 GPA scale, which is a 4-point grading scale. The table below shows a typical letter grade/GPA conversion system:

So from the table, we know that A = 4.0, B = 3.0, and C+ = 2.3. Now that we have all the information, we can have a look at how to calculate the GPA using a weighted average method:

• Sum the number of credits. 4 + 4 + 3 + 2 = 13 , that was a really easy step.
• Take the value assigned to the grade and multiply it by the number of credits. In our case, it will be:
• A · 4 credits = 4.0 · 4 = 16 ;
• B · 4 credits = 3.0 · 4 = 12 ;
• A · 3 credits = 4.0 · 3 = 12 ;
• C · 2 credits = 2.3 · 2 = 4.6 .
• Add all the values. 16 + 12 + 12 + 4.6 = 44.6 .
• Divide the sum by the total number of credits. So, for our example, it's equal to 44.6/13 = 3.43

We may write the whole weighted average formula as:

(4 · 4 + 4 · 3 + 3 · 4 + 2.3 · 2) / (4 + 4 + 3 + 2) = 3.43

Let's compare this result to an average that is not weighted. Then we don't take the credits into account, and we divide the sum of grades by its total number.

(4 + 3 + 4 + 2.3) / 4 = 3.33

Notice how the weighted average changed. Sometimes it may be a really significant difference - like a grade difference or even whether you pass or fail your course.

Let's repeat what the weighted average formula looks like:

But what does it mean? To figure out how to calculate a weighted average, we need to know the weight of each value. Typically, we present the weights in the form of a percentage or (in statistics) a probability of occurrence.

For example, let's suppose that exams, quizzes, and homework assignments all contribute to a class's grade. Each of the three exams is worth 25 percent of the grade, the quizzes are worth 15 percent, and the homework assignments are worth 10 percent. To calculate the average, you multiply the percentage by the grades and add them together. If the test scores are 75, 90, and 88, the quiz average is 70, and the homework grade is 86, the weighted average is as follows:

(0.25 · 75 + 0.25 · 90 + 0.25 · 88 + 0.15 · 70 + 0.10 · 86) / 1 = 82.35

Compare this to a non-weighted average of (75 + 90 + 88 + 70 + 86) / 5 = 81.8

In statistics, you will often encounter a discrete probability distribution that has values for x and their associated probabilities. Since the probabilities for each value of x will likely not all be the same, we can apply the weighted average formula. Simply multiply each x value by its probability of occurring and sum the values.

🙋 In case you need to estimate the geometric mean, Omni's geometric mean calculator will come in handy.

We often use a weighted average to calculate the so-called weighted GPA . It's a term that rarely appears in the context of college GPA (although college GPA is computed using a weighted average method, with courses credits as weights) but is usually used for high school GPA. Let's have a closer look at this topic.

The first thing we need to emphasize: you need to be precise about what you want to take into account during weighting - credits, course difficulty, or maybe both these factors ?

Course difficulty is taken into account in most weighted GPA calculations. It rewards you for taking classes of a higher level by adding extra points to your grade. There are a couple of types of more demanding courses which influence your weighted GPA score:

• AP Courses (Advanced Placement Courses) usually give you an additional 1 point to your standard GPA score;
• IB Courses (International Baccalaureate Courses) are also rewarded with 1 extra point;
• College Prep classes can also add 1 point to your grade; and
• Honors Courses most often give you an additional 0.5 points (although you can find examples of schools where it's awarded with 1 point).

So, what are the options for weighing in High school GPA calculations? Let's define:

Unweighted GPA , as the GPA where we DON'T care about course difficulty :

a) and we DON'T care about course credits:

High School GPA = Σ grade value / Σ courses

b) and we DO care about course credits:

High School GPA = Σ (grade value · credits) / Σ credits

Weighted GPA , as the GPA where we DO care about course difficulty :

High School GPA = Σ (weighted grade value) / Σ courses

High School GPA = Σ (weighted grade value · credits)/ Σ credits

It may look a bit overwhelming, but let's have a look at a hypothetical results sheet, and everything should be clear:

• 1 a) Unweighted GPA: we DON'T care about course difficulty and credits.

All the courses have the same grade scale and credits, no matter the course difficulty. So we may convert our grades into numbers:

Then, we can calculate the unweighted GPA as follows:

Unweighted High School GPA = Σ grade value / Σ courses

= (4.0 + 3.3 + 2.3 + 3.7) / 4 = 13.3 / 4 = 3.325 ≈ 3.33

Did you notice that it's a standard average? It's just summing all scores and dividing the result by the total number of observations (4 courses).

• 1 b) Unweighted GPA: we DON'T care about course difficulty, but we DO care about credits.

Things are getting more complicated when we consider the course credits. Some sources ignore the course's credits for unweighted GPA scores, but others keep them. So, if your classes have some credits/points, you can calculate the weighted average of grades and credits (but still, it's not the thing we usually name the weighted GPA ):

Then, the GPA will be equal to:

= (4.0 · 0.5 + 3.3 · 1 + 2.3 · 0.5 + 3.7 · 1) / (0.5 + 1 + 0.5 + 1)

= 10.15 / 3 = 3.38333… ≈ 3.38

The courses with higher credits value have better marks in our example, so the overall GPA is also higher.

• 2 a) Weighted GPA: we DO care about course difficulty and DON'T care about course credits.

Depending on the course type, the letter grades are translated to different numerical values:

Continuing with our example, now our four classes have the course type assigned:

As two courses are not standard classes, they get extra points ( A from Maths - 4.5 instead of 4.0, as it's an Honors course, A- from English - 4.7 instead of 3.7, as it's an AP course).

The formula for the calculation of weighted GPA is:

Weighted High School GPA = Σ (weighted grade value) / Σ courses

= (4.5 + 3.3 + 2.3 + 4.7) / 4 = 14.8 / 4 = 3.7 ,

where weighted grade value is a:

• grade value + 0 for Regular courses;
• grade value + 0.5 for Honors courses; and
• grade value + 1 for AP/IB/College Prep courses.

So we omitted the courses' credits, but we've considered the course's difficulty. And finally, we have

• 2 b) Weighted GPA: we DO care about course difficulty and DO care about course credits.

So if you're taking into account both credits and course difficulty, then the result is:

Weighted High School GPA = Σ (weighted grade value · credits) / Σ credits

= (4.5 · 0.5 + 3.3 · 1 + 2.3 · 0.5 + 4.7 · 1) / (0.5 + 1 + 0.5 + 1) = 11.4 / 3 = 3.8

That wasn't so hard, was it?

Now that you understood what a weighted average is let's compare different averages. We've prepared for you a table that sums up all the important information about four different means:

General formulas for means look as follows:

• Arithmetic mean:
• Geometric mean:
• Harmonic mean:
• Weighted mean

How to calculate my weighted average if my course work is worth 40%?

Assuming that your test score is worth 60% , and the coursework and test scores are expressed as fractions of 100 , follow these steps to calculate the weighted average:

• Multiply the coursework score by 2 and the test score by 3 .
• Add the results together and divide by the total of the weights: 5 .

How do I calculate weighted average?

To calculate the weighted average, follow these steps:

• Get the weight of each number.
• Multiply each number by its weight.
• Add all of the results from Step 2 together.
• Add all of the weights together.
• Divide the answer from Step 3 by the answer in Step 4 .

How do I calculate the weighted average of my purchases?

If you purchased three products of different quantities:

• 5 packs of acrylic paint at $19.99;
• 3 packs of paint brushes at $13.99; and
• 2 art canvases at $25.00.

Use the following steps to calculate the weighted average of your spending:

• Multiplying the price by the quantity: 5 × 19.99 = $99.95 3 × 13.99 = $41.97 2 × 25.00 = $50
• Find the total spent: 99.95 + 41.97 + 50 = $191.92
• Find the number of products sold: 5 + 3 + 2 = 10
• Find the weighted average: 191.92/10 = $19.19

What is the weighted averages of the cost of my stationary?

Assuming that you purchased:

• 3 packs of pencils at $5 each;
• 2 packs of paper at $10.00 each; and
• 5 packs of pens at $15.00.

Your weighted average is $11.

To calculate this, we find the total amount of money spent by following these steps:

• Find the amount of money spent.
• Find the total amount of items purchased.
• Divide the answer in Step 1 by the answer in Step 2.

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• Weighted Mean

Introduction to Weighted Mean

In Mathematics, the weighted mean is used to calculate the average value of the data. In the weighted mean calculation, the average value can be calculated by providing different weights to some of the individual values. We need to calculate the weighted mean when data is given in a different way compared to the arithmetic mean or sample mean. Different types of means are used to calculate the average of the data values. Let’s understand what is weighted mean and how to define weighted mean along with solved examples.

Weight Definition

Weight is defined as the measure of how heavy an object is. The weights cannot be negative. Some weight can be zero, but not all of them, since division by zero is not allowed.

The data elements which have a high weight will contribute more to the weighted mean as compared to the elements with a low weight.

What is Weighted Mean?

To calculate the weighted mean of certain data, we need to multiply the weight associated with a particular event or outcome with its associated outcome and finally sum up all the products together. It is very useful in calculating a theoretically expected outcome. Apart from weighted mean and arithmetic mean, there are various types of means such as harmonic mean, geometric mean, and so on.

Define Weighted Mean

The weighted mean is defined as an average computed by giving different weights to some of the individual values. When all the weights are equal, then the weighted mean is similar to the arithmetic mean. A free online tool called the weighted mean calculator is used to calculate the weighted mean for the given range of values.

Weighted Mean Formula

To calculate the weighted mean for a given set of non-negative data x 1 ,x 2 ,x 3 ,...x n with non-negative weights w 1 ,w 2 ,w 3 ,..., we use the formula given below.

\[WeightedMean\overline{( W )} = \frac {\sum_{i=1}^{n} w_{i}x_{i}}{\sum_{i=1}^{n} w_i}\]

Where \[\overline {W}\] is the weighted mean,

x is the repeating value,

n is the number of terms whose mean is to be calculated, and

w is the individual weights.

Uses of Weighted Means

Weighted means are useful in a wide variety of scenarios in our daily life. For example, a student uses a weighted mean in order to calculate their percentage grade in a course. In such a case, the student has to multiply the weighing of all assessment items in the course (e.g., assignments, exams, projects, etc.) by the respective grade that was obtained in each of the categories. 

It is used in descriptive statistical analysis, such as index numbers calculation. For example, stock market indices such as Nifty or BSE Sensex are computed using the weighted average method. It can also be applied in physics to find the center of mass and moments of inertia of an object.

It is also useful for businessmen to evaluate the average prices of goods purchased from different vendors where the purchased quantity is considered as the weight. It gives a better understanding of his expenses.

A customer's decision on whether to buy a product or not depends on the quality of the product, knowledge of the product, cost of the product, and service by the franchise. The customer allocates weight to each criterion and calculates the weighted average. This will help him to make a better decision on buying the product.

To recruit a person for a job, the interviewer looks at the personality, working capabilities, educational qualification, and team working skills. Based on the profile, different levels of importance (weights) are given, and then the final selection is made.

Important Notes

The weights can be in the form of quantities, decimals, whole numbers , fractions, or percentages.

If the weights are given in percentage, then the sum of the percentage will be 100%.

Weighted average for quantities (x) i having weights in percentage (P) i % is: Weighted average = ∑ (P) i % × (x) i

Solved Example of Weighted Mean

Suppose a marketing firm conducted a survey of 1,000 households to determine the average number of TVs each household owns. The data shows that there are more households with two or three TVs and a few numbers with one of four. Every household in the sample has at least one TV and not a household has more than four. Calculate the mean number of TVs per household.

Solution: Here most of the values in this data set are repeated multiple times, we can easily compute the sample mean as a weighted mean. Following are steps to calculate the weighted arithmetic mean.

Step 1: First assign a weight to each value in the dataset.

x 1 =1, w 1 =73

x 2 =2, w 2 =378

x 3 =3, w 3 =459

x 4 =4, w 4 =90

Step 2: Now compute the numerator of the weighted mean formula .

To calculate it, multiply each sample by its weight and then add the products together to get the final value

\[\displaystyle\sum\limits_{i=1}^4 i\] w i x i = w 1 x 1 + w 2 x 2 + w 3 x 3 + w 4 x 4

= 1 x 73 + 2 x 378 + 3 x 459 + 4 x 90

= 73 + 756 + 1377 + 360

Step 3: Now, compute the denominator of the weighted mean formula by adding their weights together.

\[\displaystyle\sum\limits_{i=1}^4 i\] w i = w 1 + w 2 + w 3 + w 4

= 73 + 378 + 459 + 90

Step 4: Finally divide the numerator value by the denominator value.

\[\frac{\displaystyle\sum\limits_{i=1}^4w_{i}x_{i}}{\displaystyle\sum\limits_{i=1}^4 w_{i}}\]

=\[\frac {2566}{1000}\]

Hence, the mean number of TVs per household in this sample is 2.566.

Note: The weighted mean can be easily influenced by an outlier in our data. If we have very high or very low values in our data set, then we cannot rely on the weighted mean.

Weighted Mean is a mean where some of the values contribute more than others. It represents the average of a given data. The Weighted mean is similar to the arithmetic mean or sample mean. Sometimes it is also known as the weighted average.

When the weights add to 1, we just have to multiply each weight by the matching value and sum it all up.

Otherwise, we have to multiply each weight w by its matching value x, the sum that all up, and divide it by the sum of weight. 

FAQs on Weighted Mean

1. What is weighted mean used for?

Weighted means are used in a wide variety of scenarios. For example, a student may use a weighted mean in order to calculate their percentage grade in a course such as a student would multiply the weighing of all assessment items in the course.

2. What are the steps involved in the calculation of weighted mean?

Following are the steps that are involved in the calculation of weighted mean.

Step 1: First list down the numbers and weights in tabular form. Representation in tabular form makes the calculations easy.

Step 2:  Multiply each number and relevant weight assigned to that number (w 1  by x 1 , w 2  by x 2 , and so on).

Step 3: Add the numbers obtained in Step 2 i.e . Σ x i   w i .

Step 4: Find the sum of the weights i.e. Σw i .

Step 5: Divide the total of the values obtained in step 3 by the sum of the weights obtained in step 4 i.e.  \[\frac {\sum x_i w_i}{\sum w_i}\]

3. How can we use the weighted mean calculator?

Following are the steps used in the weighted mean calculator :

Step 1: First enter the range values and their weight weighted mean in the input field.

Step 2: Now click the button “Solve” to display the weighted mean.

Step 3: Finally, the weighted mean will be displayed in the output field.

4. What are the characteristics of weighted mean?

Weighted Mean is the average computed to provide different weights to the individual values. 

If all the weights are equal, then the weighted mean is equal to the arithmetic mean.

It denotes the average of the given data. The weighted mean is equal to the arithmetic mean or sample mean. 

The weighted mean is calculated when data is provided in a different way, compared to the arithmetic mean or sample mean.

Weighted means are generally similar to arithmetic means and they have few counter-instinctive properties. 

Data elements with high weight contribute more to the weighted mean than the low weighted elements.

The weights cannot be negative, in some cases it may be zero, but not in all, since division by zero is not available or not allowed. 

Weighted means play an important role in the data analysis systems, weighted differential, and integral calculus.

Weighted Mean – Definition, Formula, Uses, and FAQs

Table of Contents

Introduction to Weighted Mean

Weighted Mean – Definition: The weighted mean is a type of arithmetic mean, which is calculated by multiplying each value in a data set by a weight and then adding up the results. The weighted mean is often used when the values in a data set are not all of equal importance. The weight can be thought of as a measure of the importance of each value in the data set.

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Weight Definition

There is no one-size-fits-all definition of weight, as it can mean different things to different people. In general, weight is a measure of how much mass an object has. It can calculated by multiplying an object’s mass by the gravitational force exerted on it. Weight can also thought of as a force that resists movement, or the amount of mass needed to produce a certain acceleration.

What Weighted Mean?

The weighted mean is a calculation that takes into account the relative importance of each number in a set. To calculate the weighted mean, you first need to determine the weight of each number. The weight of a number is the number’s importance divided by the total number of numbers in the set. Then, you add up the weights of all the numbers in the set, and divide the total weight by the total number of numbers. The weighted mean is the result of this calculation.

Define Weighted Mean

The weighted mean is a mathematical calculation that takes into account the relative importance of each number in a set. The calculation performed by multiplying each number in the set by a weight, and then adding the results. The weighted mean then calculated by dividing the sum by the sum of the weights.

Weighted Mean Formula

The weighted mean formula used to calculate the average of a set of numbers, where some numbers are given more weight than others. weighted mean calculated by multiplying each number in the set by its weight, and then adding all of the weighted values together. The weighted mean formula is:

(Weighted Mean = \frac{1}{N}\sum_{i=1}^{N}w_i x_i)

(Weighted Mean\) is the weighted mean

(N\) is the number of data points

(w_i\) is the weight of data point \(i\)

(x_i\) is the data point \(i\)

Uses of Weighted Means

Weighted means often used in surveys, where they can used to adjust for the non-response bias of a survey. This done by giving different weights to different respondents, depending on how likely they to respond. This can give a more accurate estimate of the population mean than would obtained if all respondents given the same weight.

Weighted means can also used to adjust for the clustering of data. This done by giving different weights to different data points, depending on how close they are to other data points. This can give a more accurate estimate of the population mean than would obtained if all data points given the same weight.

Solved Example

Example: Weighted Mean In a class of 30 students, the average (mean) score on a test is 75. The average score of the male students is 80 and the average score of the female students is 70. Find the weighted mean score.

Ans. The weighted mean score is 76.7.

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Frequently Asked Questions (FAQs)

What does weighted mean means.

Weighted mean means giving different values or items varying levels of importance or weight in a calculation.

How do you explain weighted mean in research?

In research, the weighted mean accounts for the significance of individual data points by assigning weights, often used in aggregating survey results.

How do you calculate weighted mean?

To calculate the weighted mean, multiply each value by its corresponding weight, sum these products, and divide by the total weight.

What is the weighted mean difference?

The weighted mean difference measures the impact of different factors on the mean, often used in statistical analysis to assess the influence of variables.

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• For Weighting Online Opt-In Samples, What Matters Most?
• 1. How different weighting methods work

Table of Contents

• 2. Reducing bias on benchmarks
• 3. Variability of survey estimates
• Acknowledgements
• Appendix A: Survey methodology
• Appendix B: Synthetic population dataset
• Appendix C: Adjustment procedures

Historically, public opinion surveys have relied on the ability to adjust their datasets using a core set of demographics – sex, age, race and ethnicity, educational attainment, and geographic region – to correct any imbalances between the survey sample and the population. These are all variables that are correlated with a broad range of attitudes and behaviors of interest to survey researchers. Additionally, they are well measured on large, high-quality government surveys such as the American Community Survey (ACS), conducted by the U.S. Census Bureau, which means that reliable population benchmarks are readily available.

But are they sufficient for reducing selection bias 6  in online opt-in surveys? Two studies that compared weighted and unweighted estimates from online opt-in samples found that in many instances, demographic weighting only minimally reduced bias, and in some cases actually made bias worse. 7  In a previous Pew Research Center study comparing estimates from nine different online opt-in samples and the probability-based American Trends Panel, the sample that displayed the lowest average bias across 20 benchmarks (Sample I) used a number of variables in its weighting procedure that went beyond basic demographics, and it included factors such as frequency of internet use, voter registration, party identification and ideology. 8  Sample I also employed a more complex statistical process involving three stages: matching followed by a propensity adjustment and finally raking (the techniques are described in detail below).

The present study builds on this prior research and attempts to determine the extent to which the inclusion of different adjustment variables or more sophisticated statistical techniques can improve the quality of estimates from online, opt-in survey samples. For this study, Pew Research Center fielded three large surveys, each with over 10,000 respondents, in June and July of 2016. The surveys each used the same questionnaire, but were fielded with different online, opt-in panel vendors. The vendors were each asked to produce samples with the same demographic distributions (also known as quotas) so that prior to weighting, they would have roughly comparable demographic compositions. The survey included questions on political and social attitudes, news consumption, and religion. It also included a variety of questions drawn from high-quality federal surveys that could be used either for benchmarking purposes or as adjustment variables. (See Appendix A for complete methodological details and Appendix F for the questionnaire.)

This study compares two sets of adjustment variables: core demographics (age, sex, educational attainment, race and Hispanic ethnicity, and census division) and a more expansive set of variables that includes both the core demographic variables and additional variables known to be associated with political attitudes and behaviors. These additional political variables include party identification, ideology, voter registration and identification as an evangelical Christian, and are intended to correct for the higher levels of civic and political engagement and Democratic leaning observed in the Center’s previous study .

The analysis compares three primary statistical methods for weighting survey data: raking, matching and propensity weighting. In addition to testing each method individually, we tested four techniques where these methods were applied in different combinations for a total of seven weighting methods:

Propensity weighting

• Matching + Propensity weighting
• Matching + Raking
• Propensity weighting+ Raking
• Matching + Propensity weighting + Raking

Because different procedures may be more effective at larger or smaller sample sizes, we simulated survey samples of varying sizes. This was done by taking random subsamples of respondents from each of the three (n=10,000) datasets. The subsample sizes ranged from 2,000 to 8,000 in increments of 500. 9  Each of the weighting methods was applied twice to each simulated survey dataset (subsample): once using only core demographic variables, and once using both demographic and political measures. 10  Despite the use of different vendors, the effects of each weighting protocol were generally consistent across all three samples. Therefore, to simplify reporting, the results presented in this study are averaged across the three samples.

Often researchers would like to weight data using population targets that come from multiple sources. For instance, the American Community Survey (ACS), conducted by the U.S. Census Bureau, provides high-quality measures of demographics. The Current Population Survey (CPS) Voting and Registration Supplement provides high-quality measures of voter registration. No government surveys measure partisanship, ideology or religious affiliation, but they are measured on surveys such as the General Social Survey (GSS) or Pew Research Center’s Religious Landscape Study (RLS).

For some methods, such as raking, this does not present a problem, because they only require summary measures of the population distribution. But other techniques, such as matching or propensity weighting, require a case-level dataset that contains all of the adjustment variables. This is a problem if the variables come from different surveys.

To overcome this challenge, we created a “synthetic” population dataset that took data from the ACS and appended variables from other benchmark surveys (e.g., the CPS and RLS). In this context, “synthetic” means that some of the data came from statistical modeling (imputation) rather than directly from the survey participants’ answers. 11

The first step in this process was to identify the variables that we wanted to append to the ACS, as well as any other questions that the different benchmark surveys had in common. Next, we took the data for these questions from the different benchmark datasets (e.g., the ACS and CPS) and combined them into one large file, with the cases, or interview records, from each survey literally stacked on top of each other. Some of the questions – such as age, sex, race or state – were available on all of the benchmark surveys, but others have large holes with missing data for cases that come from surveys where they were not asked.

The next step was to statistically fill the holes of this large but incomplete dataset. For example, all the records from the ACS were missing voter registration, which that survey does not measure. We used a technique called multiple imputation by chained equations (MICE) to fill in such missing information. 12  MICE fills in likely values based on a statistical model using the common variables. This process is repeated many times, with the model getting more accurate with each iteration. Eventually, all of the cases will have complete data for all of the variables used in the procedure, with the imputed variables following the same multivariate distribution as the surveys where they were actually measured.

The result is a large, case-level dataset that contains all the necessary adjustment variables. For this study, this dataset was then filtered down to only those cases from the ACS. This way, the demographic distribution exactly matches that of the ACS, and the other variables have the values that would be expected given that specific demographic distribution. We refer to this final dataset as the “synthetic population,” and it serves as a template or scale model of the total adult population.

This synthetic population dataset was used to perform the matching and the propensity weighting. It was also used as the source for the population distributions used in raking. This approach ensured that all of the weighted survey estimates in the study were based on the same population information. See Appendix B for complete details on the procedure.

For public opinion surveys, the most prevalent method for weighting is iterative proportional fitting, more commonly referred to as raking. With raking, a researcher chooses a set of variables where the population distribution is known, and the procedure iteratively adjusts the weight for each case until the sample distribution aligns with the population for those variables. For example, a researcher might specify that the sample should be 48% male and 52% female, and 40% with a high school education or less, 31% who have completed some college, and 29% college graduates. The process will adjust the weights so that gender ratio for the weighted survey sample matches the desired population distribution. Next, the weights are adjusted so that the education groups are in the correct proportion. If the adjustment for education pushes the sex distribution out of alignment, then the weights are adjusted again so that men and women are represented in the desired proportion. The process is repeated until the weighted distribution of all of the weighting variables matches their specified targets.

Raking is popular because it is relatively simple to implement, and it only requires knowing the marginal proportions for each variable used in weighting. That is, it is possible to weight on sex, age, education, race and geographic region separately without having to first know the population proportion for every combination of characteristics (e.g., the share that are male, 18- to 34-year-old, white college graduates living in the Midwest). Raking is the standard weighting method used by Pew Research Center and many other public pollsters.

In this study, the weighting variables were raked according to their marginal distributions, as well as by two-way cross-classifications for each pair of demographic variables (age, sex, race and ethnicity, education, and region).

Matching is another technique that has been proposed as a means of adjusting online opt-in samples. It involves starting with a sample of cases (i.e., survey interviews) that is representative of the population and contains all of the variables to be used in the adjustment. This “target” sample serves as a template for what a survey sample would look like if it was randomly selected from the population. In this study, the target samples were selected from our synthetic population dataset, but in practice they could come from other high-quality data sources containing the desired variables. Then, each case in the target sample is paired with the most similar case from the online opt-in sample. When the closest match has been found for all of the cases in the target sample, any unmatched cases from the online opt-in sample are discarded.

If all goes well, the remaining matched cases should be a set that closely resembles the target population. However, there is always a risk that there will be cases in the target sample with no good match in the survey data – instances where the most similar case has very little in common with the target. If there are many such cases, a matched sample may not look much like the target population in the end.

There are a variety of ways both to measure the similarity between individual cases and to perform the matching itself. 13  The procedure employed here used a target sample of 1,500 cases that were randomly selected from the synthetic population dataset. To perform the matching, we temporarily combined the target sample and the online opt-in survey data into a single dataset. Next, we fit a statistical model that uses the adjustment variables (either demographics alone or demographics + political variables) to predict which cases in the combined dataset came from the target sample and which came from the survey data.

The kind of model used was a machine learning procedure called a random forest. Random forests can incorporate a large number of weighting variables and can find complicated relationships between adjustment variables that a researcher may not be aware of in advance. In addition to estimating the probability that each case belongs to either the target sample or the survey, random forests also produce a measure of the similarity between each case and every other case. The random forest similarity measure accounts for how many characteristics two cases have in common (e.g., gender, race and political party) and gives more weight to those variables that best distinguish between cases in the target sample and responses from the survey dataset. 14

We used this similarity measure as the basis for matching.

The final matched sample is selected by sequentially matching each of the 1,500 cases in the target sample to the most similar case in the online opt-in survey dataset. Every subsequent match is restricted to those cases that have not been matched previously. Once the 1,500 best matches have been identified, the remaining survey cases are discarded.

For all of the sample sizes that we simulated for this study (n=2,000 to 8,000), we always matched down to a target sample of 1,500 cases. In simulations that started with a sample of 2,000 cases, 1,500 cases were matched and 500 were discarded. Similarly, for simulations starting with 8,000 cases, 6,500 were discarded. In practice, this would be very wasteful. However, in this case, it enabled us to hold the size of the final matched dataset constant and measure how the effectiveness of matching changes when a larger share of cases is discarded. The larger the starting sample, the more potential matches there are for each case in the target sample – and, hopefully, the lower the chances of poor-quality matches.

A key concept in probability-based sampling is that if survey respondents have different probabilities of selection, weighting each case by the inverse of its probability of selection removes any bias that might result from having different kinds of people represented in the wrong proportion. The same principle applies to online opt-in samples. The only difference is that for probability-based surveys, the selection probabilities are known from the sample design, while for opt-in surveys they are unknown and can only be estimated.

For this study, these probabilities were estimated by combining the online opt-in sample with the entire synthetic population dataset and fitting a statistical model to estimate the probability that a  case comes from the synthetic population dataset or the online opt-in sample. As with matching, random forests were used to calculate these probabilities, but this can also be done with other kinds of models, such as logistic regression. 15  Each online opt-in case was given a weight equal to the estimated probability that it came from the synthetic population divided by the estimated probability that it came from the online opt-in sample. Cases with a low probability of being from the online opt-in sample were underrepresented relative to their share of the population and received large weights. Cases with a high probability were overrepresented and received lower weights.

As with matching, the use of a random forest model should mean that interactions or complex relationships in the data are automatically detected and accounted for in the weights. However, unlike matching, none of the cases are thrown away. A potential disadvantage of the propensity approach is the possibility of highly variable weights, which can lead to greater variability for estimates (e.g., larger margins of error).

Combinations of adjustments

Some studies have found that a first stage of adjustment using matching or propensity weighting followed by a second stage of adjustment using raking can be more effective in reducing bias than any single method applied on its own. 16  Neither matching nor propensity weighting will force the sample to exactly match the population on all dimensions, but the random forest models used to create these weights may pick up on relationships between the adjustment variables that raking would miss. Following up with raking may keep those relationships in place while bringing the sample fully into alignment with the population margins.

These procedures work by using the output from earlier stages as the input to later stages. For example, for matching followed by raking (M+R), raking is applied only the 1,500 matched cases. For matching followed by propensity weighting (M+P), the 1,500 matched cases are combined with the 1,500 records in the target sample. The propensity model is then fit to these 3,000 cases, and the resulting scores are used to create weights for the matched cases. When this is followed by a third stage of raking (M+P+R), the propensity weights are trimmed and then used as the starting point in the raking process. When first-stage propensity weights are followed by raking (P+R), the process is the same, with the propensity weights being trimmed and then fed into the raking procedure.

• When survey respondents are self-selected, there is a risk that the resulting sample may differ from the population in ways that bias survey estimates. This is known as selection bias, and it occurs when the kinds of people who choose to participate are systematically different from those who do not on the survey outcomes. Selection bias can occur in both probability-based surveys (in the form of nonresponse) as well as online opt-in surveys. ↩
• See Yeager, David S., et al. 2011. “ Comparing the Accuracy of RDD Telephone Surveys and Internet Surveys Conducted with Probability and Non-Probability Samples. ” Public Opinion Quarterly 75(4), 709-47; and Gittelman, Steven H., Randall K. Thomas, Paul J. Lavrakas and Victor Lange. 2015. “ Quota Controls in Survey Research: A Test of Accuracy and Intersource Reliability in Online Samples .” Journal of Advertising Research 55(4), 368-79. ↩
• In the 2016 Pew Research Center study a standard set of weights based on age, sex, education, race and ethnicity, region, and population density were created for each sample. For samples where vendors provided their own weights, the set of weights that resulted in the lowest average bias was used in the analysis. Only in the case of Sample I did the vendor provide weights resulting in lower bias than the standard weights ↩
• Many surveys feature sample sizes less than 2,000, which raises the question of whether it would be important to simulate smaller sample sizes. For this study, a minimum of 2,000 was chosen so that it would be possible to have 1,500 cases left after performing matching, which involves discarding a portion of the completed interviews. ↩
• The process of calculating survey estimates using different weighting procedures was repeated 1,000 times using different randomly selected subsamples. This enabled us to measure the amount of variability introduced by each procedure and distinguish between systematic and random differences in the resulting estimates. ↩
• The idea for augmenting ACS data with modeled variables from other surveys and measures of its effectiveness can be found in Rivers, Douglas, and Delia Bailey. 2009. “ Inference from Matched Samples in the 2008 US National Elections .” Presented at the 2009 American Association for Public Opinion Research Annual Conference, Hollywood, Florida; and Ansolabehere, Stephen, and Douglas Rivers. 2013. “ Cooperative Survey Research .” Annual Review of Political Science 16(1), 307-29. ↩
• See Azur, Melissa J., Elizabeth A. Stuart, Constantine Frangakis, and Philip J. Leaf. 2011. “Multiple Imputation by Chained Equations: What Is It and How Does It Work?: Multiple Imputation by Chained Equations.” International Journal of Methods in Psychiatric Research 20(1), 40–49. ↩
• See Stuart, Elizabeth A. 2010. “ Matching Methods for Causal Inference: A Review and a Look Forward .” Statistical Science 25(1), 1-21 for a more technical explanation and review of the many different approaches to matching that have been developed. ↩
• See Appendix C for a more detailed explanation of random forests and the matching algorithm used in this report, as well as Zhao, Peng, Xiaogang Su, Tingting Ge and Juanjuan Fan. 2016. “ Propensity Score and Proximity Matching Using Random Forest .” Contemporary Clinical Trials 47, 85-92. ↩
• See Buskirk, Trent D., and Stanislav Kolenikov. 2015. “ Finding Respondents in the Forest: A Comparison of Logistic Regression and Random Forest Models for Response Propensity Weighting and Stratification. ” Survey Methods: Insights from the Field (SMIF). ↩
• See Dutwin, David and Trent D. Buskirk. 2017. “ Apples to Oranges or Gala versus Golden Delicious? Comparing Data Quality of Nonprobability Internet Samples to Low Response Rate Probability Samples .” Public Opinion Quarterly 81(S1), 213-239. ↩

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Weighted Mean

Statistical glossary.

Weighted Mean:

Besides its use as a descriptive statistic , the weighted mean is also used to construct filters .

See also Mean values (comparison) and the online short course Basic Concepts in Probability and Statistics

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Weighted Mean Formula

Weighted Mean is an average computed by giving different weights to some of the individual values. If all the weights are equal, then the weighted mean is the same as the arithmetic mean.

It represents the average of a given data. The Weighted mean is similar to the arithmetic mean or sample mean. The Weighted mean is calculated when data is given in a different way compared to the arithmetic mean or sample mean.

Weighted means generally behave in a similar approach to arithmetic means, they do have a few counter-instinctive properties. Data elements with a high weight contribute more to the weighted mean than the elements with a low weight.

The weights cannot be negative. Some may be zero, but not all of them; since division by zero is not allowed. Weighted means play an important role in the systems of data analysis, weighted differential and integral calculus.

Formula of weighted Mean

The Weighted mean for given set of non-negative data \(\begin{array}{l}{x_{1},\:x_{2},\:x_{3},….x_{n}}\end{array} \)   with non-negative weights \(\begin{array}{l}{w_{1},\:w_{2},\:w_{3},….w_{n}}\end{array} \)   can be derived from the formula given below.

\(\begin{array}{l}x\end{array} \) is the repeating value \(\begin{array}{l}w\end{array} \) is the number of occurrences of \(\begin{array}{l}x\end{array} \) weight \(\begin{array}{l}\overline{x}\end{array} \) is the weighted mean

Solved Example of Weighted Mean

Question: Suppose that a marketing firm conducts a survey of 1,000 households to determine the average number of TVs each household owns. The data show a large number of households with two or three TVs and a smaller number with one or four. Every household in the sample has at least one TV and no household has more than four. Find the mean number of TVs per household.

As many of the values in this data set are repeated multiple times, you can easily compute the sample mean as a weighted mean. Follow these steps to calculate the weighted arithmetic mean:

Step 1: Assign a weight to each value in the dataset:

Step 2: Compute the numerator of the weighted mean formula.

Multiply each sample by its weight and then add the products together:

= (1)(73)+(2)(378)+(3)(459)+(4)(90)

= 73 + 756 + 1377 +

Step 3: Now, compute the denominator of the weighted mean formula by adding the weights together.

= 73 + 378 + 459 + 90

Step 4: Divide the numerator by the denominator

The mean number of TVs per household in this sample is 2.566.

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Looking for a Growth Stock? 3 Reasons Why Artisan Partners (APAM) is a Solid Choice

Growth investors focus on stocks that are seeing above-average financial growth, as this feature helps these securities garner the market's attention and deliver solid returns. However, it isn't easy to find a great growth stock.

By their very nature, these stocks carry above-average risk and volatility. Moreover, if a company's growth story is over or nearing its end, betting on it could lead to significant loss.

However, it's pretty easy to find cutting-edge growth stocks with the help of the Zacks Growth Style Score (part of the Zacks Style Scores system), which looks beyond the traditional growth attributes to analyze a company's real growth prospects.

Our proprietary system currently recommends Artisan Partners Asset Management ( APAM Quick Quote APAM - Free Report ) as one such stock. This company not only has a favorable Growth Score, but also carries a top Zacks Rank.

Studies have shown that stocks with the best growth features consistently outperform the market. And for stocks that have a combination of a Growth Score of A or B and a Zacks Rank #1 (Strong Buy) or 2 (Buy), returns are even better.

Here are three of the most important factors that make the stock of this investment management firm a great growth pick right now.

Earnings Growth

Earnings growth is arguably the most important factor, as stocks exhibiting exceptionally surging profit levels tend to attract the attention of most investors. For growth investors, double-digit earnings growth is highly preferable, as it is often perceived as an indication of strong prospects (and stock price gains) for the company under consideration.

While the historical EPS growth rate for Artisan Partners is 2.1%, investors should actually focus on the projected growth. The company's EPS is expected to grow 17% this year, crushing the industry average, which calls for EPS growth of 14.8%.

Impressive Asset Utilization Ratio

Growth investors often overlook asset utilization ratio, also known as sales-to-total-assets (S/TA) ratio, but it is an important feature of a real growth stock. This metric shows how efficiently a firm is utilizing its assets to generate sales.

Right now, Artisan Partners has an S/TA ratio of 0.7, which means that the company gets $0.7 in sales for each dollar in assets. Comparing this to the industry average of 0.24, it can be said that the company is more efficient.

In addition to efficiency in generating sales, sales growth plays an important role. And Artisan Partners looks attractive from a sales growth perspective as well. The company's sales are expected to grow 11.2% this year versus the industry average of 4.5%.

Promising Earnings Estimate Revisions

Superiority of a stock in terms of the metrics outlined above can be further validated by looking at the trend in earnings estimate revisions. A positive trend is of course favorable here. Empirical research shows that there is a strong correlation between trends in earnings estimate revisions and near-term stock price movements.

The current-year earnings estimates for Artisan Partners have been revising upward. The Zacks Consensus Estimate for the current year has surged 2.3% over the past month.

Bottom Line

Artisan Partners has not only earned a Growth Score of B based on a number of factors, including the ones discussed above, but it also carries a Zacks Rank #2 because of the positive earnings estimate revisions.

You can see the complete list of today's Zacks #1 Rank (Strong Buy) stocks here .

This combination indicates that Artisan Partners is a potential outperformer and a solid choice for growth investors.

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