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HOW TO CALCULATE VARIANCE

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1 Calculating Sample Variance
2 Calculating Population Variance
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Article Summary

Co-authored by Mario Banuelos, PhD

Last Updated: April 19, 2023 References Approved

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This article was co-authored by Mario Banuelos, PhD. Mario Banuelos is an
Assistant Professor of Mathematics at California State University, Fresno. With
over eight years of teaching experience, Mario specializes in mathematical
biology, optimization, statistical models for genome evolution, and data
science. Mario holds a BA in Mathematics from California State University,
Fresno, and a Ph.D. in Applied Mathematics from the University of California,
Merced. Mario has taught at both the high school and collegiate levels.

There are 12 references cited in this article, which can be found at the bottom
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wikiHow marks an article as reader-approved once it receives enough positive
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This article has been viewed 3,003,201 times.

What is variance? Variance is a measure of how spread out a data set is, and we
calculate it by finding the average of each data point's squared difference from
the mean.[1] X Research source It's useful when creating statistical models
since low variance can be a sign that you are over-fitting your data. Once you
get the hang of the formula, you'll just have to plug in the right numbers to
find your answer. Read on for a complete step-by-step tutorial that'll teach you
how to calculate both sample variance and population variance.





STEPS

Method 1
Method 1 of 2:


CALCULATING SAMPLE VARIANCE

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    1
    Use the sample variance formula if you're working with a partial data set.
    In most cases, statisticians only have access to a sample, or a subset of
    the population they're studying. For example, instead of analyzing the
    population "cost of every car in Germany," a statistician could find the
    cost of a random sample of a few thousand cars. He can use this sample to
    get a good estimate of German car costs, but it will likely not match the
    actual numbers exactly.[2] X Research source
     * Example: Analyzing the number of muffins sold each day at a cafeteria,
       you sample six days at random and get these results: 38, 37, 36, 28, 18,
       14, 12, 11, 10.7, 9.9. This is a sample, not a population, since you
       don't have data on every single day the cafeteria was open.
     * If you have every data point in a population, skip down to the method
       below instead.
    
    
    
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    2
    Write down the sample variance formula. The variance of a data set tells you
    how spread out the data points are. The closer the variance is to zero, the
    more closely the data points are clustered together. When working with
    sample data sets, use the following formula to calculate variance:[3] X
    Research source
     * s2{\displaystyle s^{2}} = ∑[(xi{\displaystyle x_{i}} - x̅)2{\displaystyle
       ^{2}}]/(n - 1)
     * s2{\displaystyle s^{2}} is the variance. Variance is always measured in
       squared units.
     * xi{\displaystyle x_{i}} represents a term in your data set.
     * ∑, meaning "sum," tells you to calculate the following terms for each
       value of xi{\displaystyle x_{i}}, then add them together.
     * x̅ is the mean of the sample.
     * n is the number of data points.
    
    
    Advertisement
    
    
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    Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
    3
    Calculate the mean of the sample. The symbol x̅ or "x-bar" refers to the
    mean of a sample.[4] X Research source Calculate this as you would any mean:
    add all the data points together, then divide by the number of data
    points.[5] X Expert Source Mario Banuelos, PhD
    Assistant Professor of Mathematics Expert Interview. 11 December 2021.
     * Example: First, add your data points together: 17 + 15 + 23 + 7 + 9 + 13
       = 84
       Next, divide your answer by the number of data points, in this case six:
       84 ÷ 6 = 14.
       Sample mean = x̅ = 14.
     * You can think of the mean as the "center-point" of the data. If the data
       clusters around the mean, variance is low. If it is spread out far from
       the mean, variance is high.[6] X Expert Source Mario Banuelos, PhD
       Assistant Professor of Mathematics Expert Interview. 11 December 2021.
    
    
    
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    4
    Subtract the mean from each data point. Now it's time to calculate
    xi{\displaystyle x_{i}} - x̅, where xi{\displaystyle x_{i}} is each number
    in your data set. Each answer tells you that number's deviation from the
    mean, or in plain language, how far away it is from the mean.[7] X Expert
    Source Mario Banuelos, PhD
    Assistant Professor of Mathematics Expert Interview. 11 December 2021.
     * Example:
       x1{\displaystyle x_{1}} - x̅ = 17 - 14 = 3
       x2{\displaystyle x_{2}} - x̅ = 15 - 14 = 1
       x3{\displaystyle x_{3}} - x̅ = 23 - 14 = 9
       x4{\displaystyle x_{4}} - x̅ = 7 - 14 = -7
       x5{\displaystyle x_{5}} - x̅ = 9 - 14 = -5
       x6{\displaystyle x_{6}} - x̅ = 13 - 14 = -1
     * It's easy to check your work, as your answers should add up to zero. This
       is due to the definition of mean, since the negative answers (distance
       from mean to smaller numbers) exactly cancel out the positive answers
       (distance from mean to larger numbers).
    
    
    
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    5
    Square each result. As noted above, your current list of deviations
    (xi{\displaystyle x_{i}} - x̅) sum up to zero. This means the "average
    deviation" will always be zero as well, so that doesn't tell use anything
    about how spread out the data is. To solve this problem, find the square of
    each deviation.[8] X Research source This will make them all positive
    numbers, so the negative and positive values no longer cancel out to
    zero.[9] X Research source
     * Example:
       (x1{\displaystyle x_{1}} - x̅)2=32=9{\displaystyle ^{2}=3^{2}=9}
       (x2{\displaystyle (x_{2}} - x̅)2=12=1{\displaystyle ^{2}=1^{2}=1}
       92 = 81
       (-7)2 = 49
       (-5)2 = 25
       (-1)2 = 1
     * You now have the value (xi{\displaystyle x_{i}} - x̅)2{\displaystyle
       ^{2}} for each data point in your sample.
    
    
    
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    Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
    6
    Find the sum of the squared values. Now it's time to calculate the entire
    numerator of the formula: ∑[(xi{\displaystyle x_{i}} - x̅)2{\displaystyle
    ^{2}}]. The upper-case sigma, ∑, tells you to sum the value of the following
    term for each value of xi{\displaystyle x_{i}}. You've already calculated
    (xi{\displaystyle x_{i}} - x̅)2{\displaystyle ^{2}} for each value of
    xi{\displaystyle x_{i}} in your sample, so all you need to do is add the
    results of all of the squared deviations together.[10] X Expert Source Mario
    Banuelos, PhD
    Assistant Professor of Mathematics Expert Interview. 11 December 2021. [11]
    X Research source
     * Example: 9 + 1 + 81 + 49 + 25 + 1 = 166.
    
    
    
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    Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
    7
    Divide by n - 1, where n is the number of data points. A long time ago,
    statisticians just divided by n when calculating the variance of the sample.
    This gives you the average value of the squared deviation, which is a
    perfect match for the variance of that sample. But remember, a sample is
    just an estimate of a larger population. If you took another random sample
    and made the same calculation, you would get a different result. As it turns
    out, dividing by n - 1 instead of n gives you a better estimate of variance
    of the larger population, which is what you're really interested in. This
    correction is so common that it is now the accepted definition of a sample's
    variance.[12] X Research source
     * Example: There are six data points in the sample, so n = 6.
       Variance of the sample = s2=1666−1={\displaystyle s^{2}={\frac
       {166}{6-1}}=} 33.2
    
    
    
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    Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
    8
    Understand variance and standard deviation. Note that, since there was an
    exponent in the formula, variance is measured in the squared unit of the
    original data. This can make it difficult to understand intuitively.
    Instead, it's often useful to use the standard deviation. You didn't waste
    your effort, though, as the standard deviation is defined as the square root
    of the variance. This is why the variance of a sample is written
    s2{\displaystyle s^{2}}, and the standard deviation of a sample is
    s{\displaystyle s}.
     * For example, the standard deviation of the sample above = s = √33.2 =
       5.76.
    
    
    
    Advertisement
    



Method 2
Method 2 of 2:


CALCULATING POPULATION VARIANCE

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    Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
    1
    Use the population variance formula if you've collected data from every
    point in the population. The term "population" refers to the total set of
    relevant observations. For example, if you're studying the age of Texas
    residents, your population would include the age of every single Texas
    resident. You would normally create a spreadsheet for a large data set like
    that, but here's a smaller example data set:[13] X Research source
     * Example: There are exactly six fish tanks in a room of the aquarium. The
       six tanks contain the following numbers of fish:
       x1=5{\displaystyle x_{1}=5}
       x2=5{\displaystyle x_{2}=5}
       x3=8{\displaystyle x_{3}=8}
       x4=12{\displaystyle x_{4}=12}
       x5=15{\displaystyle x_{5}=15}
       x6=18{\displaystyle x_{6}=18}
    
    
    
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    Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
    2
    Write down the population variance formula. Since a population contains all
    the data you need, this formula gives you the exact variance of the
    population. In order to distinguish it from sample variance (which is only
    an estimate), statisticians use different variables:[14] X Research source
     * σ2{\displaystyle ^{2}} = (∑(xi{\displaystyle x_{i}} - μ)2{\displaystyle
       ^{2}})/n
     * σ2{\displaystyle ^{2}} = population variance. This is a lower-case sigma,
       squared. Variance is measured in squared units.
     * xi{\displaystyle x_{i}} represents a term in your data set.
     * The terms inside ∑ will be calculated for each value of xi{\displaystyle
       x_{i}}, then summed.
     * μ is the population mean
     * n is the number of data points in the population
    
    
    
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    3
    Find the mean of the population. When analyzing a population, the symbol μ
    ("mu") represents the arithmetic mean. To find the mean, add all the data
    points together, then divide by the number of data points.[15] X Research
    source
     * You can think of the mean as the "average," but be careful, as that word
       has multiple definitions in mathematics.
     * Example: mean = μ = 5+5+8+12+15+186{\displaystyle {\frac
       {5+5+8+12+15+18}{6}}} = 10.5
    
    
    
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    4
    Subtract the mean from each data point. Data points close to the mean will
    result in a difference closer to zero. Repeat the subtraction problem for
    each data point, and you might start to get a sense of how spread out the
    data is.[16] X Trustworthy Source Science Buddies Expert-sourced database of
    science projects, explanations, and educational material Go to source
     * Example:
       x1{\displaystyle x_{1}} - μ = 5 - 10.5 = -5.5
       x2{\displaystyle x_{2}} - μ = 5 - 10.5 = -5.5
       x3{\displaystyle x_{3}} - μ = 8 - 10.5 = -2.5
       x4{\displaystyle x_{4}} - μ = 12 - 10.5 = 1.5
       x5{\displaystyle x_{5}} - μ = 15 - 10.5 = 4.5
       x6{\displaystyle x_{6}} - μ = 18 - 10.5 = 7.5
    
    
    
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    href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative
    Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
    5
    Square each answer. Right now, some of your numbers from the last step will
    be negative, and some will be positive. If you picture your data on a number
    line, these two categories represent numbers to the left of the mean, and
    numbers to the right of the mean. This is no good for calculating variance,
    since these two groups will cancel each other out. Square each number so
    they are all positive instead.[17] X Research source
     * Example:
       (xi{\displaystyle x_{i}} - μ)2{\displaystyle ^{2}} for each value of i
       from 1 to 6:
       (-5.5)2{\displaystyle ^{2}} = 30.25
       (-5.5)2{\displaystyle ^{2}} = 30.25
       (-2.5)2{\displaystyle ^{2}} = 6.25
       (1.5)2{\displaystyle ^{2}} = 2.25
       (4.5)2{\displaystyle ^{2}} = 20.25
       (7.5)2{\displaystyle ^{2}} = 56.25
    
    
    
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    Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}
    6
    Find the mean of your results. Now you have a value for each data point,
    related (indirectly) to how far that data point is from the mean. Take the
    mean of these values by adding them all together, then dividing by the
    number of values.[18] X Research source
     * Example:
       Variance of the population =
       30.25+30.25+6.25+2.25+20.25+56.256=145.56={\displaystyle {\frac
       {30.25+30.25+6.25+2.25+20.25+56.25}{6}}={\frac {145.5}{6}}=} 24.25
    
    
    
 7. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/2\/29\/Calculate-Variance-Step-15.jpg\/v4-460px-Calculate-Variance-Step-15.jpg","bigUrl":"\/images\/thumb\/2\/29\/Calculate-Variance-Step-15.jpg\/aid867321-v4-728px-Calculate-Variance-Step-15.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div
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    7
    Relate this back to the formula. If you're not sure how this matches the
    formula at the beginning of this method, try writing out the whole problem
    in longhand:
     * After finding the difference from the mean and squaring, you have the
       value (x1{\displaystyle x_{1}} - μ)2{\displaystyle ^{2}},
       (x2{\displaystyle x_{2}} - μ)2{\displaystyle ^{2}}, and so on up to
       (xn{\displaystyle x_{n}} - μ)2{\displaystyle ^{2}}, where
       xn{\displaystyle x_{n}} is the last data point in the set.
     * To find the mean of these values, you sum them up and divide by n: (
       (x1{\displaystyle x_{1}} - μ)2{\displaystyle ^{2}} + (x2{\displaystyle
       x_{2}} - μ)2{\displaystyle ^{2}} + ... + (xn{\displaystyle x_{n}} -
       μ)2{\displaystyle ^{2}} ) / n
     * After rewriting the numerator in sigma notation, you have
       (∑(xi{\displaystyle x_{i}} - μ)2{\displaystyle ^{2}})/n, the formula for
       variance.
    
    
    
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   What are deviations?
   Mario Banuelos, PhD
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   Mario Banuelos is an Assistant Professor of Mathematics at California State
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   specializes in mathematical biology, optimization, statistical models for
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   the University of California, Merced. Mario has taught at both the high
   school and collegiate levels.
   Mario Banuelos, PhD
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   Expert Answer
   
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   A deviation is the distance of an observation or a data point from the mean,
   or the center of all the data points. it gives you a sense of how spread
   apart the data is from the mean.
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   Thank you for your feedback.
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 * Question
   What is the easiest way to find variance?
   Mario Banuelos, PhD
   Assistant Professor of Mathematics
   Mario Banuelos is an Assistant Professor of Mathematics at California State
   University, Fresno. With over eight years of teaching experience, Mario
   specializes in mathematical biology, optimization, statistical models for
   genome evolution, and data science. Mario holds a BA in Mathematics from
   California State University, Fresno, and a Ph.D. in Applied Mathematics from
   the University of California, Merced. Mario has taught at both the high
   school and collegiate levels.
   Mario Banuelos, PhD
   Assistant Professor of Mathematics
   Expert Answer
   
   Support wikiHow by unlocking this expert answer.
   
   First, calculate the mean or average of all of the data points. Then,
   calculate the difference between each data point and that mean. Square each
   of those differences, add them all up, then divide them by n (the total
   number of data points) minus 1.
   Thanks! We're glad this was helpful.
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 * Question
   How do I calculate the variance of four numbers?
   Community Answer
   Follow these steps: Work out the mean (the simple average of the numbers.)
   Then, for each number, subtract the mean and square the result (the squared
   difference). Finally, work out the average of those squared differences.
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TIPS

 * Using "n-1" instead of "n" in the denominator when analyzing samples is a
   technique called Bessel's correction. The sample is only an estimate of the
   full population, and the mean of the sample is biased to fit that estimate.
   This correction removes this bias. This is related to the fact that, once
   you've listed n - 1 data points, the final nth point is already constrained,
   since only certain values will result in the sample mean (x̅) used in the
   variance formula.[19] X Research source
   
   Thanks
   Helpful 0 Not Helpful 0
 * Since it is difficult to interpret the variance, this value is usually
   calculated as a starting point for calculating the standard deviation.
   
   Thanks
   Helpful 3 Not Helpful 1



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REFERENCES

 1. ↑ https://www.scribbr.com/statistics/variance/
 2. ↑
    https://www.simplilearn.com/tutorials/machine-learning-tutorial/population-vs-sample
 3. ↑ https://www.youtube.com/watch?v=VgKHjVDK0uM
 4. ↑ http://stattrek.com/statistics/notation.aspx
 5. ↑ Mario Banuelos, PhD. Assistant Professor of Mathematics. Expert Interview.
    11 December 2021.
 6. ↑ Mario Banuelos, PhD. Assistant Professor of Mathematics. Expert Interview.
    11 December 2021.
 7. ↑ Mario Banuelos, PhD. Assistant Professor of Mathematics. Expert Interview.
    11 December 2021.
 8. ↑ https://www.webpages.uidaho.edu/learn/statistics/lessons/lesson03/3_7.htm
 9. ↑ https://www.youtube.com/watch?v=sOb9b_AtwDg


More References (10)

 10. ↑ Mario Banuelos, PhD. Assistant Professor of Mathematics. Expert
     Interview. 11 December 2021.
 11. ↑ https://www.webpages.uidaho.edu/learn/statistics/lessons/lesson03/3_7.htm
 12. ↑ https://www.youtube.com/watch?v=sOb9b_AtwDg
 13. ↑ https://methods.sagepub.com/video/calculating-variance
 14. ↑ https://www.youtube.com/watch?v=VgKHjVDK0uM
 15. ↑
     https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/statistics/descriptive-statistics/variance-and-standard-deviation.html
 16. ↑
     https://www.sciencebuddies.org/science-fair-projects/science-fair/variance-and-standard-deviation
 17. ↑
     http://www.hunter.cuny.edu/dolciani/pdf_files/brushup-materials/calculating-variance-and-standard-deviation.pdf
 18. ↑
     http://www.hunter.cuny.edu/dolciani/pdf_files/brushup-materials/calculating-variance-and-standard-deviation.pdf
 19. ↑ http://www.statsdirect.com/help/default.htm#basics/degrees_freedom.htm




ABOUT THIS ARTICLE

Co-authored by:
Mario Banuelos, PhD
Assistant Professor of Mathematics
This article was co-authored by Mario Banuelos, PhD. Mario Banuelos is an
Assistant Professor of Mathematics at California State University, Fresno. With
over eight years of teaching experience, Mario specializes in mathematical
biology, optimization, statistical models for genome evolution, and data
science. Mario holds a BA in Mathematics from California State University,
Fresno, and a Ph.D. in Applied Mathematics from the University of California,
Merced. Mario has taught at both the high school and collegiate levels. This
article has been viewed 3,003,201 times.
30 votes - 77%

Co-authors: 27
Updated: April 19, 2023
Views: 3,003,201
Categories: Featured Articles | Probability and Statistics

Article SummaryX

To calculate the variance of a sample, or how spread out the sample data is
across the distribution, first add all of the data points together and divide by
the number of data points to find the mean. For example, if your data points are
3, 4, 5, and 6, you would add 3 + 4 + 5 + 6 and get 18. Then, you would divide
18 by the total number of data points, which is 4, and get 4.5. Therefore, the
mean of the data set is 4.5. Next, subtract the mean from each data point in the
sample. In this example, you would subtract the mean, or 4.5, from 3, then 4,
then 5, and finally 6 and end up with -1.5, -0.5, 0.5, and 1.5. Now, square each
of these results by multiplying each result by itself. If you square -1.5, -0.5,
0.5, and 1.5, you would get 2.25, 0.25, 0.25, and 2.25. Then, add up all of the
squared values. Here, you would add 2.25 + 0.25 + 0.25 + 2.25 and get 5.
Finally, divide the sum by n - 1, where n is the total number of data points. In
the example there are 4 data points, so you would divide the sum, which is 5, by
4 - 1, or 3, and get 1.66. Therefore, the variance of the sample is 1.66. To
learn how to calculate the variance of a population, scroll down!

Did this summary help you?YesNo




In other languages
Español:calcular la varianza
Italiano:Calcolare la Varianza
Deutsch:Varianz berechnen
Français:calculer la variance
Русский:посчитать дисперсию случайной величины
中文:计算方差
Português:Calcular a Variância
Nederlands:Variantie berekenen
Bahasa Indonesia:Menghitung Variasi
日本語:分散を計算する
ไทย:คำนวณความแปรปรวน
Tiếng Việt:Tính phương sai
العربية:حساب التباين
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Co-authored by:
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Co-authors: 27
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Andrew Raad

Jul 17, 2016

"I am currently solving a non-perfect hedge problem between grapefruit and
orange juice where I need to calculate the variance to find out how many
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