How to Calculate Variance

Write your sample dataset. In most cases, statisticians only have information about a sample, or subset, of the population they are studying. For example, instead of analyzing the “cost of every car in Germany” population as a whole, a statistician might find the cost of a random sample of a few thousand cars in size. The statistician can use this sample to get a good estimate of the cost of cars in Germany. However, most likely it will not completely coincide with the actual numbers.

• Example: When analyzing the number of muffins sold each day at a coffee shop, you take a random six-day sample and get the following results: 38, 37, 36, 28, 18, 14, 12, 11, 10.7, 9.9. This is a sample, not a population, because you don’t have data for all store opening days.
• If there are every data point in the population, go to the method below.

Write the formula for sample variance. The variance of a dataset indicates how scattered the data points are. The closer the variance is to zero, the closer the data points are grouped together. When working with sample data, use the following variance formula: ^[1]

• 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 value in your tuple.
• ∑, which means “sum”, tells you what parameters to follow for each value xi{displaystyle x_{i}} , and then add them together.
• x̅ is the mean of the sample.
• n is the number of data points.

Calculate the mean of the sample . The symbol x̅ or “x-horizontal” is used to indicate the sample mean. ^[2] Do the same calculation as you would for any mean: add up all the data points and divide the total by the number of points.

• Example: First, add the data points together: 17 + 15 + 23 + 7 + 9 + 13 = 84
  Next, divide the result obtained 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 is centered around the mean, the variance is low. If they are scattered far from the mean, the variance is high.

Subtract the mean from each data point. Now is the time to count xi{displaystyle x_{i}} – x̅, in which xi{displaystyle x_{i}} is each point in your dataset. Each result obtained will indicate the deviation from the mean of each corresponding point, or simply put, the distance from it to the mean. ^[3] .

• For example:
  xfirst{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 is very easy to check your calculation, because the results obtained must sum to 0. That is because by definition of the mean, the results are negative (distance from mean to small numbers). more) completely cancel out the positive result (distance from mean to larger numbers).

Square all results. As noted above, the current offset list ( xi{displaystyle x_{i}} – x̅) sums to 0. That is, the “mean deviation” will also always be zero and can’t say anything about the dispersion of the data. To solve this problem, we find the square of each deviation. As a result, all are positive, negative and positive values no longer cancel each other out and sum to zero. ^[4]

• For example:
  ( xfirst{displaystyle x_{1}} – x̅) 2=32=9{displaystyle ^{2}=3^{2}=9}
  (x2{displaystyle (x_{2}} – x̅) 2=first2=first{displaystyle ^{2}=1^{2}=1}
  9 ² = 81
  (-7) ² = 49
  (-5) ² = 25
  (-1) ² = 1
• Now you have ( xi{displaystyle x_{i}} – x̅) 2{displaystyle ^{2}} for each data point in the sample.

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}} ]. Great Sigma, , requires you to add the following element value for each value xi{displaystyle x_{i}} . You have calculated ( xi{displaystyle x_{i}} – x̅) 2{displaystyle ^{2}} for each value xi{displaystyle x_{i}} in the sample, so all you have to do is add the results together.

• For example: 9 + 1 + 81 + 49 + 25 + 1 = 166 .

Divide by n – 1, where n is the number of data points. Long ago, when calculating sample variance, statisticians just divided by n. That division will give you the mean of the squared deviation, which exactly matches the variance of that sample. However, keep in mind that the sample is only an estimate of a larger population. If you take another random sample and do the same calculation, you will get a different result. Turns out, dividing by n -1 instead of n gives you a better estimate of the variance of the larger population – which you really care about. This correction is so popular that it is now the accepted definition of sample variance. ^[5]

• Example: There are six data points in the sample, so n = 6.
  Sample Variance = S2=1666−first={displaystyle s^{2}={frac {166}{6-1}}=} 33.2

Understand variance and standard deviation. Note that, because there is a power in the formula, the variance is measured by the unit square of the original data. This is visually confusing. Instead, usually the standard deviation is quite useful. But you’re not wasting your time either, because the standard deviation is determined by the square root of the variance. That is why the sample variance is written as S2{displaystyle s^{2}} , and the standard deviation of a sample is S{displaystyle s} .

• For example, the standard deviation of the above sample = s = √33.2 = 5.76.