Population Variance Calculator
Enter your data values separated by commas or spaces and this calculator finds the population variance (sigma squared), population standard deviation, mean, count, and sum of squares. It then walks you through every step of the computation so you can check the arithmetic or use the working in an assignment. Use population variance when your data represents the entire group, not a sample drawn from it.
Formula
Worked example
Data set: 4, 7, 13, 2, 1. Mean = (4+7+13+2+1)/5 = 27/5 = 5.4. Squared deviations: (4-5.4)^2=1.96, (7-5.4)^2=2.56, (13-5.4)^2=57.76, (2-5.4)^2=11.56, (1-5.4)^2=19.36. Sum of squares = 93.2. Population variance = 93.2/5 = 18.64. Standard deviation = sqrt(18.64) = 4.317.
What is population variance?
Population variance is a measure of how spread out the values in a complete data set are around their mean. Denoted sigma squared, it is computed by finding the average of the squared differences between each value and the mean of the whole group. A variance of zero means every value is identical. A larger variance indicates the values are more scattered. Because variance squares the deviations, it emphasises outliers more than a simple average distance would. Taking the square root returns the population standard deviation (sigma), which is expressed in the same units as the original data and is therefore easier to interpret in context.
Population variance vs. sample variance
The distinction matters when you do not have data on every member of the group you are studying. Population variance divides the sum of squared deviations by N (the total count), giving an exact measure of spread for that complete data set. Sample variance divides by N-1 instead, applying what statisticians call Bessel's correction. This correction compensates for the fact that a sample tends to underestimate the true spread of the population it was drawn from, making sample variance an unbiased estimator of population variance. If your data is a census of the whole group, use population variance. If it is a subset drawn at random, use sample variance.
How to calculate population variance step by step
Step 1: add up all the values and divide by N to find the mean. Step 2: for each value, subtract the mean and square the result. Step 3: add up all those squared differences to get the sum of squares. Step 4: divide the sum of squares by N to get population variance. Step 5: take the square root to convert to standard deviation if needed. For example, with values 4, 7, 13, 2, 1: mean = 5.4; squared differences are 1.96, 2.56, 57.76, 11.56, 19.36; sum of squares = 93.2; population variance = 93.2 / 5 = 18.64; standard deviation = 4.317.
What the standard deviation tells you that variance does not
Variance is expressed in squared units (square centimetres if your data was in centimetres, for instance), which makes it hard to interpret directly. Standard deviation is the square root of variance, returning the result to the original unit of measurement. For roughly bell-shaped data, the empirical rule says about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This makes standard deviation easier to use when communicating spread to a non-technical audience, while variance is often preferred in mathematical and statistical derivations because it has convenient algebraic properties.
Population vs. sample variance: when to use each
| Scenario | Formula denominator | Symbol | When to use |
|---|---|---|---|
| Population variance | N | sigma-squared | You have data on every member of the group |
| Sample variance | N - 1 | s-squared | You have a subset drawn from a larger group |
| Bessel's correction | (N - 1) corrects bias | s-squared | Makes s-squared an unbiased estimator of sigma-squared |
The key difference is the denominator used to compute variance. This affects whether your result is exact or an unbiased estimate.
Frequently asked questions
When should I use population variance instead of sample variance?
Use population variance when your data covers every member of the group you want to describe, for example all students in a single class, all products produced in one batch, or all match scores from a completed tournament. Use sample variance when your data is a random subset drawn from a larger population, and you want to estimate the spread of that wider group.
Why does sample variance divide by N-1 instead of N?
Dividing by N when you have a sample systematically underestimates the true population variance, because sample values cluster closer to the sample mean than they would to the population mean. Replacing N with N-1, known as Bessel's correction, corrects for this bias and makes the sample variance an unbiased estimator of the population variance.
What is sum of squares and how is it used?
Sum of squares (SS) is the total of all squared differences between each data point and the mean. It is an intermediate step in computing variance: SS divided by N gives population variance, and SS divided by N-1 gives sample variance. SS also appears in analysis of variance (ANOVA) and regression, where it is partitioned into explained and unexplained components.
Can variance be negative?
No. Variance is always zero or positive. Each squared difference is non-negative by definition, so their sum and average are also non-negative. A variance of zero means every value in the data set is identical and there is no spread at all.
What is the relationship between variance and standard deviation?
Standard deviation is the square root of variance. If the population variance is 18.64, the population standard deviation is sqrt(18.64) = 4.317. Both measure spread, but standard deviation is in the same units as the original data (for example centimetres or dollars), while variance is in squared units, making standard deviation more interpretable for most practical purposes.