Population Variance Calculator

Population variance (sigma^2) divides the sum of squared deviations by N (the total population count), giving the exact variance of the entire population. Sample variance (s^2) divides by n-1 (Bessel's correction) to produce an unbiased estimate of sigma^2 from a sample. Use population variance only when your data is the complete population, such as all grades in a single class with no intent to generalize further. For most research and surveys, use sample variance.

Population variance sigma^2 = Sum((xi - mu)^2) / N, where xi are individual data values, mu is the population mean (Sum(xi)/N), and N is the total count. Step by step: (1) compute mu; (2) subtract mu from each xi to get deviations; (3) square each deviation; (4) sum all squared deviations to get SS; (5) divide SS by N. The population standard deviation sigma = sqrt(sigma^2).

Example with data [2, 4, 6, 8, 10]: Step 1: mu = (2+4+6+8+10)/5 = 6. Step 2: deviations = [-4, -2, 0, 2, 4]. Step 3: squared deviations = [16, 4, 0, 4, 16]. Step 4: SS = 16+4+0+4+16 = 40. Step 5: sigma^2 = 40/5 = 8.0. Step 6: sigma = sqrt(8) = 2.828. The deviation table in this calculator shows all these steps for any dataset you enter.

The coefficient of variation (CV) = (sigma / mu) x 100%. It expresses variability as a percentage of the mean, making it unit-free and comparable across datasets. For example, a sigma of 5 kg when mu = 50 kg gives CV = 10%, while sigma of 5 kg when mu = 10 kg gives CV = 50%. CV below 15% typically indicates low variability; 15-30% is moderate; above 30% is high. CV is undefined when the mean is zero.

No. Population variance cannot be negative. Because it sums squared deviations from the mean, every term in the sum is non-negative. Variance equals zero only when every data point is identical to the population mean, meaning all values are the same. If you get a negative value from a manual calculation, there is an arithmetic error in computing the squared deviations or summing them.

Use population variance when you have data for every member of the group you care about (the entire population). Examples: grades of all 30 students in a specific class, weight of every item in a fixed production batch, scores of all players in a completed tournament. Use sample variance in all other cases: opinion poll data, quality control sampling, experimental data from a subset of patients. Most statistical textbooks and software default to sample variance.

Population variance sigma^2 is fundamental to many statistical concepts. For a discrete random variable X, Var(X) = E[(X - mu)^2] = E[X^2] - mu^2. For independent random variables X and Y, Var(X+Y) = Var(X) + Var(Y), the variance addition rule. In normal distributions, the distribution is fully described by mu and sigma^2. In regression, the total sum of squares (TSS) is N times the variance of the response variable.

Grouped data population variance is used when data is presented as a frequency distribution (class intervals with counts) rather than individual values. The formula is sigma^2 = Sum(fi * (xi - mu)^2) / N, where fi are class frequencies, xi are class midpoints, and N = Sum(fi). This calculator's Grouped Frequency mode implements this formula and displays the weighted deviation table showing f*(x-mu)^2 for each class.

Population standard deviation sigma = sqrt(sigma^2). Variance sigma^2 = sigma^2 (trivially). Standard deviation is in the same units as the original data, while variance is in squared units. For a dataset with sigma = 3 metres, sigma^2 = 9 m^2. In practice, standard deviation is reported in scientific papers and news because it shares the unit with the data. Variance is preferred in mathematical derivations and ANOVA because variances of independent variables add together (SDs do not).

For a normally distributed population, sigma^2 is one of two parameters that completely define the distribution (along with the mean mu). The normal distribution N(mu, sigma^2) has 68.27% of values within one sigma, 95.45% within two sigma, and 99.73% within three sigma of the mean. Knowing sigma^2 lets you compute any probability or percentile for the population without additional data. The calculator's outputs can be used directly as inputs to normal distribution calculations.

Population variance is in squared units because it sums the squares of deviations: (xi - mu)^2. Squaring serves two purposes: (1) it makes all terms positive so deviations above and below the mean do not cancel each other; (2) it gives greater weight to values far from the mean, making variance sensitive to outliers. The squared-unit issue is resolved by taking the square root to get standard deviation, which is in the same units as the data.

Use the Coefficient of Variation (CV = sigma/mu x 100%) rather than raw variance or standard deviation. Example: Population A has mu = 100, sigma = 10, CV = 10%. Population B has mu = 10, sigma = 3, CV = 30%. Population B is more variable relative to its mean, even though its sigma is smaller. This calculator reports CV automatically whenever the mean is non-zero, allowing direct relative comparisons.