Variance and Standard Deviation Calculator

Variance is a measure of how spread out a data set is around its mean. A variance of zero means all values are identical. Higher variance indicates greater spread. It is a fundamental concept in statistics used in data analysis, finance, and science.

Population: sigma^2 = sum(xi - mean)^2 / N. Sample: s^2 = sum(xi - mean)^2 / (n-1). Here xi are individual values, mean is the arithmetic mean, and N or n is the count. The (n-1) denominator for samples is called Bessel's correction — it corrects the underestimation of population variance.

Variance is expressed in squared units (e.g. cm^2 if measuring length in cm), which makes interpretation harder. Standard deviation is the square root of variance and uses the same units as the data. In practice standard deviation is reported more often, while variance is an intermediate calculation step.

Use population (divisor N) when your data covers the entire population — e.g. all students in a class. Use sample (divisor n-1) when your data is a random sample from a larger population — e.g. a survey of 100 people. For large data sets the difference is minimal, but for small samples the corrected formula gives a better estimate.

Variance is a sum of squared deviations from the mean. Every term is non-negative (the square of any real number is >= 0), so the sum is also non-negative. Variance equals zero only when all values in the data set are identical.

1) Compute the arithmetic mean. 2) Subtract the mean from each value. 3) Square each difference. 4) Sum all the squares. 5) Divide by N (population) or n-1 (sample). Example for [2, 4, 6]: mean=4, deviations: -2, 0, 2; squares: 4, 0, 4; sum=8; population variance = 8/3 ≈ 2.67.

Variance is used in financial risk analysis (measuring price volatility of stocks), manufacturing quality control (dimension scatter), medical statistics (measurement variability), machine learning (regularisation, PCA) and social sciences. It is also the basis of ANOVA (analysis of variance) used to compare groups.

Variance is very sensitive to outliers because deviations are squared — a large deviation amplifies its effect quadratically. A single value far from the mean can greatly inflate the variance. It is advisable to check your data for outliers before computing variance, or consider more robust measures of dispersion.

The range (maximum minus minimum) is the simplest measure of spread but only uses two extreme values and is very sensitive to outliers. Variance considers all values and their distances from the mean, making it a more representative and stable measure. In data analysis variance (or standard deviation) is preferred over range.

For a population of one element variance is 0 — a single number shows no dispersion. For a sample of one element the formula requires dividing by n-1=0, which is mathematically undefined. The calculator returns 0 in this case as a safe fallback. In practice at least two observations are needed for meaningful statistical measures.