Variance Calculator

Variance measures the average squared deviation of each data point from the mean. The mathematical approach differs slightly depending on whether your data represents an entire population or just a sample. For a population, the formula is sigma squared equals the sum of the squared differences between each value and the population mean, divided by the total number of observations (N). For a sample, known as sample variance (s squared), you divide by the degrees of freedom, which is n minus 1. The standard formulas are expressed as: Population Variance sigma squared = sum from i=1 to N of (x_i - mu) squared divided by N, and Sample Variance s squared = sum from i=1 to n of (x_i - x_bar) squared divided by (n - 1).

Imagine you are evaluating test scores from a sample of five students: 70, 80, 85, 90, and 95. First, calculate the sample mean by summing these values (420) and dividing by the number of observations (5), yielding an average of 84. Next, subtract the mean from each score and square the result: (70 - 84) squared equals 196; (80 - 84) squared equals 16; (85 - 84) squared equals 1; (90 - 84) squared equals 36; and (95 - 84) squared equals 121. Sum these squared differences to get 370. Because this is a sample dataset, divide this sum by the degrees of freedom (n - 1, which is 5 - 1 = 4). Dividing 370 by 4 gives a sample variance of 92.5.

Always verify whether your dataset represents an entire population or a sample, as choosing the wrong dataset type will skew your denominator and alter your final result. Pay close attention to outliers; because variance squares the deviations from the mean, exceptionally high or low values can disproportionately inflate the variance figure. Finally, remember that variance is expressed in squared units, so if your raw data is measured in dollars, your variance is in squared dollars. To interpret spread in original units, consider taking the square root of your variance to find the standard deviation.