Population Variance of Three Numbers Calculator

Formula

Population variance = ((First-(First+Second+Third) ÷ 3)^2+(Second-(First+Second+Third) ÷ 3)^2+(Third-(First+Second+Third) ÷ 3)^2) ÷ 3

Worked example

With First: 3, Second: 6, Third: 9, population variance is 6.

Calculation method

The rule is Population variance = ((First-(First+Second+Third) ÷ 3)^2+(Second-(First+Second+Third) ÷ 3)^2+(Third-(First+Second+Third) ÷ 3)^2) ÷ 3.

Calculate the mean, square each observation’s distance from it, and divide the sum by three because the three values are treated as the full population.

For this population variance of three numbers example, changing first from 3 to 4 while keeping the other inputs fixed changes population variance from 6 to 4.222222.

Reading population variance

Variance is in squared input units; standard deviation takes its square root and returns to the original unit. A larger value means greater spread. In the worked values on this page, first 3, second 6, third 9 produce population variance 6.

Practical use

Use the measure to compare variability between small data sets while keeping individual observations in view. The specific task here is to find population variance from first, second, third.

Limits and common mistakes

Population formulas divide by n; sample formulas divide by n − 1. Choose the form that matches whether the values are the whole group or a sample. Verify first before interpreting population variance; a value measured on another basis would answer a different question.

Why the relationship works

The calculation begins with the arithmetic mean, finds each observation’s deviation from that mean, squares the deviations, and then combines them. Population variance divides by the full count; sample variance uses one fewer degree of freedom. Standard deviation takes a square root so its unit matches the original observations. The population variance of three numbers rule uses first, second, third in Population variance = ((First-(First+Second+Third) ÷ 3)^2+(Second-(First+Second+Third) ÷ 3)^2+(Third-(First+Second+Third) ÷ 3)^2) ÷ 3; read each symbol as the field named on this page.

Interpreting another set of values

For population variance of three numbers, the fields labeled first, second, third are the known quantities. The calculator solves for population variance using Population variance = ((First-(First+Second+Third) ÷ 3)^2+(Second-(First+Second+Third) ÷ 3)^2+(Third-(First+Second+Third) ÷ 3)^2) ÷ 3. The output summarizes the entered observations; it does not preserve every detail of them. Keep the original values alongside a mean or spread measure, especially when only three observations are used. When comparing two calculations, keep their units and input definitions identical.

For a second population variance of three numbers check, keep the other default inputs fixed and change first from 3 to 5. Population variance changes from 6 to 2.888889. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

For three values, inspect all three original observations as well as the summary. The same mean can describe very different spreads. A range can change when only an extreme changes, while the mean can change when any observation changes. These three-number examples demonstrate formulas but do not establish a population pattern. The reported population variance is 6 in the worked example. Keep its stated unit or counting convention when checking it against another calculation.