Pooled Variance Calculator

Pooled variance (also known as combined variance or composite variance), is a method of estimating the variance of different populations when the mean of each population can be different, but the variance of each population can be assumed to be the same.

The pooled variance is calculated as the sample covariance for a subset and for the total set.

To perform the calculation, enter two series of numbers. Then click the 'Calculate' button.

Input format

The data can be entered as a series of numbers, separated by semicolons or spaces. You can enter the data as a list (one value per line). Or from a column from Excel spreadsheet by copy & paste

Formulas for two groups of the pooled variance

\(\displaystyle S_p^2=\frac{(n-1)S_x^2+(m-1)S_y^2}{n+m-2} \)

Calculating the variance of a sample

\(\displaystyle S^2=\frac{1}{n-1} \sum^n_{i=1} (x_i-\overline{x})^2 \)

\(s^2\)	Variance
\(n\)	Number of data points
\(x_i\)	Single data point
\(\overline{x}\)	Mean of the sample

Example

data set \( \displaystyle x= 3, 5, 7, 8 \)
data set \( \displaystyle y= 10, 16, 22, 27 \)

mean \( \displaystyle x= \frac{3+ 5+ 7+ 8}{4} =5.75\)

mean \( \displaystyle y= \frac{10+ 16+ 22+ 27}{4} =18.75\)

\( \displaystyle S_x^2=\frac{1}{4-1}\cdot((3-5.75)^2+(5-5.75)^2+(7-5.75)^2+(8-5.75)^2)\)

\( \displaystyle S_x^2=\frac{1}{3}\cdot(7.5625+0.5625+1.5625+5.0625)\)

\( \displaystyle S_x^2=\frac{1}{3}\cdot 14.75 =\color{blue}{4.9167}\)

\( \displaystyle S_y^2=\frac{1}{4-1}\cdot((10-18.75)^2+(16-18.75)^2+(22-18.75)^2+(27-18.75)^2)\)

\( \displaystyle S_y^2=\frac{1}{3}\cdot(76.5625+7.5625+10.5625+68.0625)\)

\( \displaystyle S_y^2=\frac{1}{3}\cdot 162.75 =\color{blue}{54.25}\)

\( \displaystyle S_p^2= \frac{(4-1)\cdot 4.9167 +(4-1)\cdot 54.25}{4+4-2} \)

\( \displaystyle S_p^2= \frac{3\cdot 4.9167 +3\cdot 54.25}{6} \)

\( \displaystyle S_p^2= \frac{14.75 +162.75}{6} =\color{blue}{29.583}\)