Pooled Variance
Formulas and examples for the pooled variance of a data series
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.
Pooled variance formulas
\(\displaystyle S_p^2=\frac{(n-1)S_x^2+(m-1)S_y^2}{n+m-2} \)
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 \( \displaystyle y= \frac{10+ 16+ 22+ 27}{4} =18.75\)
\( \displaystyle S_x^2=\frac{1}{4-1}\cdot((3-5.75)^2+(5-5.75)^2+(7-5.75)^2+(8-5.75)^2)\)
\( \displaystyle S_x^2=\frac{1}{3}\cdot(7.5625+0.5625+1.5625+5.0625)\)
\( \displaystyle S_x^2=\frac{1}{3}\cdot 14.75 =\color{blue}{4.9167}\)
\( \displaystyle S_y^2=\frac{1}{4-1}\cdot((10-18.75)^2+(16-18.75)^2+(22-18.75)^2+(27-18.75)^2)\)
\( \displaystyle S_y^2=\frac{1}{3}\cdot(76.5625+7.5625+10.5625+68.0625)\)
\( \displaystyle S_y^2=\frac{1}{3}\cdot 162.75 =\color{blue}{54.25}\)
\( \displaystyle S_p^2= \frac{(4-1)\cdot 4.9167 +(4-1)\cdot 54.25}{4+4-2} \)
\( \displaystyle S_p^2= \frac{3\cdot 4.9167 +3\cdot 54.25}{6} \)
\( \displaystyle S_p^2= \frac{14.75 +162.75}{6} =\color{blue}{29.583}\)
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