Pooled Standard Deviation
Formulas and examples for the pooled standard deviation of two data series
The Pooled Standard Deviation is a weighted average of standard deviations for two or more groups. The individual standard deviations are averaged, with more “weight” given to larger sample sizes.
Pooled standard deviation formulas
\(\displaystyle SD_p= \sqrt{\frac{(n-1)SD_x^2+(m-1)SD_y^2}{n+m-2}} \)
Calculating the standard deviation of a sample
\(\displaystyle s=\sqrt{ \frac{1}{n-1} \sum^n_{i=1} (x_i-\overline{x})^2} \)
\(s^2\) Standard deviation \(n\) 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 \( \displaystyle y= \frac{10+ 16+ 22+ 27}{4} =18.75\)
\( \displaystyle SD_x=\sqrt{\frac{1}{4-1}\cdot((3-5.75)^2+(5-5.75)^2+(7-5.75)^2+(8-5.75)^2)}\)
\( \displaystyle SD_x=\sqrt{\frac{1}{3}\cdot(7.5625+0.5625+1.5625+5.0625)}\)
\( \displaystyle SD_x=\sqrt{\frac{1}{3}\cdot 14.75} =\sqrt{4.9167}=\color{blue}{2.217}\)
\( \displaystyle SD_y=\sqrt{\frac{1}{4-1}\cdot((10-18.75)^2+(16-18.75)^2+(22-18.75)^2+(27-18.75)^2)}\)
\( \displaystyle SD_y=\sqrt{\frac{1}{3}\cdot(76.5625+7.5625+10.5625+68.0625)}\)
\( \displaystyle SD_y=\sqrt{\frac{1}{3}\cdot 162.75} =\sqrt{54.25} =\color{blue}{7.3655}\)
\( \displaystyle SD_p= \sqrt{\frac{(4-1)\cdot 2.217^2 +(4-1)\cdot 7.37^2}{4+4-2}} \)
\( \displaystyle SD_p= \sqrt{\frac{3\cdot 4.9167 +3\cdot 54.25}{6}} \)
\( \displaystyle SD_p= \sqrt{\frac{14.75 +162.75}{6}} =\sqrt{29.583} =\color{blue}{5.44}\)
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