Pooled Standard Deviation Calculator

This page calculates the pooled standard deviation of two lists.

Die gepoolte Standardabweichung wird als Standardabweichungen für eine Teilmenge berechnet.

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

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

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

\( \displaystyle SD_x=\sqrt{\frac{1}{3}\cdot(7.5625+0.5625+1.5625+5.0625)}\)

\( \displaystyle SD_x=\sqrt{\frac{1}{3}\cdot 14.75} =\sqrt{4.9167}=\color{blue}{2.217}\)

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

\( \displaystyle SD_y=\sqrt{\frac{1}{3}\cdot(76.5625+7.5625+10.5625+68.0625)}\)

\( \displaystyle SD_y=\sqrt{\frac{1}{3}\cdot 162.75} =\sqrt{54.25} =\color{blue}{7.3655}\)

\( \displaystyle SD_p= \sqrt{\frac{(4-1)\cdot 2.217^2 +(4-1)\cdot 7.37^2}{4+4-2}} \)

\( \displaystyle SD_p= \sqrt{\frac{3\cdot 4.9167 +3\cdot 54.25}{6}} \)

\( \displaystyle SD_p= \sqrt{\frac{14.75 +162.75}{6}} =\sqrt{29.583} =\color{blue}{5.44}\)

More statistics functions

Arithmetic Mean • Contraharmonic Mean • Covariance • Empirical distribution CDF • Deviation • Five-Number Summary • Geometric Mean • Harmonic Mean • Inverse Empirical distribution CDF • Kurtosis • Log Geometric Mean • Lower Quartile • Median • Pooled Standard Deviation • Pooled Variance • Skewness (Statistische Schiefe) • Upper Quartile • Variance