Formula to calculate pooled sample standard deviation

Pooled Standard Deviation

• If it is impossible to make many repeated measurements with the same sample

      • Then precision can be estimated during longer time in the form of pooled standard deviation

      • Pooled standard deviation can be used to calculate: – Repeatability – Within-lab reproducibility

The pooled standard deviation is a method for estimating a single standard deviation to represent all independent samples or groups in your study when they are assumed to come from populations with a common standard deviation. The pooled standard deviation is the average spread of all data points about their group mean (not the overall mean). It is a weighted average of each group's standard deviation. The weighting gives larger groups a proportionally greater effect on the overall estimate. Pooled standard deviations are used in 2-sample t-tests, ANOVAs, control charts, and capability analysis.

Formula \[s_p=\sqrt{\frac{(n-1)s^2+(m-1)r^2}{n+m-2}}\]

The term ‘s_p’ represents the pooled sample standard deviation.  The term ‘n’ represents the size of the first sample, and the term ‘m’ represents the size of the second sample that is being pooled with the first sample.  The term ‘s’ represents the variance of the first sample, and ‘r’ represents the variance of the second sample.

for example 

Here n=50 m=200 s=3 r=6.8

putting these vaues i the formula we get pooled standard deviation=6.235