## Pooled Standard Deviation

Larger samples are more use than smaller samples. The hypothesis tests conducted using larger samples are more reliable and accurate. Often though, it is impossible or too expensive to carry out a large survey. An alternative is to group smaller similar surveys together. The mean of a number of surveys, each of size and mean is easily found: There exists a similar formula for the variance. The pooled sample variance of surveys, each of size and sample variance is more complicated: .

The standard deviation is the square root of this. The pooled standard deviation is most useful when used in the two sample t test, when conducting hypothesis tests for the means of two samples.

In this case The sample variance is the same as if the two samples had been pooled and the pooled data used to find the variance, with the difference that the denominator in the root is not but This is because one degree of freedom is lost for each sample.

Proof: Example: Find the pooled standard deviation of two samples of size 40, sample standard deviation 7 and sample size 55, sample size 12 respectively.  