A pooled standard deviation is simply a weighted average of standard deviations from two or more independent groups.

In statistics it appears most often in the two sample t-test, which is used to test whether or not the means of two populations are equal.

The formula to calculate a pooled standard deviation for two groups is as follows:

Pooled standard deviation = √ (n₁-1)s₁² +  (n₂-1)s₂² /  (n₁+n₂-2)

where:

• n₁, n₂: Sample size for group 1 and group 2, respectively.
• s₁, s₂: Standard deviation for group 1 and group 2, respectively.

The following examples show two methods for calculating a pooled standard deviation between two groups in R.

Method 1: Calculate Pooled Standard Deviation Manually

Suppose we have the following data values for two samples:

• Sample 1: 6, 6, 7, 8, 8, 10, 11, 13, 15, 15, 16, 17, 19, 19, 21
• Sample 2: 10, 11, 13, 13, 15, 17, 17, 19, 20, 22, 24, 25, 27, 29, 29

The following code shows how to calculate the pooled standard deviation between these two samples:

#define two samples
data1 <- c(6, 6, 7, 8, 8, 10, 11, 13, 15, 15, 16, 17, 19, 19, 21)
data2 <- c(10, 11, 13, 13, 15, 17, 17, 19, 20, 22, 24, 25, 27, 29, 29)

#find sample standard deviation of each sample
s1 <- sd(data1)
s2 <- sd(data2)

#find sample size of each sample
n1 <- length(data1)
n2 <- length(data2)

#calculate pooled standard deviation
pooled <- sqrt(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1+n1-2))

#view pooled standard deviation
pooled

[1] 5.789564

The pooled standard deviation turns out to be 5.789564.

Method 2: Calculate Pooled Standard Deviation Using a Package

Another way to calculate the pooled standard deviation between two samples in R is to use the sd_pooled() function from the effectsize package.

The following code shows how to use this function in practice:

library(effectsize)

#define two samples
data1 <- c(6, 6, 7, 8, 8, 10, 11, 13, 15, 15, 16, 17, 19, 19, 21)
data2 <- c(10, 11, 13, 13, 15, 17, 17, 19, 20, 22, 24, 25, 27, 29, 29)

#calculate pooled standard deviation between two samples
sd_pooled(data1, data2)

[1] 5.789564

Note that this matches the value that we calculated manually in the previous example.

The following tutorials provide more information on calculating a pooled standard deviation: