How to use the combined and pooled standard deviation calculator
- Add one row per group with its sample size, mean and standard deviation.
- Use the sample standard deviation, the version with n - 1 in the denominator, which is what most software reports.
- Run the calculation to merge the groups into one.
- Read the combined size, mean, variance and standard deviation, and the pooled standard deviation that t tests use.
Formula
Combined size N = sum of n_i, and combined mean = sum of n_i x mean_i / N
Within groups sum of squares = sum of (n_i - 1) x sd_i^2
Between groups sum of squares = sum of n_i x (mean_i - combined mean)^2
Combined variance = (within + between) / (N - 1), and pooled variance = within / (N - k)
The combined figure describes everyone as one group, so it includes the gaps between the group means. The pooled figure describes only the spread inside the groups, which is why it is smaller.
Two groups, 10 and 20 values
Group 1 has 10 values with mean 5 and standard deviation 2. Group 2 has 20 values with mean 8 and standard deviation 3. The combined size is 30 and the combined mean is (10 x 5 + 20 x 8) / 30 = 7.
The within groups sum of squares is 9 x 4 + 19 x 9 = 207, and the between groups sum is 10 x (5 - 7)^2 + 20 x (8 - 7)^2 = 60. The combined variance is 267 / 29 = 9.2068966, so the combined standard deviation is 3.0342868. The pooled variance is 207 / 28 = 7.3928571, so the pooled standard deviation is 2.7189809.
Combined or pooled
Use the combined value when the groups are really parts of one population and you want the spread of all the values together, such as classes that form one grade. Use the pooled value when you assume the groups have the same underlying spread and want the best estimate of it, which is what a two sample t test and an analysis of variance use.
Do not average standard deviations directly. The result ignores both the group sizes and the distance between the group means, and it is wrong whenever the means differ.
Related calculations
To find the standard deviation of raw data, use the standard deviation calculator. To turn a standard deviation and a sample size into a standard error, use the standard error calculator. The pooled value feeds the effect size in the Cohen's d calculator.
Assumptions and limits
- Each standard deviation is a sample standard deviation with n - 1 in the denominator.
- Every group has at least 2 values. A group of 1 has no spread to report.
- Groups do not overlap. A value must not be counted in two groups.
- The pooled value assumes similar spread in every group, which you should check before you use it.
Frequently asked questions
How do I combine standard deviations of two groups?
Combine the sums of squares, not the standard deviations. Add the within group sums of squares to the between group sum of squares and divide by the total size minus 1.
What is a pooled standard deviation?
It is the square root of the weighted average of the group variances, with weights of n - 1. It estimates the shared spread inside the groups.
Why is the combined value bigger than the pooled one?
The combined value also counts the gap between the group means. If the means are equal the two values are almost the same.
Can I use population standard deviations?
Convert them first. Multiply each variance by n / (n - 1) to get the sample form, or the results will be a little low.