Combined Standard Deviation Calculator (Merging Two or Three Groups)
The mean and standard deviation of several groups put together, from each group's size, mean and standard deviation — without needing the raw data.
Results
What this tool does
You cannot average standard deviations: two groups that each have a spread of 10 but whose means are 20 apart have a combined spread of about 14, because the gap between the groups adds variation of its own. The correct route adds up the sums of squares: the spread inside each group, (n − 1) s², plus the spread of the group means round the overall mean, n (m − M)². Divide by N − 1 for the sample figure or by N if the groups are the entire population. The last result tells you how much of the total spread is just differences between the groups.
Formula
mean = Σ nᵢmᵢ ÷ Σ nᵢ · SS = Σ (nᵢ − 1) sᵢ² + Σ nᵢ (mᵢ − mean)² · s = √(SS ÷ (N − 1))
Variables
| Symbol | Meaning | Unit |
|---|---|---|
n1 | Group 1 — size | — |
m1 | Group 1 — mean | — |
s1 | Group 1 — standard deviation | — |
n2 | Group 2 — size | — |
m2 | Group 2 — mean | — |
s2 | Group 2 — standard deviation | — |
n3 | Group 3 — size (0 if there are only two groups) | — |
m3 | Group 3 — mean | — |
s3 | Group 3 — standard deviation | — |
SD | Combined standard deviation (sample) | — |
N | Combined size | — |
M | Combined mean | — |
SP | Combined standard deviation (whole population) | — |
BT | Share of the spread that comes from differences between group means | % |
Worked example
- Group 1 — size30
- Group 1 — mean70
- Group 1 — standard deviation8
- Group 2 — size25
- Group 2 — mean75
- Group 2 — standard deviation10
- Group 3 — size (0 if there are only two groups)0
- Group 3 — mean0
- Group 3 — standard deviation0
- Combined standard deviation (sample)9.2265
- Combined size55
- Combined mean72.2727
- Combined standard deviation (whole population)9.1422
- Share of the spread that comes from differences between group means7.4 %
Limitations
- Standard deviation and variance are calculated for a sample (dividing by n − 1). For a full population, divide by n instead.
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.