Standard Deviation Calculator
How spread out a set of numbers is, with the subtraction and the squaring laid out line by line.
Results
Standard deviation (sample)
13.4907
The working, step by step
d = x - 18
x d d²
4 -14 196
8 -10 100
15 -3 9
16 -2 4
23 5 25
42 24 576
Σd² 910
s² = 910 ÷ 5 = 182
s = √182 = 13.4907
Variance (sample)
182
Mean (average)
18
Squared distances added up
910
If this is the whole population
12.3153
And its variance
151.667
Count
6
Range
38
What this tool does
Standard deviation answers a question the average cannot: how far from typical do the individual values sit? Two production lines can share a mean of 100 while one delivers 99–101 and the other 60–140, and only the deviation reveals it. This page shows the working rather than just the answer — each value's distance from the mean, that distance squared, the total, and the division that follows — because the formula only makes sense once you can see that it is an average of squared distances. It uses the sample formula, dividing by n − 1; the population figure is given alongside.
Formula
s = √( Σ(xᵢ − x̄)² ÷ (n − 1) )
Variables
| Symbol | Meaning | Unit |
|---|---|---|
numbers | Numbers (separated by comma, space or line) | — |
std_dev | Standard deviation (sample) | — |
WK | The working, step by step | — |
variance | Variance (sample) | — |
mean | Mean (average) | — |
SQ | Squared distances added up | — |
PS | If this is the whole population | — |
PV | And its variance | — |
count | Count | — |
range | Range | — |
Worked example
- Numbers (separated by comma, space or line)4, 8, 15, 16, 23, 42
- Standard deviation (sample)13.4907
- The working, step by stepd = x - 18 x d d² 4 -14 196 8 -10 100 15 -3 9 16 -2 4 23 5 25 42 24 576 Σd² 910 s² = 910 ÷ 5 = 182 s = √182 = 13.4907
- Variance (sample)182
- Mean (average)18
- Squared distances added up910
- If this is the whole population12.3153
- And its variance151.667
- Count6
- Range38
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.