Standard deviation calculator
Sample or population standard deviation and variance, with the full working shown line by line so you can check it against your homework.
Box = Q1 to Q3, bar = median, triangle = mean, circles = outliers.
Working, step by step
Mean x̄ = 154 ÷ 10 = 15.4. Subtract the mean from every value, square it, add the squares, divide by n − 1 = 9, then take the square root.
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 11 | −4.4 | 19.36 |
| 12 | −3.4 | 11.56 |
| 13 | −2.4 | 5.76 |
| 14 | −1.4 | 1.96 |
| 15 | −0.4 | 0.16 |
| 15 | −0.4 | 0.16 |
| 15 | −0.4 | 0.16 |
| 18 | 2.6 | 6.76 |
| 19 | 3.6 | 12.96 |
| 22 | 6.6 | 43.56 |
| Σ | 102.4 |
s = √(102.4 ÷ 9) = 3.3730962
The formulas
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
Population: σ = √( Σ(x − μ)² ÷ N )
Variance is the same expression without the square root. Because deviations are squared, variance is in squared units (cm², $²); the square root brings the SD back to the data’s own units.
What standard deviation actually measures
Each dot is a data value. The gold line is the mean; the shaded band is one standard deviation either side. Drag a dot (or focus it and use the arrow keys) away from the others and watch how one far-off value inflates the SD, because distances are squared before they are averaged.
Show me the steps
An AI tutor walks through the working for any problem you type. The answer above is computed in your browser; the explanation is optional and is the only thing that leaves your device.
Questions people ask
Should I use sample or population standard deviation?
Use the population formula (divide by N) only when your data includes every member of the group you care about, such as all 30 students in one class. If your data is a sample used to estimate a larger group, use the sample formula (divide by n − 1). When unsure, sample is the safer choice and is what most statistics courses expect.
Why divide by n − 1?
A sample’s values sit closer to their own mean than to the true population mean, so squared deviations from the sample mean are slightly too small. Dividing by n − 1 instead of n (Bessel’s correction) removes that bias from the variance.
What does the standard deviation tell me?
It is the typical distance of a value from the mean, in the same units as the data. For roughly bell-shaped data, about 68% of values fall within 1 SD of the mean and about 95% within 2 SD (the empirical rule).
What is the difference between standard deviation and standard error?
Standard deviation describes the spread of individual values. Standard error (SD ÷ √n) describes how precisely the sample mean estimates the population mean; it shrinks as the sample grows.
How do I calculate it in Excel or Google Sheets?
STDEV.S (or STDEV) for a sample and STDEV.P for a population. The results here match those functions.