What the standard deviation tells you
Standard deviation is the most common way to measure how spread out a data set is around its mean. A small value means the numbers sit close to the average; a large one means they are scattered. Teachers ask for it in statistics homework, quality engineers track it on production lines, and analysts use it to compare the volatility of anything from test scores to stock returns.
What makes this calculator different is that it shows the full worked solution, not just the answer. You get the mean, the table of deviations and squared deviations, the sum of squares, both variances and both standard deviations — population and sample — exactly the way you are expected to write them out in class. If you only need a quick summary (median, mode, range), the mean, median and mode calculator is a better fit; this one is built around the step-by-step procedure.
How to use it
- Paste or type your numbers into the box. Commas, spaces, semicolons and line breaks all work as separators, so you can copy a column straight from a spreadsheet.
- Results update instantly: the two headline numbers are the population (σ) and sample (s) standard deviations.
- Scroll through the “Step by step” section to follow every calculation, then hit “Copy results” to paste the summary into your assignment.
Everything runs locally in your browser — no data leaves your device.
The method, step by step
| Step | Operation | Formula |
|---|---|---|
| 1 | Find the mean | x̄ = Σxᵢ ÷ n |
| 2 | Subtract the mean from each value, square it | (xᵢ − x̄)² |
| 3 | Add up all the squares | Σ(xᵢ − x̄)² |
| 4 | Divide by n (population) or n − 1 (sample) | σ² or s² |
| 5 | Take the square root | σ = √σ², s = √s² |
The calculator also reports the coefficient of variation, CV = standard deviation ÷ mean × 100, which turns the spread into a percentage so you can compare data sets measured in different units.
Worked example
Data set: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8).
Step 1 — the values add up to 40, so the mean is x̄ = 40 ÷ 8 = 5.
Step 2 — subtracting 5 from each value gives −3, −1, −1, −1, 0, 0, 2, 4. Squaring them gives 9, 1, 1, 1, 0, 0, 4, 16.
Step 3 — sum of squares: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
Step 4 — population variance: σ² = 32 ÷ 8 = 4. Sample variance: s² = 32 ÷ 7 ≈ 4.5714.
Step 5 — population standard deviation: σ = √4 = 2. Sample standard deviation: s = √4.5714 ≈ 2.1381.
The population coefficient of variation is 2 ÷ 5 × 100 = 40 %, a fairly wide spread relative to the mean.
Frequently asked questions
When should I divide by n and when by n − 1?
Use n (population formula) when your data covers every member of the group you care about — say, the grades of your entire class. Use n − 1 (sample formula) when your data is a sample drawn from a larger population; the smaller denominator, known as Bessel’s correction, compensates for the fact that a sample tends to underestimate the true spread. If a textbook problem says “sample”, always use n − 1.
Why is my sample standard deviation blank with one value?
With a single data point, n − 1 equals 0 and the sample formula would divide by zero, so it is undefined. The population deviation of one value is 0, since there is nothing to deviate from.
Is variance or standard deviation better?
They carry the same information. Variance is expressed in squared units (dollars², cm²), which is awkward to interpret, while standard deviation is in the original units of the data. Report the standard deviation; keep the variance for intermediate algebra.
What counts as a “high” coefficient of variation?
There is no universal cutoff, but as a rule of thumb a CV under 10 % suggests very consistent data, while values above 30 % indicate substantial dispersion. The CV is only meaningful for data measured on a ratio scale with a mean different from zero.