Standard Deviation Calculator (Mean, Median, Mode)
- Free
- No sign-up
- Works on phones
- Shows the working
By Yasir Hafeez · Updated
This standard deviation calculator summarizes any list of numbers. Paste or type your data and it returns the count, sum, mean, median, mode, range, the population standard deviation \( \sigma \) and variance, and the sample standard deviation \( s \) and variance, plus the sorted data.
It doubles as a mean, median and mode calculator. It’s free and needs no sign-up.
How to use it
- Type or paste your numbers. Separate them with commas, spaces or new lines; decimals and negatives are fine.
- Press Calculate.
- Choose the right standard deviation: sample (s) if your data is a sample from a bigger group, population (\( \sigma \)) if it’s every member of the group.
Formulas
$$\bar{x} = \frac{\sum x}{n}, \qquad \sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}, \qquad s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}$$
The median is the middle value of the sorted data (the average of the two middle values when n is even). The mode is the most frequent value. The range is the largest minus the smallest.
Worked example
Data: 2, 4, 4, 4, 5, 5, 7, 9 (as loaded above).
Mean. \( \frac{40}{8} = 5 \).
Deviations squared. \( 9, 1, 1, 1, 0, 0, 4, 16 \), which add to 32.
Population SD. \( \sigma = \sqrt{32/8} = \sqrt4 = 2 \).
Sample SD. \( s = \sqrt{32/7} \approx 2.138 \).
Median \( = \frac{4 + 5}{2} = 4.5 \); mode = 4 (three times); range \( = 9 - 2 = 7 \).
Sample or population?
| Use | When | Divide by |
|---|---|---|
| Population \( \sigma \) | you have data for the whole group (every student in a class) | n |
| Sample s | your data is part of a larger group (a survey of 100 voters) | n − 1 |
Dividing by \( n - 1 \) (Bessel’s correction) corrects the tendency of a sample to underestimate the spread. Most real data is a sample, so s is the usual choice in statistics courses and software (Excel’s STDEV.S, Google Sheets’ STDEV).
Interpreting standard deviation
A small SD means the values cluster near the mean; a large one means they’re spread out. For roughly bell-shaped data, about 68% of values lie within 1 SD of the mean, 95% within 2 and 99.7% within 3. To see where a single value falls, convert it to a z-score with the z-score calculator.
Common mistakes
- Using the wrong SD. Check whether you have a sample or a population.
- Forgetting to square the deviations. Unsquared deviations always add to zero.
- Forgetting the square root. Without it you have the variance.
- Median of unsorted data. Sort first.
Related guides
- Standard Deviation Formula: Step by Step
- Mean, Median and Mode
- Z-Score and the Z Table
- Average Value of a Function
Further reading
- Standard deviation (Wikipedia) — definitions, interpretation and the sample correction.
- OpenStax Introductory Statistics 2e — a free statistics textbook with chapters on descriptive statistics.
FAQ
How do you calculate standard deviation?
Find the mean, subtract it from each value, square the results, add them, divide by n (population) or n − 1 (sample), and take the square root.
What is the difference between sample and population standard deviation?
The population SD divides by n; the sample SD divides by n − 1, which makes it slightly larger.
What is a good standard deviation?
There’s no universal “good” value; it depends on the units and the mean. Compare it to the mean (the coefficient of variation is SD ÷ mean) or to other data sets.
Can standard deviation be negative?
No. It’s a square root of an average of squares, so it’s zero or positive.