Standard Deviation Calculator
Paste or type a list of numbers to get the standard deviation, variance, mean, median, mode, quartiles and more, with every step of the working shown.
Frequency chart
Show step-by-step working
How to use the Standard Deviation Calculator
- Paste your numbers into the box. Separate them with commas, spaces, tabs or new lines; you can paste a column straight from Excel or Google Sheets.
- Choose sample or population. Use sample if your numbers are a sample from a bigger group (most common); use population if they are the whole group.
- Read the results: standard deviation, variance, mean, median, mode, quartiles, range and more update as you type.
- Open the working to see each value's deviation from the mean and its square, the same steps you would write by hand.
Standard deviation formulas
| Measure | Formula |
|---|---|
| Mean | x̄ = Σx ÷ n |
| Population variance | σ² = Σ(x − μ)² ÷ n |
| Sample variance | s² = Σ(x − x̄)² ÷ (n − 1) |
| Standard deviation | σ = √σ² or s = √s² |
| Standard error of the mean | SE = s ÷ √n |
| Coefficient of variation | CV = SD ÷ mean × 100% |
Worked example
Take the numbers 2, 4, 4, 4, 5, 5, 7, 9. The mean is 40 ÷ 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32.
- Population variance = 32 ÷ 8 = 4, so the population standard deviation is √4 = 2.
- Sample variance = 32 ÷ 7 ≈ 4.571, so the sample standard deviation is ≈ 2.138.
- The median is 4.5 and the mode is 4 (it appears three times).
Mean, median and mode
The mean is the average: add all values and divide by how many there are. The median is the middle value once the numbers are sorted (or the average of the two middle values). The mode is the value that appears most often. When data has extreme values, the median is usually a better "typical" value than the mean.
What the standard deviation tells you
For roughly bell-shaped (normal) data, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This is called the 68-95-99.7 rule.
Frequently asked questions
What is standard deviation?
Standard deviation measures how spread out numbers are around their mean. A small standard deviation means the values are close to the average; a large one means they are spread out.
What is the difference between sample and population standard deviation?
Population standard deviation divides the sum of squared deviations by n (the number of values). Sample standard deviation divides by n − 1, which corrects the tendency of a sample to underestimate the spread of the whole population.
How do I calculate standard deviation by hand?
Find the mean, subtract it from each value, square each result, add the squares, divide by n (population) or n − 1 (sample) to get the variance, and take the square root.
How are quartiles calculated?
This calculator uses linear interpolation between data points, the same method as Excel's QUARTILE.INC and Google Sheets' QUARTILE. Other textbooks use slightly different methods, so small differences are normal.
What if there is no mode?
If every value appears the same number of times, there is no mode. If two or more values tie for the most appearances, all of them are listed.
How are outliers detected?
Values below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR are flagged as possible outliers, the same rule used for box plots.