Standard Deviation Calculator
Population and sample standard deviation and variance, step by step.
Enter at least two numbers, separated by commas or spaces. For example: 2, 4, 4, 4, 5, 5, 7, 9.
Use the population standard deviation (σ) when your numbers are the entire group you care about, and the sample standard deviation (s) when they are a sample drawn from a larger group. Everything is calculated in your browser — nothing you enter is sent anywhere. For the basic averages first, try the average calculator.
Population vs. sample standard deviation
| Measure | Divides by | When to use it |
|---|---|---|
| Population SD (σ) | N (the number of values) | Your data is the whole group — e.g. every test score in a class |
| Sample SD (s) | N − 1 | Your data is a sample of a larger group — e.g. a survey of 50 students |
Both start the same way: find the mean, subtract it from each value, square each difference, and add the squares. The only difference is the final division — by N for a population, by N − 1 for a sample. The sample version divides by a smaller number, so it is always slightly larger.
Standard deviation calculator — population and sample SD with steps
A standard deviation calculator measures how spread out a set of numbers is around its mean, and shows every step of the working. Enter your numbers and you get the mean, the population standard deviation and variance, the sample standard deviation and variance, and a step-by-step table of each value, its deviation from the mean, and the squared deviation.
Standard deviation is the square root of the variance, which puts the answer back in the same units as your data — if your numbers are test scores out of 100, the standard deviation is in points, not points squared. A small standard deviation means the values cluster tightly around the mean; a large one means they are widely spread. For the classic example 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5, the population standard deviation is 2, and the sample standard deviation is about 2.138.
The step-by-step table is the part students find most useful: it shows (x − mean) and (x − mean)² for every value, so you can follow the calculation line by line and check it against your own homework.
How standard deviation works
How far, on average, the numbers sit from their mean. The calculator takes each number, subtracts the mean, squares the result, averages those squares (that is the variance), then takes the square root. The step-by-step table shows every step so you can follow it for homework.
Population standard deviation divides by N (use it when your numbers are the whole group). Sample standard deviation divides by N-1 (use it when your numbers are a sample of a bigger group). For [2, 4, 4, 4, 5, 5, 7, 9] the population SD is exactly 2.
Need just the centre of your numbers? Use the average calculator.
How to use the standard deviation calculator
- Enter your numbers in the box, separated by commas or spaces — for example, 2, 4, 4, 4, 5, 5, 7, 9.
- Enter at least two numbers; a sample standard deviation needs at least two values.
- Press Calculate standard deviation to see the mean, population SD and variance, and sample SD and variance.
- Read the step-by-step table to see each value's deviation from the mean and its square.
- Pick the right result: population (σ) if your numbers are the whole group, sample (s) if they are a sample of a larger group.
Frequently asked questions
What is the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9?
The population standard deviation is 2 (variance 4), and the sample standard deviation is about 2.138. The mean is 5, and the squared deviations add up to 32: 32 ÷ 8 = 4 for the population variance, and 32 ÷ 7 ≈ 4.571 for the sample variance.
What is the difference between population and sample standard deviation?
Population SD divides the squared-deviation total by N (all values); sample SD divides by N − 1. Use population SD when your data is the entire group, and sample SD when it is a sample of a larger group. The sample value is always slightly larger.
What does standard deviation tell you?
How spread out values are around the mean. A small standard deviation means the values cluster close to the mean; a large one means they are widely spread. It is in the same units as your data, unlike variance, which is in squared units.
How do you calculate standard deviation step by step?
Find the mean, subtract it from each value, square each difference, add the squares, divide by N (population) or N − 1 (sample), and take the square root. The step-by-step table above shows each of these stages for your numbers.
Can standard deviation be zero?
Yes — when every value is identical, each deviation is 0, so the variance and standard deviation are both 0. That means there is no spread at all in the data.
Why do you square the deviations?
Plain deviations always add up to zero (positive and negative cancel out), so they cannot measure spread. Squaring makes every deviation positive, so they add up to a meaningful total — and taking the square root at the end puts the answer back in the original units.
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