Standard Deviation Calculator – Sample and Population, With Steps
Paste a list of numbers to get the standard deviation, variance, mean, median, mode, range and standard error — with a step-by-step table showing how it was calculated.
Standard deviation measures how spread out numbers are around the mean. For the data 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. Use “sample” unless your data includes every member of the group.
How to use Standard Deviation Calculator
- Type or paste your numbers, separated by commas, spaces or new lines.
- Choose “Sample” or “Population”.
- Read the standard deviation and the other statistics.
- Open “Show the steps” to see each deviation.
- Copy the result for your report or homework.
What standard deviation tells you
The standard deviation describes how far values typically are from the average. A small standard deviation means the numbers sit close to the mean; a large one means they are widely spread. Two classes can both average 70 points, but if one has a standard deviation of 4 and the other 15, their results look very different.
The formulas
Population: σ = √( Σ(x − μ)² ÷ N )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
First find the mean, subtract it from every value, square the differences and add them up. Divide by N for a population or by n − 1 for a sample, and take the square root. The value before the square root is the variance.
Worked example
| x | x − mean | (x − mean)² |
|---|---|---|
| 2 | −3 | 9 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
The mean is 40 ÷ 8 = 5 and the squared deviations add up to 32. Population variance = 32 ÷ 8 = 4, so σ = 2. Sample variance = 32 ÷ 7 ≈ 4.571, so s ≈ 2.138.
Sample or population?
Use population only when your data contains every member of the group you care about — for example the marks of all 25 students in one class. Use sample when your numbers are a subset used to estimate a larger group, such as 200 survey answers about a whole city. Dividing by n − 1 (Bessel’s correction) corrects the tendency of a sample to underestimate the spread. When in doubt, use sample.
The 68–95–99.7 rule
For roughly normal data, about 68 % of values lie within one standard deviation of the mean, 95 % within two and 99.7 % within three. That makes standard deviation useful for spotting unusual values and for quality control.
Other statistics shown
The calculator also gives the median, mode, range, sum, standard error (s ÷ √n), which tells you how precise the mean is, and the coefficient of variation. For other maths tasks, try the GPA Calculator or the Slope Calculator.
You may also find these free tools useful: GPA Calculator, LCM & GCF Calculator and Slope Calculator.
Frequently asked questions
How do you calculate standard deviation?
Find the mean, square each value’s distance from it, 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 sample version divides by n − 1 instead of n, which gives a slightly larger value to correct for estimating from a subset.
What is a good standard deviation?
There is no universal good value; it depends on the data. Compare it with the mean, for example using the coefficient of variation.
How is variance related to standard deviation?
The variance is the standard deviation squared. Standard deviation is easier to read because it uses the same unit as the data.
How do I enter my data?
Separate numbers with commas, spaces, semicolons or new lines. Decimal commas like 2,5 work when values are separated by spaces or lines.
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