Standard Deviation Calculator
Calculate the sample or population standard deviation of your data instantly — with the variance, mean, a distribution chart, and a full step-by-step solution.
Separate values with commas, spaces, or new lines.
- Variance
- 4.5714
- Mean (x̄)
- 5.0000
- Count (n)
- 8
- Sum (Σx)
- 40.0000
- Sum of squares
- 32.0000
- Std. error
- 0.7559
- Min / Max
- 2.0000 / 9.0000
- Coeff. of variation
- 42.7618%
Using the sample formula (divide by 7).
Distribution
n = 8How to use this calculator
- Paste or type your numbers into the data box — separated by commas, spaces, or new lines.
- Choose whether your numbers are a sample (divide by n − 1) or a whole population (divide by n).
- Read the standard deviation, variance, and summary statistics on the right, and open the step-by-step solution to see the full working.
What is standard deviation?
Standard deviation measures how spread out a set of numbers is around their mean. A small standard deviation means the values cluster tightly around the average; a large one means they are more scattered. It is expressed in the same units as the data, which makes it easy to interpret.
It is the square root of the variance. Where variance is in squared units, taking the square root brings the measure of spread back to the original scale of your data.
Standard deviation formula
The population standard deviation (σ) divides the sum of squared deviations by the number of values, N:
The sample standard deviation (s) divides by n − 1 instead of n. This is Bessel’s correction — it makes s an unbiased estimator of the population spread when you only have a sample:
How to calculate standard deviation step by step
- Find the mean (average) of your data.
- Subtract the mean from each value to get its deviation.
- Square each deviation so negatives and positives don’t cancel.
- Add up all the squared deviations (the “sum of squares”).
- Divide by n − 1 for a sample, or by N for a population, to get the variance.
- Take the square root of the variance — that is the standard deviation.
Sample vs. population standard deviation
Use the population formula (÷ N) only when your data includes every member of the group you care about. Use the sample formula (÷ n − 1) when your data is a sample drawn from a larger population — which is by far the most common case in research and coursework.
| Population (σ) | Sample (s) | |
|---|---|---|
| Divides by | N | n − 1 |
| Use when | You have every value | You have a subset |
| Excel function | STDEV.P | STDEV.S |
How to interpret standard deviation
For roughly bell-shaped (normal) data, the empirical rule says about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. The shaded band in the chart above marks ±1 standard deviation around the mean.
There is no universally “good” or “bad” value — it depends on context. To compare spread between datasets measured on different scales, use the coefficient of variation, which expresses the standard deviation as a percentage of the mean.
Standard deviation in Excel
In Excel or Google Sheets, use =STDEV.S(range) for a sample and =STDEV.P(range) for a population. For example, =STDEV.S(A1:A8) returns the sample standard deviation of the values in cells A1 to A8.
Frequently asked questions
What is the difference between sample and population standard deviation?
The population standard deviation divides the sum of squared deviations by N (the total number of values), while the sample standard deviation divides by n − 1. Dividing by n − 1 corrects the bias that arises when you estimate the spread of a whole population from just a sample.
Is standard deviation the same as variance?
No. Variance is the average of the squared deviations from the mean, so it is in squared units. The standard deviation is the square root of the variance, which returns the measure of spread to the original units of the data.
Can standard deviation be negative?
No. Standard deviation is a square root of a sum of squares, so it is always zero or positive. It equals zero only when every value in the data set is identical.
What does a high standard deviation mean?
A high standard deviation means the values are spread out over a wider range away from the mean. A low standard deviation means they are clustered closely around the mean.
How do I calculate standard deviation in Excel?
Use =STDEV.S(range) for a sample or =STDEV.P(range) for a population. For example, =STDEV.S(A1:A20) gives the sample standard deviation of cells A1 through A20.