Standard Deviation Calculator
Skip the manual arithmetic and calculate your own numbers.
Step 1: find the mean
Add all values in the data set and divide by the number of values. This mean becomes the reference point for the next step.
Step 2: find each deviation from the mean
Subtract the mean from each value. Some deviations will be positive and others negative. If you added these raw deviations together, they would cancel, which is why the next step squares them.
Step 3: square the deviations
Square each deviation and add the squared values. This total describes the overall squared distance from the mean.
Step 4: calculate variance
For a population, divide the sum of squared deviations by N, the number of values. For a sample, the common formula divides by n − 1.
Sample variance = Σ(x − x̄)² ÷ (n − 1)
Step 5: take the square root
Variance is expressed in squared units. Taking the square root returns the measure to the original units of the data.
Population vs. sample standard deviation
Use the population formula when your data represents the complete population you want to describe. Use the sample formula when the data is a sample intended to estimate variability in a larger population. The sample formula uses n − 1, which usually produces a slightly larger value for the same data.
How to interpret the result
Standard deviation has meaning relative to the scale of the data. A standard deviation of 5 may be small for values measured in hundreds but large for values clustered around 10. When the distribution is approximately normal, standard deviation is also useful for describing how observations are distributed around the mean.
Common mistakes
- Using the sample formula when the data is the full population, or vice versa.
- Forgetting to square deviations before adding them.
- Taking the square root too early.
- Comparing standard deviations across data sets with very different scales without context.
- Assuming standard deviation alone describes the shape of a distribution.
Frequently asked questions
What is the difference between variance and standard deviation?
Variance is the average squared deviation from the mean under the chosen population or sample formula. Standard deviation is the square root of variance.
Why does sample standard deviation use n minus 1?
Using n − 1 is a standard correction that improves estimation of population variance from a sample.
Can standard deviation be negative?
No. Variance and standard deviation cannot be negative because they are based on squared deviations.