Standard Deviation Calculator

Calculate standard deviation, variance, and mean from any data set. Free online calculator supporting both population and sample standard deviation, with full step-by-step solutions.

What is Standard Deviation Calculator?

Standard deviation measures how spread out a set of numbers is from its average (mean). A low standard deviation means the values cluster tightly around the mean; a high standard deviation means they're spread over a wider range.

There are two versions: population standard deviation (σ), used when your data represents an entire population, and sample standard deviation (s), used when your data is a sample drawn from a larger population. The sample version divides by n − 1 instead of n — a correction called Bessel's correction — which produces a slightly larger, less biased estimate of the true population spread.

Formula

Population: σ = √( Σ(xᵢ - x̄)² / n )

Sample: s = √( Σ(xᵢ - x̄)² / (n - 1) )

Where x̄ is the mean, xᵢ is each individual value, and n is the count of values.

How to Calculate

Enter your numbers, separated by commas, spaces, or new lines.
Choose whether your data is a full population or a sample.
The calculator finds the mean, then each value's deviation from it.
Review the full step-by-step calculation of variance and standard deviation.

Example

For the data set [4, 8, 15, 16, 23, 42]: the mean is 18. The sample standard deviation (n-1) is approximately 13.62, while the population standard deviation (n) is approximately 12.44.

Key Benefits

Supports both population and sample standard deviation
Shows variance, mean, sum, and count alongside standard deviation
Full step-by-step solution, not just the final number
Accepts flexible input formats (commas, spaces, or new lines)

Common Mistakes to Avoid

Using population standard deviation (n) when the data is actually a sample, which understates the true spread
Forgetting to square the deviations before summing them, which would let positive and negative deviations cancel out to zero
Confusing variance (squared units) with standard deviation (same units as the original data)
Applying the formula to fewer than two data points, where standard deviation is undefined

Pro Tips

If you're not sure whether to use population or sample mode, sample is the safer default for most real-world data collection
A standard deviation close to zero indicates highly consistent, low-variability data
Compare standard deviation to the mean (as a percentage) to judge relative — not just absolute — variability across different data sets
Outliers have an outsized effect on standard deviation because deviations are squared before averaging

Key Terms Explained

Variance: The average of the squared deviations from the mean
Mean: The arithmetic average of all values in the data set
Bessel's Correction: Dividing by n-1 instead of n to reduce bias when estimating from a sample
Deviation: The difference between a single value and the mean

Common Use Cases

Analyzing test scores or survey results to understand data spread
Quality control — measuring consistency in manufacturing measurements
Finance — measuring the volatility or risk of an investment's returns
Scientific research — reporting the spread of experimental measurements

Frequently Asked Questions

What's the difference between population and sample standard deviation?

Population standard deviation (σ) is used when your data represents the entire group you care about, and divides by n. Sample standard deviation (s) is used when your data is only a subset drawn from a larger population, and divides by n-1 to correct for bias — this correction is called Bessel's correction.

Why do we square the deviations before averaging?

Squaring makes every deviation positive, so they don't cancel each other out when summed (which would always give zero, since deviations are defined relative to the mean). Squaring also weights larger deviations more heavily, which is useful for measuring spread.

What does a high standard deviation mean?

A high standard deviation means the data points are spread far from the mean — the data set has high variability. A low standard deviation means most values are close to the mean, indicating more consistency.

How is standard deviation related to variance?

Standard deviation is simply the square root of variance. Variance is expressed in squared units (e.g., dollars²), while standard deviation converts it back to the original units (e.g., dollars), making it easier to interpret.