Standard Deviation Calculator

Calculate the standard deviation of any dataset — choose between population (σ) and sample (s) formulas. Full step-by-step working included.

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Enter Your Data

Use sample when your data is a subset of a larger group. Use population when you have all data for the group.

What Is Standard Deviation?

Standard deviation tells you how spread out the values in a dataset are from the mean. A small standard deviation means the data points are clustered close to the mean. A large standard deviation means the data is more spread out.

Real-World Examples

Example 1 — Test Scores: If the mean score in your class is 70 and the standard deviation is 5, most students scored between 65 and 75. If the standard deviation were 20, scores would be much more spread out (some very high, some very low).

Example 2 — Manufacturing: A factory making bolts of length 10 cm wants a small standard deviation — this means each bolt is close to 10 cm and quality is consistent.

Interpreting the Result

Std Dev Meaning
Close to 0 Data points are very close to the mean (low variability)
Equal to mean Coefficient of variation = 100% (moderate variability)
Much larger than mean Data is very spread out (high variability)

Frequently Asked Questions

Population standard deviation (σ) is used when you have data for every member of the group. Sample standard deviation (s) is used when your data is a sample taken from a larger population. The sample formula divides by (n−1) instead of n to correct for the fact that a sample tends to underestimate the true population spread.
If we just summed the differences (xᵢ − x̄), positives and negatives would cancel out, always giving zero. By squaring them, all values become positive. Taking the square root at the end brings the result back to the original units.
No. Standard deviation is always zero or positive. It equals zero only when all data values are identical (no spread at all).