What is the difference between sample and population standard deviation?

Population standard deviation divides the summed squared differences by n (the count) and is used when your data covers the entire group. Sample standard deviation divides by n − 1 and is used when your data is a sample drawn from a larger population — the n − 1 correction (Bessel's correction) reduces bias in the estimate.

How is standard deviation calculated?

Find the mean, subtract it from each value and square the result, sum those squared differences, divide by n (population) or n − 1 (sample) to get the variance, then take the square root to get the standard deviation.

What is variance versus standard deviation?

Variance is the average of the squared differences from the mean. Standard deviation is the square root of the variance, which puts the spread back into the same units as your original data.

Why does sample standard deviation need at least two values?

Because it divides by n − 1. With only one value, n − 1 is zero, so the formula is undefined. Population standard deviation needs only one value.

What input formats are accepted?

Numbers separated by commas or spaces (or both). Non-numeric tokens are ignored, so a stray label or blank will not break the result.

Is my data uploaded anywhere?

No — the calculation runs entirely in your browser, and the numbers you paste are never uploaded.

Standard Deviation Calculator

Standard deviation measures how spread out a data set is — the typical distance of each value from the mean. A small value means the numbers cluster tightly; a large one means they are widely scattered. Paste your numbers, choose sample or population, and this calculator returns the standard deviation, variance, and mean.

How it works

The calculator follows the textbook steps:

Mean = sum of values ÷ count (n).
For each value, compute (value − mean)² and sum them — the sum of squared differences.
Variance = that sum ÷ n for a population, or ÷ (n − 1) for a sample.
Standard deviation = √variance.

Sample mode requires at least two values (because it divides by n − 1); population mode needs at least one.

Example

For the data set 10, 12, 23, 23, 16, 23, 21, 16 (n = 8):

Mean = 144 ÷ 8 = 18
Sum of squared differences = 64 + 36 + 25 + 25 + 4 + 25 + 9 + 4 = 192
Sample variance = 192 ÷ 7 ≈ 27.43, sample SD ≈ 5.24
Population variance = 192 ÷ 8 = 24, population SD ≈ 4.90

Measure	Sample (n − 1)	Population (n)
Variance	27.43	24.00
Std deviation	5.24	4.90
Mean	18	18

Everything is computed in your browser, so nothing you enter is uploaded.