SigmaCalc

Standard Deviation Calculator

σ and s from raw numbers.

Dataset (comma / space / new-line separated)

Count: 10
Mean μ: 12.7
Sample σ (n−1): 4.449
Population σ (n): 4.22
Variance (sample): 19.789
Variance (population): 17.81

What you entered

Dataset: 10 values

1
Count of values
n= 10
2
Mean
sum ÷ n= 12.7
3
Sum of squared deviations from the mean
Σ(xᵢ − 12.7)²= 178.1
4
Sample variance (÷ n−1)
178.1 ÷ 9= 19.7889
5
Sample standard deviation
√19.7889= 4.4485
6
Population variance (÷ n)
178.1 ÷ 10= 17.81
7
Population standard deviation
√17.81= 4.2202

Result

Sample standard deviation: 4.449

With a mean of 12.7, this dataset has a sample standard deviation of 4.449 — most values fall within about that distance of the mean.

Standard deviation and variance, sample versus population

Sample figures divide by n minus 1, population figures divide by n. Which you want depends on whether the data is your whole group or a sample from a larger one.

Dataset	Mean	Sample SD	Population SD	Sample variance	Population variance
2, 4, 4, 4, 5, 5, 7, 9	5.00	2.138	2.000	4.571	4.000
10, 12, 23, 23, 16, 23, 21, 16	18.00	5.237	4.899	27.429	24.000
1, 2, 3, 4, 5	3.00	1.581	1.414	2.500	2.000
5, 10, 15, 20, 25, 30	17.50	9.354	8.539	87.500	72.917
70, 75, 80, 85, 90, 95, 100	85.00	10.801	10.000	116.667	100.000
3, 7, 7, 19, 24, 30, 32, 45	20.88	14.672	13.724	215.268	188.359

The sample figure is always the larger of the two, and the gap narrows as the dataset grows - it is 7% on the five-value set in row three and about 6.5% on the eight-value sets. Dividing by n minus 1 rather than n is Bessel's correction, and it exists because a sample's own mean sits closer to its own values than the true population mean does, which would otherwise understate the spread. Use the sample figure unless you genuinely have every member of the population. Standard deviation is the square root of variance, which is why it is in the same units as the data while variance is in squared units.

How spread out is your data?

Standard deviation measures the typical distance each value sits from the mean — a small standard deviation means values cluster tightly; a large one means they're spread out widely.

Sample vs. population — why it matters

Dividing by n−1 instead of n (Bessel's correction) is used when your data is a sample meant to estimate a larger population — it slightly inflates the result to correct for the fact that a sample tends to underestimate true variability.

Frequently asked questions

My class test scores are 72, 85, 90, 68, 95. What's the standard deviation?

Mean = 82. Deviations: −10, 3, 8, −14, 13. Squared: 100, 9, 64, 196, 169. Sum = 538. Sample variance (÷4) = 134.5. Sample SD = √134.5 = 11.6. This tells you the typical score sits about 11.6 points from the class average.

How is standard deviation different from variance?

Standard deviation is the square root of variance. Variance (134.5 points²) is in squared units, which isn't intuitive. Standard deviation (11.6 points) is in the same units as your data, making it directly interpretable. Use the variance calculator for detailed variance breakdowns.

What does a 'high' or 'low' standard deviation mean?

It depends on context. Test scores with SD of 5 out of 100 = tightly clustered (consistent class). SD of 25 = widely spread (mixed abilities). Compare SD to the mean: a coefficient of variation (SD÷mean×100%) above 30% generally signals high variability.

When should I use standard deviation vs. interquartile range?

SD works best for symmetric, bell-shaped data without extreme outliers. IQR (the middle 50% spread) is more robust when data is skewed or has outliers — like income or house prices. For skewed datasets, see the quartiles & IQR calculator.