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Standard deviation measures how "spread out" a data set is: if the values cluster around the mean it's small, if they scatter over a wide range it's large. Two classes can have the same average score, but if one ranges 45–55 and the other 20–80, standard deviation is what tells them apart. It is the most fundamental measure of dispersion in statistics.
Enter your numbers separated by commas, and the mean, variance and standard deviation (both sample and population) are calculated instantly.
Standard Deviation Calculator: step-by-step worked example
Five students’ test scores: 70, 80, 90, 75, 85. How spread out are they?
- Numbers
- 70, 80, 90, 75, 85
- 1
Find the mean
x̄ = Σx ÷ n
(70 + 80 + 90 + 75 + 85) ÷ 5 = 80
- 2
Each value’s deviation from the mean
Deviation = x − x̄
= -10, 0, 10, -5, 5
- 3
Sum of squared deviations
Σ(x − x̄)²
100 + 0 + 100 + 25 + 25 = 250
- 4
Sample variance
s² = Σ(x − x̄)² ÷ (n − 1)
250 ÷ 4 = 62.5
- 5
Sample standard deviation
s = √s²
√62.5 = 7.9057
- 6
Population standard deviation
σ = √(Σ(x − x̄)² ÷ n)
√(250 ÷ 5) = 7.0711
- 70100
- 800
- 90100
- 7525
- 8525
Good to know: Use the sample formula (divide by n − 1) when your data is a sample of a larger group; use the population formula only when you have every member. In Excel: STDEV.S vs STDEV.P.
Standard Deviation Calculator — how it works
The steps
1. Find the mean: sum of all values ÷ count. 2. Square each value's difference from the mean (squaring removes negatives and emphasises large deviations). 3. The average of those squares is the variance. 4. The square root of the variance is the standard deviation — bringing the measure back to the data's original units.
Sample or population?
There are two versions with different denominators: - Population: sum of squares ÷ n. Use this when you have the entire data set. - Sample: sum of squares ÷ (n−1). Use this when working with a sample drawn from a larger population; the (n−1) divisor makes the estimate unbiased.
The tool reports both; which to use depends on whether your data is the whole population or a sample.
Why the square root?
Variance keeps units squared (e.g. "points²"); the square root converts the result back to the original unit (points), making it interpretable. That is why standard deviation, not variance, is what people quote.
Where it is used
- Testing/measurement: the spread of scores, how far a student is from the mean (z-score).
- Finance: the volatility of returns (a risk measure).
- Quality control: manufacturing tolerances, process variability.
Worked example
Take the data set 2, 4, 4, 5, 7: the mean is (2+4+4+5+7) ÷ 5 = 4.4. The squared differences from the mean are (−2.4)²+(−0.4)²+(−0.4)²+0.6²+2.6² = 5.76+0.16+0.16+0.36+6.76 = 13.2. The population variance is 13.2 ÷ 5 = 2.64 and the standard deviation √2.64 ≈ 1.62. In the sample version the denominator is (n−1) = 4: variance 13.2 ÷ 4 = 3.3, standard deviation √3.3 ≈ 1.82. If this is the whole data set use 1.62; if it is a sample of a larger population use 1.82.
Frequently asked questions
How is standard deviation calculated?+
Find the mean, square each value's difference from the mean, average those squares (the variance), and take the square root. Just enter the numbers separated by commas.
What is the difference between sample and population standard deviation?+
The population version divides the sum of squares by n; the sample version divides by (n−1). Use population if you have the whole data set, sample if you drew it from a larger population.
How do standard deviation and variance relate?+
Standard deviation is the square root of variance. Variance is in squared units (points²); the square root returns it to the original unit (points), making it interpretable.
What does a large standard deviation mean?+
That the data is spread over a wide range around the mean — high variability (or, in finance, high risk). A small standard deviation means the values cluster near the mean.
If two data sets have the same mean, are their standard deviations equal?+
No — the same mean can hide very different spreads. A set ranging 45–55 and one ranging 20–80 can share a mean but have very different standard deviations.
Why does standard deviation use squares and a square root?+
Squaring stops deviations above and below the mean from cancelling out and emphasises large deviations; the square root then converts the result back to the original unit.
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About this page
- Publisher:
- Kalkoro — Teknolojig Bilişim Teknolojileri A.Ş., İstanbul
- Written by:
- Ahmet Tur — Editor (Math, date & time, unit conversion, physics & chemistry, everyday tools)
- Reviewed by:
- Kübra Yıldız — Editor
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