Standard Deviation Calculator
Find the sample or population standard deviation of a data set, with the mean, variance, and squared-deviation steps used on paper.
Sample s = 13.4907375632
- Count the values and find the mean: n = 6, sum = 108, mean = 108 / 6 = 18.
- (4 − 18)² = −14² = 196.
- (8 − 18)² = −10² = 100.
- (15 − 18)² = −3² = 9.
- (16 − 18)² = −2² = 4.
- (23 − 18)² = 5² = 25.
- (42 − 18)² = 24² = 576.
- Sum of squared deviations = 910.
- Variance = 910 / n − 1 = 910 / 5 = 182.
- s = √182 = 13.4907375632.
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How to Use the Standard Deviation Calculator
Paste your numbers, choose sample or population, then select Calculate. The tool shows the standard deviation, the matching variance, the mean, and each squared deviation used on paper.
- Enter the data set. Commas, spaces, and new lines all work.
- Choose Sample (n − 1) or Population (n). Reset restores the six-value sample example.
- Read the steps to see the mean, the sum of squares, the divisor, and the square root.
The Mean, Median, Mode Calculator on SolverCalc already reports a population standard deviation as a side metric. This page is only about spread: it compares s and σ and writes the substitution steps. Standardizing one score with a known μ and σ belongs on the Z-Score Calculator. Reading that z as a left-tail percentile belongs on the Percentile Calculator.
Sample vs. Population
Both formulas start the same way. Find the mean, subtract it from each value, square those differences, and add them. That sum of squares is SS. The last two steps differ: a sample divides SS by n − 1, then takes the square root. A population divides by n.
Dividing a sample by n − 1 is Bessel’s correction. It keeps s from systematically underestimating the spread of the larger group the sample came from. The variance is the quantity under the square root: s² or σ².
Standard Deviation Examples
The six-value classroom set
For 4, 8, 15, 16, 23, 42 the mean is 18 and SS = 196 + 100 + 9 + 4 + 25 + 576 = 910. Sample variance is 910 / 5 = 182, so s = √182 = 13.4907375632. Population variance is 910 / 6, so σ = 12.3153021346. That population value is the side metric on the Mean, Median, Mode Calculator.
Every value the same
For 5, 5, 5, 5 the mean is 5 and every deviation is 0, so both s and σ are 0. Spread is zero only when the list does not vary.
Two values
For 2 and 8 the mean is 5 and SS = 18. The sample standard deviation is √18 = 4.24264068712. The population value is √9 = 3.
A single number
One value has no sample standard deviation, because n − 1 would be 0. The population value is 0: that lone number is the whole population and does not vary.
Test scores
For 90, 85, 78, 92, 88, 76, 95 the mean is about 86.29 and the sample standard deviation is 7.08788369248. The scores sit a few points from the mean, not tens of points.
When to Use Sample or Population
Use the population formula when the list is the entire group you care about: every score in one class, every day in a short experiment you will not generalize. Use the sample formula when the list stands in for a larger group: a survey, a homework “sample of size n,” or any set you will treat as evidence about more data.
This page does not compute a confidence interval, a z-table, or a weighted standard deviation. For z = (x − μ) / σ from a single score, use the Z-Score Calculator. It is educational, not a professional statistics package.
Common Standard Deviation Mistakes
Using n when the problem asked for n − 1
A homework “sample” almost always wants s. Using n gives a smaller number, here 12.32 instead of 13.49 on the six-value set.
Forgetting to square the deviations
The deviations themselves add to zero. Squaring makes every gap positive before you average them.
Stopping at variance
Variance is in squared units. Standard deviation is the square root, so it is back in the original units.
Calling range a substitute for standard deviation
Range uses only the two extremes. Standard deviation uses every value. A single outlier moves both, but not by the same amount.
Asking for a sample value from one number
n − 1 = 0 is undefined. Enter a second value or switch to the population formula.
Standard Deviation FAQs
What is the difference between s and σ?
s is the sample standard deviation and divides by n − 1. σ is the population standard deviation and divides by n.
What is the sample standard deviation of 4, 8, 15, 16, 23, 42?
13.4907375632, because SS = 910 and 910 / 5 = 182.
What is the population standard deviation of that same set?
12.3153021346, because 910 is divided by 6 instead of 5.
Can standard deviation be zero?
Yes, when every value equals the mean. The All equal preset shows this.
Does this find the mean, median, and mode?
It reports the mean because the formula needs it. For median and mode, use the Mean, Median, Mode Calculator.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
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Disclaimer
This calculator is for general educational use. It reports unweighted sample and population standard deviation only. It is not a substitute for a statistics package or professional analysis.