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Standard Deviation Calculator

Compute mean, variance and standard deviation for a set of numbers.

Inputs

6 valid values

Results

Mean of 6 values
18
Sample std dev s = 13.4907 · Population σ = 12.3153
Count (n)
6
Sum (Σx)
108
Mean (x̄)
18
Median
15.5
Min
4
Max
42
Range
38
Population variance (σ²)
151.6667
Sample variance (s²)
182
Population std dev (σ)
12.3153
Sample std dev (s)
13.4907
Coefficient of variation
74.9485%
Formulas used
  • σ² = Σ(x − μ)² ÷ n
  • s² = Σ(x − x̄)² ÷ (n − 1)
  • σ = √σ²
  • s = √s²

Sorted data

#ValueDeviation (x − x̄)(x − x̄)²
14-14196
28-10100
315-39
416-24
523+525
642+24576
Sum of squares (Σ(x − x̄)²)910

What is the Standard Deviation Calculator?

A standard deviation calculator is a free online tool that computes the mean, median, variance, standard deviation, range, and other descriptive statistics for a list of numbers. It reports both population (σ) and sample (s) standard deviation so you can choose the right one for your dataset.

Standard Deviation Calculator formula

Mean μ = (Σxᵢ)/n. Population variance σ² = Σ(xᵢ − μ)² / n, population std dev σ = √σ². Sample variance s² = Σ(xᵢ − x̄)² / (n − 1), sample std dev s = √s². The sample formulas use n − 1 (Bessel's correction) to give an unbiased estimate when your data is a sample of a larger population. Coefficient of variation CV = (σ / μ) × 100%.

How to use the Standard Deviation calculator

  1. 1Paste or type your data set, separated by spaces, commas, semicolons, or newlines.
  2. 2The calculator parses every numeric token, ignores non-numeric junk, and shows the count it used.
  3. 3Read the summary: n, sum, mean, median, min, max, range, population and sample variance, and population and sample standard deviation.
  4. 4Choose population (σ) when your data is the entire population, or sample (s) when it's a sample of a larger population.
  5. 5Datasets up to 10,000 values are supported; anything beyond is ignored with a visible notice.

Example calculation

For the data set 2, 4, 4, 4, 5, 5, 7, 9: n = 8, sum = 40, mean μ = 5. Sum of squared deviations = (9+1+1+1+0+0+4+16) = 32. Population variance σ² = 32/8 = 4, so σ = 2. Sample variance s² = 32/7 ≈ 4.571, so s ≈ 2.138. The classic textbook example returns the textbook answer.

Frequently asked questions

What is the difference between sample and population standard deviation?
Population standard deviation σ uses divisor n and is correct when your data IS the entire population. Sample standard deviation s uses divisor n − 1 (Bessel's correction) and is the unbiased estimator when your data is a SAMPLE of a larger population. Use s for surveys, experiments, and any data drawn from a bigger group; use σ only when you have every member of the population.
When is standard deviation undefined?
Sample standard deviation requires at least 2 data points because the formula divides by n − 1; with n = 1 the divisor is 0. The calculator shows '—' for sample variance and sample std dev when there's only one value (population std dev is correctly 0, since a single value has no spread).
How do I enter my data?
Separate numbers with any of: spaces, commas, semicolons, pipes, tabs, or newlines. Non-numeric tokens are ignored and a count badge shows how many valid numbers were used, so you can safely paste a column from a spreadsheet even if it has headers or blanks.
What is the coefficient of variation (CV)?
CV = (σ / μ) × 100% — the standard deviation expressed as a percentage of the mean. It's a unitless measure of relative spread that lets you compare variability across datasets with different units or scales. The calculator shows '—' when the mean is 0 or the sample is too small.
How is the median computed?
The data is sorted ascending. If n is odd, the median is the middle value; if n is even, it's the average of the two middle values. For 2, 4, 4, 4, 5, 5, 7, 9 (n = 8), the two middle values are 4 and 5, so the median is 4.5.

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