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Standard Deviation Calculator

Calculate population or sample standard deviation, variance, mean, SEM, and confidence-interval error bars from a list of numbers.

The data is the entire population — variance divides by N.

ExampleSample values — edit any field to see your result.

Separate values with commas, spaces, or new lines.

Results update as you type.

Standard Deviation, σ

4.8989794855664

Estimated result

Count, N
8
Sum, Σx
144
Mean, μ
18
Variance, σ²
24
Standard Error, σx̄
1.7320508075689

Margin of Error (Confidence Interval)

Confidence LevelMargin of Error
68.3%, σx̄18 ±1.732 (±9.62%)
90%, 1.645σx̄18 ±2.849 (±15.83%)
95%, 1.960σx̄18 ±3.395 (±18.86%)
99%, 2.576σx̄18 ±4.462 (±24.79%)
99.9%, 3.291σx̄18 ±5.7 (±31.67%)
99.99%, 3.891σx̄18 ±6.739 (±37.44%)
99.999%, 4.417σx̄18 ±7.65 (±42.50%)
99.9999%, 4.892σx̄18 ±8.473 (±47.07%)

Frequency Table

ValueFrequency
101 (12.5%)
121 (12.5%)
162 (25%)
211 (12.5%)
233 (37.5%)
Standard Deviation, σ4.8989794855664

Paste a list of numbers to get the mean, variance, standard deviation, and standard error of the mean. Choose Population when the list is the whole group, or Sample when it is a draw from a larger population. Repeated values also produce a frequency table, and each result includes z-based confidence error bars around the mean.

Formula

The mean is the average of the N values:

mean = Σxi / N

Population variance and standard deviation divide by N:

σ² = Σ(xi − μ)² / N
σ  = √σ²

Sample variance and standard deviation divide by N − 1:

s² = Σ(xi − x̄)² / (N − 1)
s  = √s²

The standard error of the mean is:

SEM = SD / √N

Confidence error bars use SEM times a normal z multiplier (1, 1.645, 1.960, 2.576, 3.291, 3.891, 4.417, 4.892) and report mean ± margin as a percent of the mean.

Use sample mode for almost every real-world dataset that is not an exhaustive census. Dividing by N − 1 slightly inflates the spread so a small sample does not systematically understate the population SD. Sample mode needs at least two values.

Examples

Default list as a population

For 10, 12, 23, 23, 16, 23, 21, 16, N is 8, the sum is 144, and the mean is 18. Population variance is 24, so σ is 4.8989794855664 and SEM is 1.7320508075689. The 95% error bar is 18 ± 3.395 (±18.86%).

The same list as a sample

The mean is still 18, but variance uses N − 1 = 7 and becomes 27.428571428571. Sample SD is 5.2372293656638 and SEM is 1.8516401995451.

Odd numbers 1, 3, 5, 7, 9

The mean is 5. As a population, variance is 8, σ is 2.8284271247462, and SEM is 1.2649110640674. As a sample, variance is 10, s is 3.1622776601684, and SEM is 1.4142135623731. Every value is unique, so there is no frequency table.

Frequently asked questions

What is the difference between population and sample standard deviation?
Population standard deviation (σ) treats the list as every member of the group and divides the sum of squared deviations by N. Sample standard deviation (s) uses Bessel's correction and divides by N − 1, which is the usual estimator when the list is a sample from a larger population.
How should I enter the numbers?
Paste or type values separated by commas, spaces, or new lines. Negatives and decimals are allowed. Sample mode needs at least two numbers; population mode accepts one or more.
What is the standard error of the mean (SEM)?
SEM is the standard deviation divided by the square root of N. It estimates how much the mean itself would vary across repeated samples of the same size, and it is the building block for the confidence-interval error bars.
How are the confidence-interval error bars calculated?
Each row is mean ± z × SEM. The z multipliers are 1, 1.645, 1.960, 2.576, 3.291, 3.891, 4.417, and 4.892, corresponding to about 68.3%, 90%, 95%, 99%, 99.9%, 99.99%, 99.999%, and 99.9999% under a normal approximation. The percent in parentheses is the margin as a share of the mean.
When does the frequency table appear?
When at least one value appears more than once. Rows are sorted by value and show how often each distinct number occurs, both as a count and as a percent of N.

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