Standard Deviation Calculator

Compute sample and population standard deviation and variance from a pasted list of numbers, with the mean and formulas shown.

Inputs

Paste numbers separated by commas, spaces, or new lines.

Result

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How to use this calculator

  • Paste your numbers separated by commas, spaces, or new lines.
  • Read the sample standard deviation (the usual choice) front and center.
  • Use the population standard deviation only if your data is the whole population.
  • Check the variance and sum of squared deviations in the breakdown for the working.

About this calculator

Standard deviation measures how spread out a set of numbers is around their mean: a small standard deviation means the values cluster tightly around the average, a large one means they are widely scattered. It is the square root of the variance, which is the average of the squared distances from the mean. There are two versions. The population standard deviation divides by N and is used when your data covers the entire population. The sample standard deviation divides by n โˆ’ 1 (Bessel's correction) and is the right choice when your data is a sample drawn from a larger population โ€” the most common situation. This calculator reports both, along with the variance, mean, and sum of squared deviations, from a list you paste in any common format.

How it works โ€” the formula

Mean ฮผ = ฮฃx รท n Population: ฯƒยฒ = ฮฃ(xโˆ’ฮผ)ยฒ รท N, ฯƒ = โˆšฯƒยฒ Sample: sยฒ = ฮฃ(xโˆ’xฬ„)ยฒ รท (nโˆ’1), s = โˆšsยฒ

Both square the distance of each value from the mean, average those squares, and take the root. The only difference is dividing by N (population) versus nโˆ’1 (sample).

Source: NIST/SEMATECH e-Handbook โ€” Standard Deviation

Worked examples

Example 1
2,4,4,4,5,5,7,9
Inputs:
data=2,4,4,4,5,5,7,9
Output:
mean 5, pop SD 2, sample SD 2.138
Example 2
10,12,14,16,18
Inputs:
data=10,12,14,16,18
Output:
mean 14, pop SD 2.828, sample SD 3.162
Example 3
5,5,5,5
Inputs:
data=5,5,5,5
Output:
SD 0 (no spread)

Limitations

  • Sample SD requires at least two values.
  • Treats the input as raw, unweighted data.
  • Very large datasets are summarized in full precision but rounded for display.

Reports the spread of the entered data; inferential use requires a representative sample.

Frequently asked

What is the difference between sample and population standard deviation?+
They differ only in the divisor. Population SD divides the sum of squared deviations by N (the full population size). Sample SD divides by n โˆ’ 1, which corrects the bias that arises when estimating a population's spread from a sample. Use sample SD unless you literally have every data point in the population.
Why divide by n โˆ’ 1 for a sample?+
Because the sample mean is itself estimated from the data, the squared deviations slightly underestimate the true spread. Dividing by n โˆ’ 1 instead of n (Bessel's correction) compensates, giving an unbiased estimate of the population variance.
What is variance versus standard deviation?+
Variance is the average squared deviation from the mean; standard deviation is its square root. Standard deviation is usually preferred for reporting because it is in the same units as the data, whereas variance is in squared units.
How do I interpret the standard deviation?+
It is the typical distance of a value from the mean. For roughly bell-shaped (normal) data, about 68% of values fall within one standard deviation of the mean and about 95% within two โ€” the empirical rule.
Can standard deviation be negative?+
No. It is a square root of a sum of squares, so it is always zero or positive. It equals zero only when every value is identical (no spread at all).
How many values do I need?+
Population SD can be computed from one value (it is zero). Sample SD needs at least two values, because dividing by n โˆ’ 1 is undefined for a single data point.

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