Standard Deviation Calculator

Spread, variance, and the units of your data

Standard deviation measures spread in the same units as your inputs. Variance is its square. A result of zero means all values are identical. The squared-deviation table shows the first 100 observations; the sum of squares and final result include every observation. The alternate-convention card uses the same data with the other divisor. A single population observation has zero spread, while its sample counterpart is undefined.

To compare the mean with the median and most frequent values, use the Mean, Median & Mode Calculator with the same dataset.

The formula

For a sample, subtract the mean x̄ from each observation, square those differences, add them, and divide by n − 1 before taking the square root. For a complete population, use n as the divisor instead.

How to use this calculator

1. Choose individual numbers to paste observations, or values with frequencies to enter each number beside its positive whole count.
2. Choose Sample if the values represent part of a larger group, or Population if they describe the entire group.
3. Calculate to view standard deviation and supporting statistics.

Should you choose sample or population?

Use population standard deviation when your dataset contains every member of the group you want to describe. Use sample standard deviation when your observations are a sample used to estimate spread in a larger group.

The n − 1 divisor corrects the downward bias in the estimate of population variance. It does not make the square root an exactly unbiased estimator of population standard deviation.

Interpreting the amount of spread

A larger standard deviation means observations tend to sit farther from the mean. Its importance depends on the units and context: a spread of 5 means something different for daily temperatures and annual revenue.

Outliers can strongly affect standard deviation because differences are squared. The familiar 68–95–99.7 rule applies approximately to normal distributions; do not assume it fits every dataset.

Entering repeated observations

Frequency mode expands each count into that many equal-weight observations. You can use editable rows or paste two spreadsheet columns. Up to 200 frequency rows are accepted, totaling at most 10,000 observations, with values within ±1,000,000,000,000. Duplicate value rows add together. Counts must be positive integers; percentages, fractional weights, and grouped class intervals are outside this mode.

The squared-deviation table shows the first 100 observations; the sum of squares and final result include every observation. The alternate-convention card uses the same data with the other divisor. A single population observation has zero spread, while its sample counterpart is undefined.

Assumptions & limitations

What this calculation assumes

• Every observation has equal weight.
• Comma characters separate values rather than mark thousands or decimals.
• Sample results use n − 1; population results use n.
• Each frequency is an exact whole count of identical observations.

What to keep in mind

• Sample standard deviation needs at least two observations.
• Datasets are limited to 10,000 values with absolute magnitudes up to 1 trillion.
• JavaScript floating-point arithmetic has finite precision, especially for differences smaller than the precision of the original inputs.
• Frequency entry accepts up to 200 rows totaling 10,000 observations; it does not estimate data from class intervals.

Common questions