Standard Deviation Calculator Guide

What This Calculator Does

Paste a list of numbers and select sample or population scope to calculate count, mean, squared deviations, variance and standard deviation. Expand the steps table to inspect the first 50 observations; all observations contribute to the final result, including repeats.

When to Use It

Use it to describe the spread of measurements in the same unit. Sample mode estimates population variance from a sample; population mode treats the supplied values as the complete population of interest. To standardize a value see z-score. The P-value tool has separate test assumptions.

Inputs Explained

Separate numbers with commas, spaces, semicolons or line breaks. Use dot decimals: 1,5 means two values, not 1.5. Negative numbers, zero and scientific notation such as 1e3 are accepted. Limit: 10000 values and 200000 characters. Every value must be finite with absolute magnitude at most 1e100. Unit suffixes, currency symbols, thousands grouping and missing-value labels are rejected.

How the Calculation Works

Mean=Σx/n. Squared deviations=Σ(x−mean)². Population variance=squared deviations/n. Sample variance=squared deviations/(n−1). Standard deviation is the square root of variance. Sample mode needs at least two values. A one-value population has zero variance and standard deviation.

Example

For 2, 4, 6: count=3, mean=4, deviations=−2, 0, 2, squared deviations=4, 0, 4, and total=8. Population variance=8/3≈2.66666667 and standard deviation≈1.63299316. Sample variance=8/2=4 and standard deviation=2. Try example selects sample mode; changing scope clears the result until recalculated.

For 5, 5, 5, every deviation is zero, so both scopes return zero variance and standard deviation. This means no spread, not no data. Values 1000000000002, 1000000000004 and 1000000000006 still have sample variance 4; centering helps retain their small differences.

How to Understand the Result

Standard deviation has the original unit; variance has its square. For centimetres, those units are cm and cm². Constant lists have zero spread. Steps are limited to 50 rows for readability, without reducing the data used. Up to eight decimal places are displayed, with scientific notation for very small or large values.

Common Mistakes

Do not enter 1000 as 1,000: the comma separates values. Do not confuse variance with its square root. A one-value sample is invalid rather than assigned zero spread. Extreme observations influence the result; no automatic outlier exclusion or missing-value imputation occurs.

FAQ

What is the difference between n and n−1?

Population variance divides by n; the sample estimator divides by n−1.

Can I enter negative numbers or exponent notation?

Yes, within the finite supported range.

Does the 50-row step limit change my result?

No. All entered values contribute.

Notes

The algorithm centers around the first value and uses compensated two-pass summation to reduce cancellation. JavaScript floating-point representation still limits extremely small differences between large values. NIST’s measures of scale defines variance and standard deviation. No weighted variance, grouped frequencies, confidence intervals or inference is supplied. Inputs stay in page state.