Calculators
Weighted Standard Deviation Calculator
Calculate frequency-weighted mean, variance, and population or sample standard deviation privately in your browser.
Measure spread when each value has a repeat count.
Weighted standard deviation uses weights as frequency counts. The weighted mean is Σ(w × x) / Σw. Population variance is Σ(w × (x − μ)²) / Σw, while sample variance divides the same weighted sum of squared deviations by Σw − 1. For values 2, 4, and 7 with weights 1, 2, and 1, the weighted mean is 4.25, population standard deviation is about 1.78535710714, and sample standard deviation is about 2.06155281281. Use weights to represent repeated observations; for reliability weights that already sum to one, the sample formula differs. Values may be signed or zero. Weights must be non-negative and at least one must be positive; zero-weight rows are ignored. The sample result is undefined unless total weight exceeds 1. Enter up to 10000 pairs, one per line, separating value and weight with a comma, semicolon, or spaces. Scientific notation is accepted; omit thousands separators and use consistent units. Compensated summation limits rounding error, but browser floating-point arithmetic and extreme magnitudes can still affect the last digits. Results display up to 12 significant digits. All processing stays in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
How is weighted standard deviation calculated?
Compute the weighted mean, Σ(w × x) / Σw, then the weighted sum of squared deviations, Σ(w × (x − mean)²). Population variance divides that sum by total weight; sample variance divides by total weight minus 1. The standard deviation is the square root of the variance.
What do the weights represent?
This calculator treats weights as frequency counts, as if each value were repeated that many times. Zero weights are ignored. For reliability weights normalized to sum to one, the sample denominator differs, so use the population result or a reliability-weight formula instead.
Why is the sample result sometimes undefined?
The sample formula uses total weight minus 1 in the denominator. It is undefined when total weight is 1 or less. Add more positive weight to obtain a sample standard deviation.
Which values and weights are accepted?
Values may be positive, negative, or zero. Weights must be non-negative finite numbers with at least one positive weight. Enter up to 10000 pairs, one per line, with a comma, semicolon, or space between value and weight.
Is the calculation private?
Yes. Everything runs in your browser and no data is uploaded. Compensated summation improves accuracy, but floating-point limits can still affect extreme magnitudes and the last displayed digits.