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Weighted Standard Deviation Calculator

Enter values and matching weights to find the weighted standard deviation.

How it is calculated

You enter a list of values xi and a list of weights wi of the same length, separated by commas or spaces. The weighted mean is x̄w = Σ wixi ÷ Σ wi. The weighted variance is Σ wi(xi − x̄w)² ÷ Σ wi, and the weighted standard deviation is its square root. The divisor is the sum of the weights, not that sum minus one, so this is the population-style figure with no small-sample correction. The two lists must have equal length, no weight may be negative, and the weights must add up to more than zero; otherwise there is no result. A weight of 0 simply removes that value from the calculation, and all weights equal gives the ordinary population standard deviation.

Worked example

Values 70, 80, 90 with weights 1, 2, 3. The weights add up to 6. Weighted mean: (1×70 + 2×80 + 3×90) ÷ 6 = 500 ÷ 6 = 83.33. Deviations squared and weighted: 1×(70 − 83.33)² = 177.78, 2×(80 − 83.33)² = 22.22, 3×(90 − 83.33)² = 133.33, total 333.33. Variance: 333.33 ÷ 6 = 55.56. Standard deviation: √55.56 = 7.45. Result: weighted standard deviation 7.45 around a weighted mean of 83.33.

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Common mistakes

  • Giving a different number of weights than values. Every value needs exactly one weight, and a mismatch returns no result.
  • Using percentages and expecting them to matter. Weights of 10, 20, 30 and 1, 2, 3 give the same answer, because only the ratios between weights count.
  • Expecting the sample correction of n − 1. The tool divides by the sum of the weights, so for a small sample the figure is lower than a sample standard deviation.
  • Entering a negative weight to mean less trust. A weight must be zero or positive; a negative one is rejected and shows no result, as do weights that are all zero.
  • Separating numbers with a decimal comma, such as 2,5, inside the list. The comma separates items, so write 2.5 for a decimal.

About this calculator

The weighted standard deviation measures spread around a weighted mean rather than a plain average, giving each data point influence in proportion to a weight you assign it, a reliability score, sample size, or measurement precision, instead of treating every value as equally trustworthy.

This matters whenever the data points being combined aren't equally reliable: meta-analyses in research combine results from studies of different sizes by weighting larger, more precise studies more heavily; survey analysts weight responses to correct for over- or under-sampled demographic groups; and lab scientists combine repeated measurements with different precisions using weights based on their measurement uncertainty. Using an unweighted standard deviation in these cases would let noisy or unreliable data points distort the result just as much as trustworthy ones.

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