Weighted Variance Calculator
Solve weighted variance problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Formula
Weighted Variance = Sum(w_i * (x_i - weighted mean)^2) / Sum(w_i)
Where w_i is the weight of each observation, x_i is the value, and the weighted mean = Sum(w_i * x_i) / Sum(w_i). This measures dispersion accounting for the relative importance of each data point.
Worked Examples
Example 1: Weighted Exam Score Variance
Problem: A student scores 85, 90, 78, 92, 88 on exams with weights 10%, 20%, 15%, 30%, 25%. Find weighted variance.
Solution: Weights: 0.10, 0.20, 0.15, 0.30, 0.25 (sum = 1.00)\nWeighted mean = 85(0.10) + 90(0.20) + 78(0.15) + 92(0.30) + 88(0.25)\n= 8.5 + 18 + 11.7 + 27.6 + 22 = 87.8\nWeighted variance = 0.10(85-87.8)^2 + 0.20(90-87.8)^2 + 0.15(78-87.8)^2 + 0.30(92-87.8)^2 + 0.25(88-87.8)^2\n= 0.10(7.84) + 0.20(4.84) + 0.15(96.04) + 0.30(17.64) + 0.25(0.04)\n= 0.784 + 0.968 + 14.406 + 5.292 + 0.010 = 21.46
Result: Weighted Mean: 87.80 | Weighted Variance: 21.46 | Weighted Std Dev: 4.63
Example 2: Portfolio Return Variance
Problem: Three assets have returns of 8%, 12%, 5% with portfolio weights 40%, 35%, 25%. Find the weighted variance of returns.
Solution: Weighted mean return = 8(0.40) + 12(0.35) + 5(0.25) = 3.2 + 4.2 + 1.25 = 8.65%\nWeighted variance = 0.40(8-8.65)^2 + 0.35(12-8.65)^2 + 0.25(5-8.65)^2\n= 0.40(0.4225) + 0.35(11.2225) + 0.25(13.3225)\n= 0.169 + 3.928 + 3.331 = 7.428\nWeighted std dev = sqrt(7.428) = 2.7255%
Result: Weighted Mean: 8.65% | Weighted Variance: 7.43 | Weighted Std Dev: 2.73%
Frequently Asked Questions
What is weighted variance and how is it different from regular variance?
Weighted variance is a measure of data dispersion that accounts for the relative importance or frequency of each data point through assigned weights. Regular (unweighted) variance treats all data points equally, computing the average of squared deviations from the mean. Weighted variance multiplies each squared deviation by its corresponding weight before averaging. The formula is: weighted variance = sum of (w_i times (x_i - weighted mean) squared) divided by the sum of all weights. This is useful when some observations are more reliable, more frequent, or more important than others. For example, in portfolio analysis, asset returns are weighted by their allocation percentages, giving larger positions more influence on the overall variance calculation.
How do you calculate the weighted mean?
The weighted mean is calculated by multiplying each value by its corresponding weight, summing all these products, and then dividing by the total sum of the weights. The formula is: weighted mean = (sum of w_i times x_i) / (sum of w_i). For example, if you have values 10, 20, 30 with weights 1, 3, 2, the weighted mean is (10 times 1 + 20 times 3 + 30 times 2) / (1 + 3 + 2) = (10 + 60 + 60) / 6 = 130 / 6 = 21.67. Notice how the value 20 has the most influence because it has the highest weight. The weighted mean is a prerequisite for computing weighted variance, as variance measures spread around this central value. It is commonly used in GPA calculations, survey analysis, and financial portfolio returns.
What is the difference between population and sample weighted variance?
Population weighted variance divides by the total sum of weights, assuming the data represents the entire population. Sample weighted variance applies a correction factor to account for the fact that a sample underestimates the true population variance. For frequency weights, the denominator becomes (sum of weights - 1) instead of (sum of weights). For reliability weights, the correction uses the formula: sum_w - (sum_w_squared / sum_w), known as the V1-V2 correction. The sample variance is always larger than the population variance because the correction factor in the denominator is smaller. In practice, use population variance when your data covers every member of the population, and sample variance when your data is a subset drawn from a larger population you want to make inferences about.
When should you use weighted variance instead of regular variance?
Weighted variance should be used whenever data points have different levels of importance, reliability, or frequency. Common scenarios include financial portfolio analysis where assets have different allocation percentages, survey data where respondents represent different population sizes, scientific measurements with varying precision levels, and grade calculations where assignments have different point values. In meta-analysis, study results are weighted by sample size or inverse variance to give more precise studies greater influence. In time series analysis, recent observations may be weighted more heavily than older ones. Using regular variance when weights are appropriate would give misleading results by treating a highly reliable measurement the same as an unreliable one, or a heavily invested asset the same as a minor holding.
What is the coefficient of variation and how does it relate to weighted variance?
The coefficient of variation (CV) is the ratio of the standard deviation to the mean, expressed as a percentage: CV = (standard deviation / mean) times 100. For weighted data, it uses the weighted standard deviation and weighted mean: CV = (weighted std dev / weighted mean) times 100. The CV provides a dimensionless measure of relative variability, making it useful for comparing the spread of datasets with different units or vastly different means. A CV of 20% indicates moderate variability, while a CV above 50% suggests high variability. For example, comparing the variability of stock returns (mean 10%, std dev 15%, CV = 150%) versus bond returns (mean 5%, std dev 3%, CV = 60%) shows stocks are relatively more variable even though the absolute comparison might suggest otherwise.
How do you interpret weighted standard deviation?
Weighted standard deviation is the square root of weighted variance and provides a measure of spread in the same units as the original data. It quantifies how far typical data points deviate from the weighted mean, accounting for the importance of each observation. A small weighted standard deviation indicates that the data points (especially heavily weighted ones) are clustered closely around the weighted mean. A large weighted standard deviation means the data is more spread out. In practical terms, for normally distributed weighted data, approximately 68% of the weighted observations fall within one weighted standard deviation of the weighted mean, and about 95% fall within two weighted standard deviations. This interpretation makes it intuitive for risk assessment in finance and quality control in manufacturing.