Weighted Variance Calculator
Solve weighted variance problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Weighted Variance Calculator
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Formula: Weighted Variance = Sum(w_i * (x_i - weighted mean)^2) / Sum(w_i)
Worked example — Weighted Mean: 87.80 | Weighted Variance: 21.46 | Weighted Std Dev: 4.63
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) Weighted mean = 85(0.10) + 90(0.20) + 78(0.15) + 92(0.30) + 88(0.25) = 8.5 + 18 + 11.7 + 27.6 + 22 = 87.8 Weighted 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 = 0.10(7.84) + 0.20(4.84) + 0.15(96.04) + 0.30(17.64) + 0.25(0.04) = 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% Weighted variance = 0.40(8-8.65)^2 + 0.35(12-8.65)^2 + 0.25(5-8.65)^2 = 0.40(0.4225) + 0.35(11.2225) + 0.25(13.3225) = 0.169 + 3.928 + 3.331 = 7.428 Weighted 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.
What types of weights are commonly used in weighted variance calculations?
There are several types of weights used depending on the application context. Frequency weights represent how many times each value occurs in the dataset, essentially expanding the dataset. Reliability weights (also called analytic or inverse variance weights) indicate the precision or importance of each observation. Probability weights account for unequal sampling probabilities in survey designs. Normalized weights sum to 1 and represent proportions. In finance, portfolio weights represent the percentage allocation to each asset. In physics, measurements might be weighted by the inverse of their measurement error squared (inverse variance weighting). The choice of weight type affects whether you should use the population or sample variance formula and which correction factor to apply for unbiased estimation.
Can weighted variance be negative?
No, weighted variance can never be negative when calculated correctly. Since variance is computed as the weighted sum of squared deviations from the mean, and squares are always non-negative, and weights must be non-negative, the entire sum is guaranteed to be non-negative. The only way to get a variance of exactly zero is if all data values are identical (every deviation from the mean is zero). If you encounter a negative variance in calculations, it typically indicates a programming error, an incorrect formula implementation, or the use of negative weights which are not standard. Negative weights can arise in certain advanced statistical techniques like contrast weights, but these require special handling and different variance formulas. For standard weighted variance applications, always ensure all weights are positive.
How does weighted variance apply to financial portfolio risk?
In finance, weighted variance is used to calculate portfolio risk (volatility). Each asset in a portfolio has a weight equal to its proportion of total investment. The portfolio variance accounts not only for individual asset variances weighted by their allocation but also for the covariances between all pairs of assets. The simplified formula for two assets is: portfolio variance = w1 squared times var1 + w2 squared times var2 + 2 times w1 times w2 times cov(1,2). For multiple assets, this extends to a matrix calculation involving the covariance matrix. This framework, developed by Harry Markowitz in Modern Portfolio Theory, shows that diversification (spreading weights across uncorrelated assets) reduces overall portfolio variance. Portfolio managers optimize weight allocations to minimize variance for a given expected return level.
What are common mistakes when computing weighted variance?
Several common errors occur in weighted variance calculations. First, forgetting to use the weighted mean instead of the unweighted mean leads to incorrect squared deviations. Second, confusing the denominator between population (sum of weights) and sample (sum of weights minus correction) formulas produces biased estimates. Third, not normalizing weights when they should sum to 1 can cause scaling errors. Fourth, using negative weights without proper handling violates the assumptions of standard variance formulas. Fifth, mixing up frequency weights and reliability weights leads to incorrect Bessel correction application. Sixth, rounding intermediate calculations too early introduces cumulative error. Always verify your calculation by checking that the variance is non-negative, the standard deviation squared equals the variance, and the results make intuitive sense given the data spread.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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