Weighted Percentiles and Interquartile Ranges

Why Weighted Percentiles Matter

A survey’s weighted mean and standard deviation summarize location and spread, but they can be sensitive to extreme observations. Percentiles provide another view: a percentile identifies a point in the outcome scale below which a specified share of the estimated population lies. The interquartile range (IQR), the distance between the 75th and 25th percentiles, summarizes the spread of the central half of that distribution.

In a survey, observations can represent different numbers of population members. A percentile calculated without weights describes the distribution of the observed sample records; a weighted percentile instead describes the estimated population distribution represented by those records and weights. This tutorial uses a precise inverse-cumulative-distribution convention, so the calculation is reproducible even when weights are unequal or outcomes are tied.

Weighted Percentiles

To calculate it, sort the records by outcome, add their weights cumulatively, and find the first outcome at which the cumulative weight reaches at least \(p\) of the total weight. Equivalently, if weights are normalized to sum to 1, find the first cumulative normalized weight at least \(p\). The cutoff is compared with the fraction of total weight, not with the fraction of records.

This inverse convention returns an observed outcome, rather than interpolating between adjacent values. Other quantile conventions may interpolate, so software and reports should identify the convention when exact agreement matters. With this convention, a tied outcome contributes its full weight at that outcome: the estimated CDF jumps by the combined weight of all records with that value. The cutoff may fall within a jump, but the estimated percentile remains the outcome where the jump occurs.

Interquartile Range and Central Asymmetry

The IQR describes the spread of the estimated distribution’s middle half. It does not say that exactly half of the weighted records lie strictly between the quartiles: ties at either quartile can place weight on the boundary, and the estimated CDF can jump over the percentile cutoff. The IQR is nonnegative because the quantile definition orders the quartiles, so \(\widehat{Q}_1\leq\widehat{Q}_3\).

Quartiles can also describe whether the middle half is balanced around the median. Compare the distance above the median, \(\widehat{Q}_3-\widehat{Q}_2\), with the distance below it, \(\widehat{Q}_2-\widehat{Q}_1\). A longer upper distance indicates that the upper half of the central interval is more spread out; a longer lower distance indicates more spread below the median. A standardized summary of this comparison is Bowley’s quartile coefficient of skewness:

When the IQR is positive, \(\widehat{B}_w\) lies between \(-1\) and 1. A positive value indicates a longer upper quartile distance from the median; a negative value indicates a longer lower distance. A value near zero says the two distances are similar, not that the entire distribution is symmetric. This is a descriptive measure of the central half only; it does not characterize the tails or establish a population shape by itself.

A Short Justification of the Cumulative-Weight Rule

The calculation rule follows directly from the definition. After sorting the observed values, let \(x_{(1)}\leq\cdots\leq x_{(n)}\) denote the ordered outcomes, with their corresponding weights \(w_{(i)}\). At an observed value \(x_{(j)}\), the estimated CDF includes the weights of all observations at or below that value. If values are distinct, this is the cumulative weight through position \(j\), divided by total weight. Therefore,

The generalized inverse selects the first \(x_{(j)}\) for which this ratio is at least \(p\). Multiplying both sides of the cutoff condition by the positive total weight gives

That inequality is exactly the cumulative-weight rule. If several records share \(x_{(j)}\), their weights are all included at that value; grouping tied records or retaining their separate rows gives the same CDF jump and percentile.

Interpretation, Scope, and Common Pitfalls

A weighted percentile is a point estimate based on the observed outcomes and their survey weights. It is meaningful for the intended population only when the records included, the outcome values, and the weights are appropriate for that target. Before calculating, specify the analytic domain and denominator: records missing the outcome or outside the target group should not silently be treated as valid observations with zero values. The total in the weighted empirical distribution is the sum of weights for the eligible records used in that calculation.

The IQR and quartile asymmetry coefficient are descriptive summaries, not measures of sampling uncertainty. Their numerical values alone do not provide a standard error, confidence interval, or test of a population hypothesis. In complex surveys, uncertainty requires methods that account for the sampling design; the weighted point estimate should not be mistaken for a guarantee that the corresponding population percentile is known exactly.

• Using record counts as the cutoff. The wrong reasoning is, “The 75th percentile is at the fourth of five records, regardless of weights.” The correction is to locate 75% of the total eligible weight, which can occur at a different ordered value.
• Interpolating without stating a convention. The wrong reasoning is to assume every percentile must lie between two observed values. Under the generalized-inverse convention used here, the estimate is the first observed value whose weighted CDF reaches the target; interpolation is a different convention.
• Calling the IQR a full measure of distribution shape. The IQR describes central spread, and Bowley’s coefficient compares the two central quartile distances. Neither captures all tail behavior or proves overall symmetry.
• Reporting a number without context. State the outcome, the population or domain, the percentile convention when relevant, and the units. For example, “the estimated weighted IQR of weekly exercise time is 6 hours” is more informative than “the IQR is 6.”

These summaries are useful when a population’s typical range and central balance matter more than its mean alone. The IQR is resistant to a small number of extreme values, while the quartile distances show whether the upper or lower side of the middle half extends farther from the median. Interpreting them alongside other summaries gives a compact, weight-aware account of distributional spread without implying more detail than three quartiles can support.