Preferential median of a set of intervals

If “ranking”, “Borda rule” or “Kemeny rule” are unfamiliar, start with the Theory page: the foundations of the method and a demo where you can vote yourself.

The problem

There are m results of measuring one quantity. Each has a centre x_k and a standard uncertainty u_k, i.e. an interval

We need one value — a consensus — and its uncertainty. The results may contradict each other: some are biased, others understate their uncertainty.

The idea in a minute

Each interval “votes”: it considers values inside it more plausible than values outside. The votes of all intervals are aggregated into a common ranking by the Borda rule. The winners are the values preferred by the largest number of intervals; their median is the preferential median (PM).

An important consequence: each interval casts exactly one vote, however narrow it is. A result with an understated uncertainty does not get a large weight, unlike in the weighted mean.

The algorithm step by step

Watch it live: press “Next”, and drag the intervals with the mouse at any step.

1. Range of actual values (RAV) A = {a₁ < a₂ < … < a_n} — n equally spaced discrete values from a₁ = min_k x_k^low to a_n = max_k x_k^up, where x_k^low = x_k − u_k and x_k^up = x_k + u_k are the interval's lower and upper bounds. Adjacent values are one partition norm h = (a_n − a₁) / (n − 1) apart. By default the number of values is chosen so that every interval contains at least one: n = max(⌈(a_n − a₁) / min_k 2u_k⌉, 11).
2. Interval-induced rankings. The interval I_k splits the RAV into A_k = {a_i ∈ A | a_i ∈ I_k} and the remaining values A \ A_k. Values in A_k are strictly preferred to the rest, and values within each group are tied: λ_k = A_k ≻ A \ A_k. Such a ranking is called an inranking (interval-induced ranking), and the set Λ(m, n) = {λ₁, …, λ_m} is the profile.
3. Aggregation by the Borda rule. The weighted n × n tournament matrix S = [s_ij] is built: s_ij = Σ_k=1^m C_ij^k, where C_ij^k = 2 if a_i ≻ a_j in λ_k; 1 if a_i ~ a_j; 0 if a_j ≻ a_i; s_ii = 0. The Borda score of a_i is the row sum z_i = Σ_j=1ⁿ s_ij. For profiles of inrankings the Kemeny and Borda rules give the same PM, so the polynomial Borda rule is used.
4. Winners. A_best = {a_i | z_i = max_j z_j}, p = |A_best|. If the winners' indices are not consecutive, the PM is undefined for this n: n is increased by one and steps 1–4 are repeated. In the implementations on this site n stops at 201; then the widest block of consecutive winners is taken.
5. Preferential median — the sample median of the winners sorted in ascending order: x_pm = a_(p+1)/2 for odd p and (a_p/2 + a_p/2+1) / 2 for even p (indices within A_best).

Fast computation

Let the coverage (summary indicator) c_i be the number of intervals containing a_i. Then

The first term does not depend on i. So the Borda winners are the values covered by the most intervals, and the score takes O(m·n) instead of O(m·n²) — without building the matrix S. Both forms give the same result; in the calculator the step-by-step breakdown shows c_i and z_i for your data.

Uncertainty of the result

The PM is one of the discrete RAV values or the midpoint of two adjacent ones, so its precision is limited by the partition norm. A conservative interval estimate is the bound ±0.5h. A GUM-style standard uncertainty follows if the error is approximated by a symmetric triangular distribution with half-width h:

The estimate is built from a finite sample, so the implementations on this site add in quadrature the standard error of the median of the interval centres — our addition, not part of the published method:

where MAD is the median absolute deviation of the centres x_k from their median; 1.4826·MAD is a robust estimate of the standard deviation, and 1.2533 = √(π/2) turns it into the standard error of the median.

Versions of the method

The method develops in two directions: which rule aggregates the profile (the exact but NP-hard Kemeny rule or the polynomial Borda rule) and whether the result is refined again.

β is the consensus ranking; u* is the bound of the interval uncertainty of the fusion result x*; W is the number of completed self-refinement cycles (set by the stopping condition); h₀ is the partition norm of the initial RAV. The Borda rule does not guarantee the exact consensus ranking β, but for profiles of inrankings it gives the same preferential median as the Kemeny rule. The standard uncertainty of the PM is u_pm ≈ 0.41h (see above).

Self-refinement

Cycles are numbered w = 1, 2, …; quantities of cycle w carry the index w, the initial computation is cycle 0.

1. A new RAV around the result. a₁^w = x*_w−1 − 0.5h_w−1, a_n^w = x*_w−1 + 0.5h_w−1; it is split into n_w = 11 values, so h_w = 0.1h_w−1.
2. A new profile. Intervals that do not intersect the new RAV are removed, the rest are renumbered (m = m_w) and represented again by inrankings λ_k^w, giving the profile Λ^w.
3. The refined result. The consensus ranking of the new profile gives the result x*_w. Cycles go on while the stopping condition holds — it compares the length of the new RAV with the lengths of its intersections with the intervals. After W cycles the result is x*_W with uncertainty ±0.5h_W = ±0.5·10^−Wh₀.

Notation: papers on the preferential median denote the result x_pm and its uncertainty u_pm; earlier papers and presentations denote the fusion result x* and its uncertainty u*.

Pseudocode

function preferential_median(x[1..m], u[1..m]):
    lo = x − u;  hi = x + u                   # bounds of the intervals I_k
    a1 = min(lo);  an = max(hi)               # bounds of the RAV
    n  = max(ceil((an − a1) / min(hi − lo)), 11)
    loop:
        h = (an − a1) / (n − 1)               # partition norm
        a[i] = a1 + (i − 1)·h,  i = 1..n
        c[i] = number of k with lo[k] ≤ a[i] ≤ hi[k]   # coverage
        A_best = { i : c[i] = max(c) }          # Borda winners
        if A_best is consecutive or n ≥ 201:
            (if not consecutive — the widest block of consecutive indices)
            x_pm = median(a[A_best])
            u    = sqrt((0.41·h)² + (1.2533·1.4826·MAD(x)/sqrt(m))²)
            return x_pm, u
        n = n + 1

This is Algorithm 2 of Muravyov et al., Int. J. Data Sci. Anal., 2026 (doi:10.1007/s41060-026-01222-6) in a shortened form. In the paper the winners are found from the row sums z_i of the tournament matrix S; here from the coverage c_i. This is the same: for inrankings z_i = Σ_k(n − |A_k| − 1) + n·c_i, and the first term is the same for all i. The limit n ≤ 201 and the sampling component of the uncertainty are additions of the implementations on this site; in the paper u_pm = 0.41h.

Ready implementations in five languages are on the Code page.

From norms to a schedule

IF&PA is also used where the result is not a measurement but a norm for planning. The chain is: history of works → labour norm → resource demand of every part of the work → a calendar plan of minimum duration. An error at the first step carries over into the plan, so the norm must be robust to a contaminated history.

1. A labour norm from the history

The norm b is how many shifts a unit of volume takes. For each observation the actual duration T_k and the volume done V_k are known; their ratio T_k/V_k is a partial estimate of the norm. The ratios are usually close together, but delays (bad weather, downtime, waiting for materials) give rare large values, always in the same direction. The mean and the estimators that start from the median are biased upwards by such one-sided contamination.

In IF&PA each ratio becomes an interval [T_k/V_k − u; T_k/V_k + u] with half-width u = κ·MAD, where MAD is the median absolute deviation of the ratios and κ is the width multiplier. The norm is the preferential median of these intervals. The norm depends on the month: in winter the same works take longer, so it is estimated separately for every “work × month” pair.

2. A month is a knapsack problem

A work is split into packages — parts on sections of the route. Package j needs r_j = b_j·V_j shifts. In a month the crews can work R shifts — the capacity of the “knapsack”. The decision x_j = 1 if the package is taken into the month:

There are usually several resources — crews, machinery, materials — and the packages must fit each of them: Σ_j r_j^lx_j ≤ R^l for every resource type l. This is the multidimensional knapsack. The problem is NP-hard, but for sizes typical of a project it is solved exactly by branch and bound.

3. A project is a chain of month-knapsacks

A calendar plan is a set of knapsacks, one for every month t = 1, …, H, linked by the order of works. The decision x_jt = 1 puts package j into month t; the demand r_jt^l depends on the month through the seasonal norm. The smallest duration

and the order is kept: a package of the next work on a section not before the previous one. Feasibility for a given H is checked by an integer solver, and the smallest H is found by enumeration or bisection from the resource lower bound.

4. Why an accurate and unbiased norm matters

• An understated norm. More work “fits” into a month than the crews can actually do: the plan is shorter, but in execution the months are overloaded and the deadline is missed.
• An overstated norm. The plan is feasible but longer than necessary — a hidden reserve nobody sees or uses.

So the norm estimator is chosen not only by its spread but also by its bias. A step-by-step explanation with the knapsack is in the interactive lecture; the full model with several resources, seasonal norms and an exact solver is in the Planner.

Notation

How the PM differs from similar estimators

Which estimator is more accurate on which data — check in the applicability configurator.

Primary sources

• Muravyov S.V., Khudonogova L.I., Emelyanova E.Yu. Interval data fusion with preference aggregation. Measurement, 2018, 116, 621–630. doi:10.1016/j.measurement.2017.08.045
• Muravyov S.V., Khudonogova L.I., Ho M.D. Analysis of heteroscedastic measurement data by the self-refining method of interval fusion with preference aggregation – IF&PA // Measurement. 2021. Vol. 183. Art. 109851. DOI: 10.1016/j.measurement.2021.109851.
• Muravyov S.V., Khudonogova L.I., Shabramov V.S., Ignatyev V.D. Comparative studies of preferential and weighted medians accuracy in robust estimation of central tendency // International Journal of Data Science and Analytics. 2026. Vol. 22, No. 1. Art. 262. DOI: 10.1007/s41060-026-01222-6.
• Muravyov S.V., Khudonogova L.I., Shabramov V.S., Ignatyev V.D., Andreyev D.I. Kemeny and Borda rules in constructing central tendency estimators based on the preferential median. Proc. AISP'25, 2025. DOI: 10.1109/AISP68263.2025.11396113.

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