(min + max) ÷ 2, with an honest account of what it ignored.
Two numbers, whatever n is
The midrange is 105.5 — further from the data than every observation except one. Nine of eleven values were ignored.
min 11.000000, max 200.000000, n = 11
Midrange = 105.500000
Computed from exactly 2 of your 11 values. The other 9 — 81.8% of the sample — could be any numbers between the min and the max without changing this answer.
Midrange
105.500000
(min + max) / 2
Mean
30.454545
all n values
Median
14.000000
the middle value
Range
189.000000
max − min
Values used
2 of 11
min and max only
Sample ignored
81.8%
no effect on the answer
Breakdown point
0%
one value can move it anywhere
Midrange − median
+91.500000
how far the extremes pulled it
Multiply the largest value by 10.000000
Midrange becomes
1,005.500000
was 105.500000
Mean becomes
194.090909
was 30.454545
Median becomes
14.000000
unchanged
Midrange moved by
900.000000
exactly half the corruption
The midrange absorbs exactly half of whatever you do to the largest value, because that value is half the formula. Push it to a billion and the midrange follows, linearly, forever. The mean dilutes the damage by n; the median does not move at all until the corruption reaches the middle of the sample.
Where the midrange wins
1,200 simulated samples of 20 from each population, with a known centre. Lower error is better; 1.00 marks the winner of each row.
Relative error of four location estimators under three population shapes
Population
Mean
Median
20% trimmed
Midrange
Uniformflat with hard bounds
1.899×
3.046×
2.484×
1.000×
Normalthe usual assumption
1.000×
1.207×
1.062×
1.773×
5% contaminated1 in 20 from a ten-times-wider normal
2.259×
1.126×
1.000×
15.229×
The ranking changes completely between rows. On uniform data the midrange wins outright; on normal data the mean does; on contaminated data the trimmed mean does and the midrange is an order of magnitude behind. No estimator is best everywhere, and “robust” is a property relative to a population rather than a virtue in itself.
For a uniform population the midrange beats the sample mean by a factor of 3.8500 at n = 20. That figure is exact, not simulated: the variance ratio is (n+1)(n+2)/6n, so the advantage grows without bound as the sample grows. It is 1.25 at n = 4, 3.85 at n = 20 and 17.17 at n = 100. The midrange is the minimum-variance unbiased estimator of a uniform centre, which is the one situation where it is not just defensible but optimal.
Everywhere else, its breakdown point is zero. Every other location estimator survives some fraction of arbitrary corruption: the median survives just under half, a 20% trimmed mean survives 20%. The midrange survives none. A single bad reading — a sensor glitch, a typo, a decimal point in the wrong place — moves it as far as that reading goes, and no sample size protects you.
What this tool shows
The midrange reads exactly two of your values, whatever n is. On the first example that means 81.8% of the sample has no effect on the answer. The tool says so, shows what one corrupted reading does to it against the mean and median, and then simulates the one population where it beats both — a uniform, where it wins by a factor of 17 at n = 100.
The midrange, with the mean, median and range beside it
How many of your values entered the answer, and what fraction did not
A corruption test: multiply the maximum and watch all three estimators
A seeded simulation of four estimators across three population shapes
The exact uniform-population advantage, (n+1)(n+2)/6n
The breakdown point, which for this estimator is zero
Two values used Breakdown point 0 Uniform MVUE Simulated ranking
Updated 12 September 2026 · Works in any browser, no installation
The midrange is (min + max) ÷ 2 — the point exactly halfway between the smallest and largest observations. It is the fastest measure of centre to compute and the least resistant to a bad value there is, because the two numbers it uses are the two most likely to be wrong. The median is its opposite in every respect.
At a glance
Formula shown
Midrange = (minimum + maximum) ÷ 2. The count of observations appears nowhere in the formula, so a sample of 10,000 and a sample of 2 are treated identically — only the extremes are read. Its breakdown point is 0: the fraction of arbitrary corruption it survives is none, and moving the maximum by d moves the midrange by d/2. For a Uniform(a, b) population, Var(midrange) = (b−a)²/(2(n+1)(n+2)) against Var(mean) = (b−a)²/(12n), so the mean’s variance is (n+1)(n+2)/6n times larger.
Scenario support
A daily mean temperature, which is the midrange of the high and low by definition; the centre of a bounded measurement range; the midpoint of a class interval in grouped data; quick mental arithmetic on a small clean set; and teaching what a breakdown point means.
Educational estimate
Planning support from the values you enter — not professional advice.
Two values, whatever n is
The formula contains no n. That is not a simplification — it is the whole behaviour of the statistic, and it is the thing every definition leaves you to notice for yourself.
On the first example, eleven values go in and two are read. The other nine — 81.8% of the sample — could be replaced by any numbers between the minimum and the maximum without changing the answer by a thousandth. The tool prints that fraction rather than leaving it implied.
So collecting more data does not improve a midrange in the way it improves a mean. It does something stranger: a larger sample pushes the observed minimum and maximum further apart, which for most populations makes the estimate WORSE. The mean converges as n grows; the midrange wanders further out.
Unless the population has hard bounds. Then the extremes converge onto those bounds instead of running away, and everything reverses — which is the section below and the one real case for this statistic.
Compare the alternatives on the same data. The mean reads all n values, the median reads one or two but chooses them by rank rather than by extremity, and a 20% trimmed mean reads the middle 60%. Only the midrange picks its two values precisely because they are the furthest out.
A breakdown point of zero, demonstrated
The breakdown point of an estimator is the fraction of the sample that can be replaced by arbitrary values before the result can be dragged anywhere at all. The median’s is just under 50%. A 20% trimmed mean’s is 20%. The mean’s is 0%, and the midrange’s is 0% in a worse way.
Multiply the largest of eleven values by ten and watch all three. The midrange goes from 105.5 to 1005.5. The mean goes from 30.45 to 194.09. The median does not move.
The midrange absorbs exactly half of the corruption, every time. Move the maximum by d and the midrange moves by d/2, because that value is half the formula. The mean at least divides the damage by n; the midrange divides it by 2 regardless of how many clean observations surround it.
And it is linear without limit. The corruption factor in the tool goes to a million; the suite checks the relationship out to a billion times the original value. There is no sample size at which a single bad reading stops mattering.
Which makes the failure modes ordinary rather than exotic. A decimal point in the wrong place, a sensor spike, a placeholder like 9999 left in a column, a unit mix-up in one row: each of these moves a midrange by half its magnitude, and none of them is visible in the output.
The one population where it is the best estimator there is
Everything above argues against the midrange. This section is why it is still in the textbooks, and it is a stronger argument than most people who dismiss the statistic realise.
For a uniform population, the midrange is the minimum-variance unbiased estimator of the centre. Not merely acceptable — optimal. Nothing unbiased does better, including the sample mean.
The advantage is exact and it grows. Var(mean) ÷ Var(midrange) = (n+1)(n+2) ÷ 6n, which is 1.25 at n = 4, 3.85 at n = 20, and 17.17 at n = 100. Linear in n, so the gap widens without bound as the sample grows.
The formula is checked against simulation rather than quoted. Twenty thousand simulated uniform samples of 20 give a variance ratio within a few per cent of the theoretical 3.85.
The mechanism is the one from the section above, running the other way. A uniform distribution has hard edges, so the sample minimum and maximum converge onto them as n grows, and they carry far more information about the centre than the interior points do. The extremes are the informative observations, not the dangerous ones.
The crossover is between n = 3 and n = 4. At n = 3 the mean is still ahead; from n = 4 the midrange wins on uniform data and never gives the lead back.
The ranking is a fact about populations, not about estimators
The simulation panel runs four estimators against three population shapes with a known centre. Reading the rows against each other is the point.
On uniform data the midrange wins outright and the sample mean is roughly twice as far from the truth. The median, which throws away the informative extremes, is the worst of the four.
On normal data the mean wins and the midrange is well behind, because a normal distribution has no bounds for the extremes to converge onto — they simply drift outward with n.
On 5%-contaminated data the 20% trimmed mean wins and the midrange is an order of magnitude behind everything else. One observation in twenty drawn from a ten-times-wider normal is enough to make it useless.
So “robust” is not a virtue an estimator has. It is a relationship between an estimator and a population. Each of these four is the best choice for one of these three rows and a poor choice for another, and no amount of data tells you which row you are in — that is a judgement about where your numbers came from.
The simulation is seeded, so it is reproducible. The same sample size gives the same table on every visit, and the verification suite checks both that this is true and that a different seed gives different numbers — so it is genuinely simulating rather than returning a memorised constant.
Where it is the right answer anyway
Three cases where the midrange is not a compromise, and one where it hides in plain sight.
Daily mean temperature IS a midrange. The figure in almost every climate record is (daily high + daily low) ÷ 2, not the average of hourly readings. That is a midrange by definition, it has been the convention for well over a century, and the continuity of the record is worth more than the improvement a different formula would bring.
Class midpoints in grouped data are midranges. When only intervals and counts survive, each interval is represented by the midpoint of its bounds. Every statistic computed from grouped data rests on a pile of midranges, which is why grouped estimates go wrong when observations pile against one edge of a class.
Bounded instruments and specifications. A gauge reading between hard limits, a tolerance band, a graded scale with a floor and a ceiling: these are the uniform-like cases where the extremes carry the information, and the midrange is doing real work.
And it is half of a statistic people use without noticing. The range is max − min and the midrange is (max + min) ÷ 2 — the same two numbers, subtracted instead of added. Anyone quoting a range has already computed everything the midrange needs.
Reporting a midrange without misleading anyone
Four habits, and the first one does most of the work.
Call it the midrange. Not “the average”. A reader who sees “average” will assume the arithmetic mean, and on skewed data the two are far apart in a way that changes conclusions.
Give the minimum and maximum beside it. They are the entire input. Printing them costs two numbers and lets a reader reconstruct and check the result, which no other measure of centre allows.
Say why, if the data is not bounded. On an unbounded population the midrange needs a justification — convention, continuity with a historical series, or genuine constraints on the instrument. “It was quick” is not one.
Pair it with a robust check. The median or a trimmed mean beside it costs nothing and reveals immediately whether the extremes are pulling the answer. A large gap between the midrange and the median is a signal to look at the data, not a result to report.
And never use it for screening or control. Anything that has to detect an unusual value cannot be built on a statistic that a single unusual value controls.
Method. The uniform-population advantage is the exact variance ratio (n+1)(n+2)/6n rather than a simulated approximation, and the suite checks that closed form against 20,000 simulated uniform samples. The corruption test is verified as a linear relationship out to a billion times the original value, alongside the median staying fixed across the same sweep. The efficiency table uses a seeded generator so it is reproducible on every render and inside the suite, which checks both that the same seed gives identical numbers and that a different seed does not — the second half being what proves it is simulating rather than returning a constant. That engine is verified on every change against 130 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Mean, Median and ModeAll three centres marked on your own data, every mode rather than just the first, and the mean-median gap read as a direct measure of skew.
Trimmed MeanTrimmed and Winsorized means at any fraction, plotted across the full sweep — because a 0% trim is the arithmetic mean and a 50% trim is exactly the median, so the trim fraction is a dial between them.
Standard DeviationSample and population standard deviation, plus variance, mean, median, quartiles, z-scores, outliers, and confidence intervals.
OutlierRuns four outlier rules together and says when each is failing — the 3-sigma rule cannot flag anything below n=11, two outliers hide each other, and the quartile convention alone can decide the answer.
VarianceSample and population variance from your data, with a live simulation that shows exactly how much the wrong divisor costs — 20% low at n = 5, closing as the sample grows.
PercentileThe value at any percentile and the percentile rank of any value, with all nine standard methods — and the round trip that shows the two directions are not inverses.
An educational tool. The midrange has a breakdown point of zero — one erroneous reading moves it by half that error’s size regardless of sample size — so it should not be used on data of uncertain provenance, and never for screening or process control.
Published a midrange tool that reports how many of your values actually entered the answer — always two, whatever n is — and what fraction of the sample therefore had no effect. On the shipped example that is 81.8%.
Demonstrated a breakdown point of zero directly: multiplying the largest of eleven values by ten moves the midrange from 105.5 to 1005.5, the mean from 30.45 to 194.09, and the median not at all. The midrange absorbs exactly half of whatever is done to the maximum, linearly, verified out to a billion times the original.
Added the fact that makes the midrange worth keeping: for a uniform population it is the minimum-variance unbiased estimator of the centre, beating the sample mean by exactly (n+1)(n+2)/6n — 1.25 at n = 4, 3.85 at n = 20, 17.17 at n = 100. The exact formula is checked against 20,000 simulated samples.
A seeded simulation compares four estimators across three population shapes at the user's chosen n. The ranking reverses completely: the midrange wins on uniform data, the mean on normal data, and the 20% trimmed mean on 5%-contaminated data where the midrange is an order of magnitude behind.
The simulation is reproducible by construction — identical seed, identical table — and the suite verifies that a different seed gives different numbers, so it is genuinely simulating rather than returning a constant.
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