Range vs IQR: which spread to report

Range vs IQR: the short answer

Both are measures of spread, and they differ in which part of the data they read. The range is one subtraction, the maximum minus the minimum, so two observations decide it and everything between them is invisible to it. The interquartile range, or IQR, is $Q_3 - Q_1$, the width of the middle 50% of the sorted data, so the extremes are exactly what it leaves out.

Two consequences follow, and they point the same way. First, the IQR is resistant and the range is not: raise the largest value by 90 units and the range rises by exactly 90 while the IQR does not move at all. Second, and less familiar, the range grows with the sample size on its own. A bigger sample gives the tails more chances to produce a far-out value, so two ranges computed from samples of different sizes are not comparable even when both samples came from the same population. The IQR has a fixed target and settles onto it rather than drifting, which is why it is the spread you can defend once you are past a handful of values.

What the range measures

The range is the distance from the smallest observation to the largest:

It carries the units of the data, it can never be negative, and it is a single number. "The range is 22 to 130 seconds" reports the minimum and the maximum, which is a different statement; the range is 108 seconds.

Only those two observations enter the subtraction. The values in between could sit in two clumps at the ends or spread evenly across the middle, and the range would report the same number either way. That is why topic 1.7 of the AP course lists the range among the nonresistant measures of variability, next to the standard deviation. The IQR and the median are the resistant pair, as what is a resistant statistic sets out in full.

The range still earns a place as context. It is the fastest spread to compute, and quoting it next to the IQR tells a reader how far the tails reach beyond the middle half. Trouble starts when it is asked to do a job it cannot do: summarize typical variability, or compare two data sets that do not have the same number of values.

What the IQR measures

The IQR is the width of the middle half of the data:

$Q_1$ ("Q one") is the first quartile, the value about a quarter of the way through the sorted data, and $Q_3$ ("Q three") is the third quartile, about three quarters of the way through. This site locates them with the median-excluded convention that the TI-84 uses, which how to find quartiles by hand works through step by step.

Both quartiles are themselves medians, of the lower half and the upper half, so they answer to positions rather than to sizes. Increase the maximum and the largest value is still the largest, the order inside the upper half is unchanged, and $Q_3$ stays exactly where it was. The IQR therefore cannot be moved by pushing an existing value further out at either end. Adding observations is a different matter, because a new value shifts every position in the sorted list beneath it. That resistance is what makes the IQR, and not the range, the quantity that sets the fences in the 1.5 times IQR outlier rule and draws the box in a boxplot.

The resistance is not unlimited. A value pulled outward from inside one of the halves can nudge the quartile on that side, and if a large share of the data sits far out on one side, that quartile shifts in earnest. What no single value at the top or the bottom can do is change it.

The differences side by side

Notice what the second row does not say. Neither of these reads the whole data set the way the standard deviation does: both are computed from a handful of positions in the sorted list. The difference is which positions. The range takes the two that are most exposed to whatever the tail happens to produce, and the IQR takes two that pushing an existing value further out cannot reach.

Why the range grows with the sample size

This is the property that decides most real comparisons, and the one students meet least often.

Adding an observation can never shrink the range. The new value either lands between the current minimum and maximum, leaving the range alone, or lands outside them, widening it. So as data accumulate the range can only hold still or grow, and the more values you draw, the more chances a far-out one has to appear. On average, a large sample reports a larger range than a small sample from the same population, with nothing about the population having changed.

To see the size of the effect, hold the population fixed: bag fill weights following a normal model with mean 300 grams and standard deviation 10 grams. Drawing 200,000 samples at each size and averaging gives this.

From 10 values to 500 the average range roughly doubles, 30.8 grams to 60.7 grams, while the average IQR stays between 13.1 and 13.5 grams. The IQR has a fixed target and converges on it: for a normal model the population IQR is $1.349\sigma$ ("sigma", the population standard deviation), here 13.49 grams, and the average IQR in the last four rows is within 1% of it every time. The range has no target to converge on. Push the simulation further and it keeps climbing, to about 64.8 grams at 1000 values and 80.4 grams at 20,000.

The first row is the one place in the table where the IQR is noticeably off its target, and the reason is mechanical. With five values the median-excluded rule leaves two values in each half, so $Q_1$ is the average of the two smallest and $Q_3$ the average of the two largest. At that size the IQR is partly built out of the extremes themselves and inherits some of the range's behavior: 16.6 grams against a target of 13.49 is about 23% high.

Read that middle column as what it is, an average over 200,000 samples rather than a promise about the one sample you have. At 20 values a single sample IQR has a standard deviation of about 3.3 grams around that average, so it lands within 1% of 13.49 grams only about 3 times in 100; at 500 values the standard deviation is down to about 0.7 grams. Where the average sits and how far one sample strays from it are separate questions, and only the first is what this table answers. The approach is not perfectly smooth either, because the median-excluded rule reads different positions depending on whether $n$ is odd or even: the average IQR is about 1% low at $n = 20$ and about 5% high at $n = 21$.

One line of practical advice comes out of the table: do not compare the range of one data set to the range of another unless the two have the same number of values. Compare IQRs instead. Their dependence on the sample size is a bias that shrinks toward zero, while the range's grows without limit.

When to use which

Report the IQR when you need a measure of spread you can defend. It pairs with the median, it survives outliers, and it compares across data sets of different sizes far better than the range does. It is not entirely free of the sample size: with five values the median-excluded rule builds the quartiles out of the extremes themselves and the average IQR runs about 23% high. What it does not do is grow without limit the way the range does. On a free-response question, a distribution that is skewed or has an outlier calls for the median and the IQR, and the reason belongs on the page next to the choice, not just the choice.

Use the range as context rather than as the answer. It makes a good second sentence: "the middle half of the bags spans 14 grams, and the full set spans 38 grams" tells a reader how the bulk behaves and how far the tails reach. It makes a poor first sentence, because one mistyped value or one unusual bag can set it by itself.

The range does have a narrow job it does better than the IQR, and it is worth naming: questions about the worst case. A tolerance check on a machine, or the gap between the coldest and warmest forecast, is a question about the extremes, and the range answers it directly while the IQR deliberately hides it. That is a question about reach, not about typical variability.

When a distribution is roughly symmetric with no outliers, the live choice is not between range and IQR at all. It is between the IQR and the standard deviation, which uses every value and feeds the z-scores and inference procedures later in the course. That comparison is standard deviation vs IQR.

The classic mix-ups and how to avoid them

Three errors do most of the damage.

Reporting the range as an interval. "The range is 283 to 321 grams" is ordinary English and the wrong statistic. The range is one number, 38 grams. Two numbers are the minimum and the maximum, which a boxplot already shows.

Comparing ranges across samples of different sizes. A class of 30 will usually report a wider range of heights than a class of 12, even if the two classes are alike in every respect. The comparison measures the sample sizes as much as the spreads. Comparing IQRs takes almost all of that away.

Reading a large range as high variability. One value can create it, and so can a large sample. Compute both, then judge the ratio against what your sample size produces on its own: the table two sections up divides out to about 2.3 at ten values, 3.7 at a hundred, and 4.5 at five hundred, and that is a clean normal population with no outlier anywhere in it. A ratio well above the figure for your own $n$ is the signal that the tails, not the bulk, are doing the work, and the 1.5 times IQR rule is what tells you whether the value at the end is formally an outlier.

The habit that prevents all three is to quote the IQR first and the range second, with units on both. The five-number summary calculator gives you the minimum, $Q_1$, the median, $Q_3$, and the maximum in one step, so both spreads are one subtraction away, and dragging a point outward in the descriptive statistics sandbox holds the IQR readout and the box still while the point walks past the fence and gets circled as an outlier.