The Interquartile Range as Middle-Half Spread

Introduction

A summary of a data set should tell us not only where observations tend to fall, but also how spread out they are. The interquartile range, usually shortened to IQR, describes spread by focusing on the span from the first quartile to the third quartile. It is useful when the question is specifically about how wide the middle-half portion of the data is.

The IQR is a difference between two quartiles, not a measure of the location of the data. Its value is reported in the same units as the observations. To calculate it, first obtain \(Q_1\) and \(Q_3\) using a stated quartile convention; then subtract the first quartile from the third.

Finding and Interpreting the IQR

Quartiles divide ordered data into four sections. The first quartile marks the lower-quarter boundary, and the third quartile marks the upper-quarter boundary. The interval from \(Q_1\) to \(Q_3\) spans the middle half of the distribution. For a finite data set, the exact quartile values depend on the convention used to calculate them. Use the convention specified for the data, and do not switch conventions partway through a calculation.

The IQR calculation itself is simple once the quartiles are known: subtract the lower endpoint from the upper endpoint. The result is a distance, so it cannot be negative when the quartiles are correctly ordered. For example, if a time measurement has \(Q_1=12\) minutes and \(Q_3=19\) minutes, the interval from the quartiles is 7 minutes wide.

Why Subtracting the Quartiles Gives a Width

The definition follows from measuring the distance between two points on a number line. If an interval begins at a lower endpoint \(a\) and ends at an upper endpoint \(b\), its width is \(b-a\). The quartile interval begins at \(Q_1\) and ends at \(Q_3\), so its width is \(Q_3-Q_1\).

Because the first quartile is at or below the third quartile, \(Q_1\leq Q_3\). Subtracting the lower endpoint from the upper endpoint therefore gives a nonnegative width. This also explains why reversing the subtraction would not give the IQR: \(Q_1-Q_3\) would be zero or negative, rather than the distance from the first quartile to the third.

The quartiles indicate the endpoints of the middle-half span, while the IQR reports only its width. Knowing the width does not, by itself, identify where that span is located. For instance, an IQR of 6 minutes could describe a quartile interval from 4 to 10 minutes or from 20 to 26 minutes.

Comparing Middle-Half Spans

When two data sets measure the same variable in the same units, their IQRs can be compared as widths. A larger IQR means a wider span between the first and third quartiles. It does not tell us which group has the larger typical value, because the IQR measures spread rather than center. To describe where a group’s middle-half span lies, report its quartile endpoints as well as its IQR.

The comparison also requires matching units and a clear quartile convention. An IQR of 8 minutes can be compared with another IQR in minutes, but not directly with an IQR in kilometers as though the numbers measured the same thing.

What the IQR Does and Does Not Say

The IQR is a compact way to describe the width of the middle-half span, but it does not provide a complete description of a data set. Two data sets can have the same IQR even when their quartile intervals are located in different places. They can also have the same quartiles but differ in the way values are arranged within or outside that interval. The IQR retains a particular piece of information—middle-half width—not every feature of the distribution.

There is also a small but important qualification to the phrase “the middle half.” Quartiles are boundaries, and data can contain ties or values exactly equal to a boundary. Consequently, the number of observations that fall inside the inclusive interval from \(Q_1\) to \(Q_3\) need not be exactly half the sample. The interpretation is that the quartiles mark the span associated with the middle half under the chosen convention, not that precisely 50% of the observed values must lie strictly between the endpoints.

Common Mistakes

• Subtracting in the wrong order. Writing \(Q_1-Q_3\) reverses the endpoints. The IQR is \(Q_3-Q_1\), which gives the nonnegative width from the lower quartile to the upper quartile.
• Calling the IQR an interval. If \(Q_1=7\) and \(Q_3=12\), the interval is from 7 to 12, while the IQR is the width \(12-7=5\). Report the interval and the width distinctly.
• Leaving out the units. If the measurements are in minutes, then the IQR is in minutes. An answer of “5” is incomplete when “5 minutes” is meaningful in context.
• Treating the IQR as a center. A difference between quartiles describes width, not a typical observation. State the quartile endpoints if the location of the span matters.
• Using inconsistent quartile conventions. Calculate both quartiles according to the same specified rule. Different conventions can yield different quartiles and therefore different IQRs for a finite data set.