Box-and-whisker plots — quartiles, IQR: Question bank

Definition The objects these traps lean on

Before the questions, pin down the exact objects being tested — every trap below assumes only these, so we build each one here rather than assuming the parent note:

== $Q_{2}$ (the median)== — the middle value of the sorted data (or the average of the two middle values if the count $n$ is even). Half the data lies below it, half above.
== $Q_{1}$ (lower quartile)== — the median of the lower half, roughly the value one-quarter of the way up the sorted queue.
== $Q_{3}$ (upper quartile)== — the median of the upper half, roughly three-quarters of the way up.
IQR (interquartile range) $= Q_{3} - Q_{1}$ . It is the width of the box: the span covered by the central 50% of the data, from $Q_{1}$ up to $Q_{3}$ .
Fences are the cutoff thresholds for calling a point an outlier, measured one-and-a-half box-widths beyond each edge: $Lower fence = Q_{1} - 1.5 IQR, Upper fence = Q_{3} + 1.5 IQR$ A point strictly outside a fence is flagged as an outlier; the whisker stops at the last real data value inside the fence, never at the fence itself.

Here is the picture every trap below refers to — the sorted queue split into four equal-count chunks, with the box, whiskers, and one lone outlier dot:

And here is what a skewed box looks like beside a symmetric one — notice where the median line sits inside each box:

True or false — justify

Recall Reveal set A

The median always sits exactly in the middle of the box. ::: False. The median $Q_{2}$ is the middle value, but the box spans $Q_{1}$ to $Q_{3}$ ; if data is skewed the median can hug one edge (look at the right-hand box in the figure). Its off-centre position is exactly what reveals Skewness. A wider box always means more data points. ::: False. Box width is the IQR (a spread of values), not a count. Every box always holds the same middle 50% of the data regardless of how wide it is drawn. If two data sets have the same IQR they have the same range. ::: False. IQR ignores the outer 25% on each side; two sets can share an IQR yet have wildly different min/max, hence different range. Removing an outlier changes the IQR a lot. ::: False (usually). That's the whole point of a robust measure — outliers live outside the quartiles, so deleting one rarely moves $Q_{1}$ or $Q_{3}$ . See Outliers and Robust Statistics. The whiskers always reach the minimum and maximum of the data. ::: False. Whiskers stop at the most extreme value inside the fences. If outliers exist, min/max are drawn as separate dots (see the lone dot in the figure) and the whisker ends short of them. Exactly 25% of data points lie strictly below $Q_{1}$ . ::: Roughly, not exactly. In a large tie-free set it's close to 25%, but with ties many points can equal $Q_{1}$ (so fewer than a quarter are strictly below), and with small $n$ the count can't divide into an exact quarter — e.g. with $n = 7$ the lower half has 3 points, which is $3/7 \approx 43%$ of that half, and only whole points can fall below the cut. Quartiles are positions that split counts as evenly as the data allows, not an exact-percentage guarantee. A box plot can show whether the data is symmetric. ::: True. Equal whisker lengths and a centred median suggest symmetry; unequal ones suggest Skewness — the plot is a visual skew detector.

Spot the error

Recall Reveal set B

"I found $Q_{1}$ by taking the value at position $n /4$ ." ::: The $n /4$ index often lands between points and depends on convention. The unambiguous school method is median of the lower half (Tukey's hinges), not a raw quarter-position count. "For $n = 7$ I included the median in both halves when finding $Q_{1}$ and $Q_{3}$ ." ::: When $n$ is odd the single middle value belongs to neither half. Including it double-counts a point and shifts both quartiles inward. "The upper fence is at $Q_{3} + 1.5 \times Q_{3}$ ." ::: The yardstick is the IQR $= Q_{3} - Q_{1}$ , not $Q_{3}$ itself: upper fence $= Q_{3} + 1.5 IQR$ . Using $Q_{3}$ makes the fence scale with location instead of spread. "A point sitting exactly on the upper fence is an outlier." ::: The rule flags points strictly outside the fences. A value equal to the fence is the boundary case and is not flagged. "I drew the whisker all the way out to the fence." ::: Fences are invisible cutoffs, not plotted lines. The whisker ends at the last real data value inside the fence, which is usually short of it. "The data was $9, 2, 5, 1$ so the median is $5$ ." ::: You must sort first: $1, 2, 5, 9$ gives median $\frac{2 + 5}{2} = 3.5$ . Quartiles are defined by position, meaningless on unsorted data. "IQR came out negative, so I subtracted wrong — let me flip it to $Q_{1} - Q_{3}$ ." ::: On correctly sorted data $Q_{3} \geq Q_{1}$ always, so IQR $= Q_{3} - Q_{1} \geq 0$ . A negative result means a computation slip, not a formula that needs flipping. "My calculator gave $Q_{1} = 3.25$ but the textbook says $3$ — one of us is wrong." ::: Not necessarily. Software uses different quartile-interpolation conventions (see the Why-questions set); mismatches of this kind are convention differences, not errors, as long as each method is applied consistently.

Why questions

Recall Reveal set C

Why do we split at the median before taking quartiles, instead of quartering the count directly? ::: The median-of-halves rule is convention-independent and always lands on well-defined values, whereas raw quarter-indices disagree between textbooks and software. Why do different software packages report different quartiles for the same data? ::: They differ on two choices: (1) whether the median itself is included in each half ("inclusive", Tukey's hinges) or excluded ("exclusive") when $n$ is odd, and (2) how they interpolate between points when the quarter-position isn't a whole index. Excel's QUARTILE.INC, QUARTILE.EXC, and R's nine "types" are all legitimate; school syllabi standardise on the median-of-halves method to avoid ambiguity. Why does the IQR beat the plain range as a measure of spread? ::: Range uses only the two most extreme (least reliable) points, so one wild value inflates it. IQR discards the outer 25% each side, making it robust — the spread-sibling of the median. Why is the multiplier in the fence rule $1.5$ and not, say, $1$ or $3$ ? ::: It's a tuned convention. On roughly normal data the quartiles sit at about $\pm 0.6745 σ$ , so a fence at $Q_{3} + 1.5 IQR$ lands at about $0.6745 σ + 1.5 (2 \times 0.6745) σ \approx 2.698 σ$ from the centre. The tail beyond $2.698 σ$ on one side is about $0.35%$ , so both tails together flag roughly $0.7%$ of points — rare enough to be notable, not so tight that everything trips it. A multiplier of $3.0$ marks the far-out "extreme" outliers instead. Why does an off-centre median inside the box tell us about skew? ::: If the median sits close to $Q_{1}$ , values pile up on the low side and stretch out high (right skew); the geometry of the box mirrors the shape of the distribution — see Skewness and the right-hand box in the opening figure. Why is the median a "fairer centre" than the mean for incomes? ::: The mean is dragged by one billionaire; the median only cares about who stands in the middle of the queue, so extreme values can't distort it. See Median and Measures of Central Tendency. Why do quartiles correspond to specific percentiles? ::: $Q_{1}, Q_{2}, Q_{3}$ are exactly the 25th, 50th, and 75th percentiles — quartiles are just the special percentiles that cut the data into four.

Edge cases

Recall Reveal set D

What happens to $Q_{1}$ and $Q_{3}$ if every value in the data set is identical (e.g. all $5$ )? ::: All quartiles equal that value, IQR $= 0$ , and the box collapses to a single line. The plot correctly shows "no spread". Can $Q_{1} = Q_{2} = Q_{3}$ even when the data isn't all identical? ::: Yes. If many values are tied at the centre (e.g. $1, 5, 5, 5, 5, 5, 9$ ) all three quartiles can coincide while min and max still differ, giving a zero-width box with long whiskers. With a single data point ( $n = 1$ ), what are the quartiles? ::: Every quartile collapses onto that one value: $Q_{1} = Q_{2} = Q_{3} =$ the point, IQR $= 0$ , and there is no spread to describe. The "box plot" is a single dot — a degenerate but consistent case. With only $n = 2$ data points, what are the quartiles? ::: The median is their average; each "half" is a single value, so $Q_{1}$ and $Q_{3}$ are just the smaller and larger point. The box spans the whole (tiny) data set. If the lower fence comes out negative but all data is positive, does that mean an error? ::: No. A negative fence simply means no value is low enough to be a low-outlier; the fence is a threshold, not a data value, and can legitimately fall below the minimum. Can a single extreme value be an outlier on one side while the whisker on the other side is perfectly normal? ::: Yes. Fences are computed independently per side; a right-skewed set may dot outliers only on the high end while the low whisker reaches its minimum normally. If exactly half the data lies at the maximum value, where does the median fall? ::: The median can equal that maximum, so $Q_{2}$ and $Q_{3}$ may both sit at the right edge — the box is heavily lopsided, signalling strong left-to-right pile-up.

Connections

Box-and-whisker plots — quartiles, IQR — parent topic these traps stress-test.
Median and Measures of Central Tendency — why the median resists outliers.
Range and Spread — the range these traps contrast against IQR.
Outliers and Robust Statistics — the fence rule and robustness live here.
Percentiles and Quantiles — quartiles as the 25/50/75 percentiles.
Skewness — reading box asymmetry.
Normal Distribution — origin of the $1.5$ multiplier's tail percentage.