The five-number summary
A box plot is drawn from five values: the minimum, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum. Together these are the five-number summary, and they split the sorted data into four parts that each hold roughly a quarter of the observations.
The box spans Q1 to Q3, with a line inside it at the median, so it covers the middle 50% of the data, and its length is the interquartile range, IQR = Q3 − Q1. The lines extending from each end are the whiskers, and any point drawn beyond a whisker is a flagged outlier.
Note what is not on the plot: the mean and the standard deviation. A box plot is built entirely from order statistics — positions in the sorted data — which is why it stands up so well to extreme values. One enormous observation shifts the maximum and gets flagged, but barely moves the median or the quartiles.
- Five-number summary: minimum, Q1, median, Q3, maximum.
- The box runs Q1 to Q3 and holds the middle 50% of the data; the line inside it is the median.
- Box plots show no mean and no standard deviation — they are built from ranks, not arithmetic on values.
Computing quartiles and the IQR
The median is the middle value of the sorted data, or the average of the two middle values when the count is even. Quartiles are the medians of the halves: Q1 for the lower half, Q3 for the upper half. Software disagrees about the details, so state your convention. StatRise uses the median-of-halves method and excludes the overall median from both halves when n is odd — the convention used throughout AP Statistics, sometimes called the exclusive or Tukey method.
Take an invented dataset of twelve delivery times in minutes, sorted: 12, 15, 17, 18, 20, 21, 22, 24, 26, 29, 31, 58. With n = 12 the median is the average of the sixth and seventh values, (21 + 22) ÷ 2 = 21.5. The lower half is 12, 15, 17, 18, 20, 21, giving Q1 = (17 + 18) ÷ 2 = 17.5; the upper half is 22, 24, 26, 29, 31, 58, giving Q3 = (26 + 29) ÷ 2 = 27.5.
That gives IQR = 27.5 − 17.5 = 10 minutes: the middle half of these deliveries spans a ten-minute window. Because the IQR ignores the top and bottom quarters entirely, it is a resistant measure of spread — the 58-minute delivery contributes nothing to it, though it would inflate the range and the standard deviation.
- Q1 is the median of the lower half, Q3 the median of the upper half; IQR = Q3 − Q1.
- Conventions differ over whether the overall median joins the halves when n is odd — always say which you used.
- In the example: Q1 = 17.5, median = 21.5, Q3 = 27.5, so IQR = 10.
The 1.5 × IQR rule and where the whiskers stop
Outliers are identified with fences, not with the whiskers themselves. The lower fence sits at Q1 − 1.5 × IQR and the upper fence at Q3 + 1.5 × IQR, and any observation outside those boundaries is flagged and plotted as an individual point.
This is the step almost everyone gets wrong: the whiskers do not extend to the fences. They extend to the most extreme actual data values still inside the fences. The fences are invisible cutoffs used to make the decision; the whisker ends are real observations.
Continue the example. IQR = 10, so the fences are 17.5 − 15 = 2.5 and 27.5 + 15 = 42.5. No value falls below 2.5, so the lower whisker reaches the smallest observation, 12. Above, 58 exceeds 42.5 and is plotted separately as an outlier; the largest value still inside the fence is 31, so that is where the upper whisker stops — not at 42.5, and not at 58.
The 1.5 multiplier is a convention, not a law: for roughly normal data it flags only a small percentage of observations, and some analyses add outer fences at 3 × IQR to separate mild outliers from extreme ones. Either way, 'outlier' means 'unusual for this batch and worth inspecting', never 'wrong' or 'delete it'.
- Fences: Q1 − 1.5 × IQR and Q3 + 1.5 × IQR; anything outside them is flagged as an outlier.
- Whiskers stop at the most extreme data values inside the fences, not at the fences.
- In the example the fences are 2.5 and 42.5, the whiskers run to 12 and 31, and 58 is the lone outlier.
Reading shape from the picture
Once the parts are labelled, the plot tells you about shape. A median near the middle of the box with roughly equal whiskers means the distribution is fairly symmetric. A median closer to Q1 with a longer upper whisker means right skew — a long tail of large values. The mirror image, median near Q3 with a long lower whisker, is left skew.
The delivery-time example is right-skewed: the median at 21.5 sits slightly below the box's midpoint of 22.5, and the upper side stretches to 31 with an outlier at 58 while the lower side stops at 12. That asymmetry is a signal about the process, not noise to trim away.
Box plots are at their best side by side, since plotting groups on a shared axis makes differences in centre and spread directly comparable. Treat separated boxes as a prompt for a formal test, not a substitute for one: overlapping boxes do not prove equality and separated boxes do not establish significance.
- Median centred with balanced whiskers suggests symmetry.
- Median near Q1 with a long upper whisker means right skew; the reverse means left skew.
- Side-by-side box plots compare groups well, but they suggest differences rather than test them.
What box plots hide, and what to do about it
A box plot's compression is also its weakness. Reporting only five numbers, it cannot show multimodality: two distinct clusters can produce a box plot indistinguishable from a single broad hump. It also hides sample size — a box drawn from eight observations looks as authoritative as one drawn from eight hundred.
The fix is to pair it with something that shows individual values. A histogram reveals modes and gaps, a jittered strip of raw points works well for small samples, and annotating n beside each box costs nothing. When the box plot and the histogram disagree, believe the histogram.
Finally, resist letting the 1.5 × IQR rule make decisions for you. A flagged point may be a data-entry error, a genuine rare event, or evidence of a heavy-tailed population. Investigate it, decide deliberately, and report what you did — silently dropping flagged points biases every summary that follows. StatRise's Quartiles and Interquartile Range calculators show each step on your own data; they sit in the descriptive-statistics category, which is free, as is the lesson module covering skewness and spread.
- Box plots cannot show multimodality and do not reveal sample size.
- Pair them with a histogram or the raw points, and label n.
- A flagged outlier is a prompt to investigate, not a licence to delete.