Five numbers describe a whole data set. A box draws them.
Read the box
A box plot has minimum 12, Q1 20, median 26, Q3 34 and maximum 50. Roughly what percentage of the data lies between 20 and 34?
20 is Q1 and 34 is Q3, and the box between them always holds the middle 50% of the data. A answers a different question — 34 − 20 = 14 is the width of the box in units, not a percentage. D is the share below Q3, not between the quartiles. The length of a box tells you nothing about how many values it contains: it is always half of them.
Minimum · Q1 · Median · Q3 · Maximum
4, 7, 8, 11, 13, 15, 18, 21, 27
sorted first — always
median = 13
the 5th of 9 values
Q1 = 7 + 82 = 7.5
middle of 4, 7, 8, 11
Q3 = 18 + 212 = 19.5
middle of 15, 18, 21, 27
Every quartile is a position, so an unsorted list gives nonsense. This is the single most common reason a box plot comes out wrong.
Reading the picture: the box runs from Q1 (20) to Q3 (34), the line inside the box marks the median (26), and the whiskers reach out to the minimum (12) and the maximum (50). So the box together holds the middle 50% of the data.
| Region | Holds |
|---|---|
| left whisker | the lowest 25% |
| left half of the box | the next 25% |
| right half of the box | the next 25% |
| right whisker | the highest 25% |
Each of the four regions holds a quarter of the data, no matter how long or short it looks. A long whisker means those values are spread out, not that there are more of them.
In the plot above the median sits left of centre and the right whisker is very long, so the data is skewed right — a few high values pulling away from the crowd.
Compare centre with median, spread with the box width.
| Section | Min | Q1 | Median | Q3 | Max |
|---|---|---|---|---|---|
| A | 40 | 58 | 66 | 74 | 92 |
| B | 52 | 62 | 66 | 70 | 80 |
Same median of 66, so neither section is typically better. But A's box spans 16 marks against B's 8, so A is far more spread out — more very strong and very weak results. That distinction is invisible in an average alone.
The same five numbers; each case reads them a new way.
An even list: the summary, range and IQR of 6, 9, 10, 14, 17, 20, 23, 31
median = 14 + 172 = 15.5
8 values: two middles
Q1 = 9 + 102 = 9.5, Q3 = 20 + 232 = 21.5
halves of four
range = 31 − 6 = 25, IQR = 21.5 − 9.5 = 12
whole width vs box width
Different: the range uses the whisker ends; the interquartile range uses only the box.
Three quarters: the summary is 30, 42, 50, 61, 88. What percent of the data is at least 42?
42 is Q1
25% lies below it
100% − 25% = 75%
both box halves plus the right whisker
Different: count quarters, not lengths — the long right whisker (61 to 88) still holds only 25%.
A median on the box's edge: 3, 5, 5, 5, 8, 12, 14
median = 5, lower half 3, 5, 5 → Q1 = 5
repeated values
upper half 8, 12, 14 → Q3 = 12
the box runs 5 to 12
median line on the left edge: skewed right
the extreme case of "median near Q1"
Different: Q1 and the median can be equal, so a box can have no left half at all.
Plot at the upper boundary of each class, then join with a smooth curve.
| Score | Frequency | Cumulative frequency |
|---|---|---|
| 1–10 | 4 | 4 |
| 11–20 | 9 | 13 |
| 21–30 | 14 | 27 |
| 31–40 | 8 | 35 |
| 41–50 | 5 | 40 |
n = 40
the final cumulative total
median at n2 = 20
read across from 20 on the curve
Q1 at n4 = 10, Q3 at 3n4 = 30
the quartiles, the same way
The 20th value falls inside the 21–30 class, since the running total reaches 13 at 20 and 27 at 30. So the median is somewhere in 21–30.
Expect to read the five numbers off a plot, give the range or IQR, say what percent lies in a region, describe the skew, or compare two plots. The fastest route is to remember that each of the four regions holds 25%, whatever its length. The 10-second check: min ≤ Q1 ≤ median ≤ Q3 ≤ max, and percentages come in multiples of 25.
Find the median
A cumulative frequency table for 60 learners shows running totals of 5, 18, 34, 51 and 60 for the score classes 1–10, 11–20, 21–30, 31–40 and 41–50. Which class contains the median?
The median sits at position n2 = 30. The running total is 18 by the end of 11–20 and 34 by the end of 21–30, so the 30th value falls in that class. C uses the class where the total passes 51 — that is where Q3 lives, not the median. Read the cumulative column, not the frequency column.
Hint: find n2 first, then walk down the cumulative column until it first reaches or passes that number.
