Data set A has the 20 values 0, 3, 6, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 32, 34, 37…

Step 1. Set A. Lower half $0, 3, 6, 8, 9$, $11, 13, 15, 17, 19$: $Q_1 = \dfrac{9 + 11}{2} = 10$. Median $\dfrac{19 + 21}{2} = 20$. Upper half $21, \ldots, 40$: $Q_3 = \dfrac{29 + 31}{2} = 30$.

Step 2. Set B. Lower half $0, 4, 6, 8, 9$, $11, 12, 13, 13, 14$: $Q_1 = \dfrac{9 + 11}{2} = 10$. Median $\dfrac{14 + 26}{2} = 20$. Upper half $26, \ldots, 40$: $Q_3 = \dfrac{29 + 31}{2} = 30$.

Step 3. Means. Each total is 400, so each mean is $\dfrac{400}{20} = 20$.

Step 4. Near the median. In A, the values 15, 17, 19, 21, 23 and 25 lie from 15 to 25: six values. In B, none do.

What it tells you. The two sets have the same five-number summary, the same box plot and even the same mean. Yet A is flat and B has two clusters with an empty middle. Its median, 20, is a value that no member of B is near. A box plot cannot show this. A histogram can.