Quartiles when n + 1 is not divisible by 4: the order is a fraction, so go part of the way between two values
When n + 1 is not a multiple of 4 the place (n + 1) ÷ 4 is a fraction, such as 3.25. There is no 3.25th value, so we go part of the way from the 3rd value to the 4th: a quarter of the gap, because 0.25 = one quarter. The rule: quartile = previous value + (next value − previous value) × (place − previous place). For place 3.25 the previous place is 3 and the next is 4, so we add 0.25 of the gap. The median is the same rule with Q₂ in place (n + 1) ÷ 2: for an even n it is half way between the two middle values. The answer always lies between the previous and the next value, which is a good check. Always order the data first and write the three places before you start.
| Place | Previous value | Next value | Part of the gap | Quartile |
|---|---|---|---|---|
| 3.25 | 3 | 4 | 0.25 | 3 + 0.25 × 1 = 3.25 |
| 6.5 | 6 | 7 | 0.5 | 6 + 0.5 × 1 = 6.5 |
| 9.75 | 9 | 10 | 0.75 | 9 + 0.75 × 1 = 9.75 |
| Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| Value | 38 | 41 | 43 | 45 | 47 | 50 | 52 | 55 | 60 |
Use the 12 delays of the first example. Take away the largest delay (15) and find the three quartiles of the 11 values left. Which quartile changed most, and why?