Cutting ordered data into four equal parts: Q1, Q2 (the median) and Q3, and the case when n + 1 is divisible by 4
The median cuts ordered data into two equal halves, but sometimes we want four levels, such as weak, fair, good and excellent. Three values do this: the lower quartile Q₁ has a quarter of the data before it, Q₂ is the median, and the upper quartile Q₃ has three quarters before it. First put the data in order. Suppose there are n values and n + 1 is a multiple of 4. Then n is odd and the three quartiles are values of the data, in the places (n + 1) ÷ 4, (n + 1) ÷ 2 and 3(n + 1) ÷ 4. Why? The median is in place (n + 1) ÷ 2. Before it there are (n − 1) ÷ 2 values, an odd number, and Q₁ is the middle of those, in place (n + 1) ÷ 4. Q₃ is the middle of the values after the median. Quartiles are not defined in one way only; this is the rule of our book, and we use it all the way. The picture shows 11 invented counts of passengers on a bus route, in order.
| Order | Value | Quartile |
|---|---|---|
| 1 | 21 | |
| 2 | 23 | |
| 3 | 25 | Q₁ |
| 4 | 27 | |
| 5 | 29 | |
| 6 | 31 | Q₂ |
| 7 | 34 | |
| 8 | 36 | |
| 9 | 38 | Q₃ |
| 10 | 40 | |
| 11 | 44 |
| Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Value | 9 | 11 | 12 | 14 | 15 | 18 | 20 |
| Quartile | Q₁ | Q₂ | Q₃ |
Write down the number of pages you read on each of 7 days, or the number of goals in 11 matches (invent them if you wish). Order them, check that n + 1 is divisible by 4 and find Q₁, Q₂ and Q₃.