Quartiles of lists and stem-and-leaf plots

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.