Summary Statistics and Box Plots

Quartiles split ordered data into four equal parts. Here are eleven values, already in order. The median is the middle one, the sixth value, which is eleven. With an odd count, leave the median out and split what is left into two halves of five. The lower half is three, five, six, eight, nine, and its middle value is six. That is the lower quartile, Q one. The upper half is twelve, fourteen, fifteen, seventeen, twenty, and its middle value is fifteen. That is the upper quartile, Q three. The interquartile range is Q three minus Q one, fifteen minus six, which is nine.

With an even count, the median sits between the two middle values. Here there are ten values, so the median is halfway between the fifth and sixth: nine plus ten, halved, is nine point five. Now split the data straight down the middle: the first five values and the last five. The lower half is two, four, four, seven, nine, so Q one is four. The upper half is ten, twelve, thirteen, sixteen, eighteen, so Q three is thirteen. The interquartile range is thirteen minus four, which is nine. If a half has an even number of values, its median is the mean of its middle two, just as before.

The five-number summary is the minimum, Q one, the median, Q three and the maximum. For the eleven values from the first step, that is three, six, eleven, fifteen and twenty. A box plot draws these five numbers against a scale. The box runs from Q one to Q three, with a line inside it at the median. The whiskers run out from the box to the minimum and the maximum. The width of the box is the interquartile range, nine, and the whole plot from end to end is the range, seventeen.

An outlier is a value that sits unusually far from the rest. The rule is: a value is an outlier if it is more than one and a half times the interquartile range below Q one, or above Q three. For these ten values, Q one is twelve and Q three is eighteen, so the interquartile range is six, and one and a half times six is nine. The fences are twelve minus nine, which is three, and eighteen plus nine, which is twenty seven. Two is below three and thirty one is above twenty seven, so both are outliers. On the box plot they are drawn as dots, and the whiskers stop at the last values inside the fences, eleven and twenty.

Each of the four parts of a box plot holds about a quarter of the data. In these test scores, the lower whisker from ten to twenty five holds about twenty five percent, and so does the short section from thirty five to forty. The long whisker from forty to sixty also holds only twenty five percent: it is long because those scores are spread out, not because there are more of them. So about seventy five percent of students scored twenty five or more, and the middle half scored between twenty five and forty. To turn a percentage into a number of students, multiply by the total. Quiz and exercises now.