Comparing Data Sets

Parallel box plots draw two data sets against the same scale, so you can compare them directly. Compare the centre first. Class B has the higher median, sixty six against sixty two, so B did a little better on average. Then compare the spread. The interquartile ranges are fifteen and fourteen, almost the same, so the middle halves are equally spread. But the ranges are forty five and thirty, so Class A had more extreme marks at both ends. A good comparison names both: Class B scored higher on average, and Class A's marks were less consistent.

A back-to-back stem-and-leaf plot puts two data sets on either side of one stem. On the right, read as usual: stem three with leaves three, four, eight means thirty three, thirty four and thirty eight. On the left, the leaves grow outwards from the stem, so the smallest leaf is closest to the stem. Stem one with leaves eight, four means fourteen and eighteen. Team X's seven scores in order are fourteen, eighteen, twenty two, twenty five, twenty five, thirty one, thirty six, so its median is the fourth, twenty five. Team Y's median is thirty three. The ranges are twenty two and twenty three, so Team Y scored more on average with a similar spread.

Take the data four, five, six, seven, eight. The mean is thirty divided by five, which is six, and the median is also six. Now change the eight to thirty eight, an outlier. The total goes up by thirty, to sixty, so the mean becomes sixty divided by five, which is twelve. The mean has doubled. But the middle value is still six, so the median has not moved at all. An outlier drags the mean towards it, while the median resists it. When a data set has outliers, the median and the interquartile range describe it better than the mean and the range.

When a mean changes, work with totals. Six values with a mean of fifteen have a total of fifteen times six, which is ninety. Add a new value of twenty nine: the total becomes one hundred and nineteen and there are now seven values, so the mean is one hundred and nineteen divided by seven, which is seventeen. You can also work backwards. For the seven values to have a mean of sixteen, their total must be sixteen times seven, one hundred and twelve, so the new value must be one hundred and twelve minus ninety, which is twenty two. A new value above the old mean raises it, and a value below it lowers it.

Here are summaries of two sets of delivery times. Set Q's mean, twenty seven, is well above its median, twenty two. That gap is a sign that a few very long times, outliers, are pulling the mean up. So compare the medians instead: twenty three and twenty two, almost the same typical time. Then compare spread with the interquartile range, which ignores outliers: six for P and ten for Q. So P and Q are about equally quick on a typical day, but P is more consistent. Quiz and exercises now.