Cumulative frequency and box plots

Cumulative frequency is a running total of the frequencies.

1. Add a running total column to the table.
2. Plot each running total at the upper bound (top end) of its class.
3. Start at (lowest bound, 0), and join the points with a smooth curve.

With n values, read across from these cumulative frequencies to the curve, then down:

• median: n ÷ 2
• lower quartile (LQ): n ÷ 4
• upper quartile (UQ): 3n ÷ 4.

The interquartile range is IQR = UQ − LQ. It is the spread of the middle half of the data, so outliers don't affect it.

Example: use the curve (n = 40) to estimate the median, the quartiles and the IQR.

1. Median: across from 20, then down: about 24 minutes
2. LQ: across from 10: about 17 minutes
3. UQ: across from 30: 30 minutes
4. IQR = 30 − 17 = about 13 minutes

A box plot shows five values on a number line:

1. the smallest value, at the end of the left whisker
2. the lower quartile, at the left end of the box
3. the median, as a line inside the box
4. the upper quartile, at the right end of the box
5. the largest value, at the end of the right whisker.

Example: the shortest time was 3 minutes and the longest was 48. Describe the box plot for the times.

1. Whiskers: from 3 to 17, and from 30 to 48
2. Box: from 17 (LQ) to 30 (UQ)
3. A line inside the box at 24, the median

The box holds the middle half of the data.

To compare two box plots, compare:

1. the medians: which is higher on average
2. the IQRs: which is more consistent (a smaller IQR means more consistent).

Write each in context. Use the median and IQR rather than the range, because they aren't affected by outliers.

Example: group A's times have a median of 24 minutes and an IQR of 13 minutes. Group B's have a median of 28 minutes and an IQR of 8 minutes. Compare the groups.

1. On average, group B took longer, because its median (28) is higher than group A's (24)
2. Group B's times were more consistent, because its IQR (8) is smaller than group A's (13)