Dispersion measures; construct box plot

Absolute and relative measures of dispersion

• Absolute measure: expresses the spread in the same unit as the data. Examples: range, quartile deviation, mean deviation, standard deviation.
• Relative measure: a unit-free ratio or percentage used to compare the spread of different data sets. Examples: coefficient of range, coefficient of quartile deviation, coefficient of variation.

Box plot from grouped data

N = 50. Cumulative frequencies: 2, 8, 30, 43, 50. Class width = 10.

Q₁: position N/4 = 12.5, class 30-40 (cf before = 8, f = 22)

Q1 = 30 + (12.5 − 8)/22 × 10 = 32.05

Median: position 25, class 30-40

Median = 30 + (25 − 8)/22 × 10 = 37.73

Q₃: position 3N/4 = 37.5, class 40-50 (cf before = 30, f = 13)

Q3 = 40 + (37.5 − 30)/13 × 10 = 45.77

The smallest and largest values are not known exactly, so the whiskers are drawn to the ends of the first and last classes: 10 and 60.

IQR = 45.77 − 32.05 = 13.72.

 10    20    30    40    50    60
  |     |     |     |     |     |
  10---------[32.05==37.73==45.77]-----------60
  min         Q1    median   Q3              max

The median (37.73) is nearer to Q₁ (distance 5.68) than to Q₃ (distance 8.04), which hints at a slight right skew. However, the lower whisker (10 to 32.05, length 22.05) is longer than the upper whisker (45.77 to 60, length 14.23), which points the other way. Because the whiskers are drawn only to the class ends, they are approximate, so the marks are roughly symmetric and no clear skew can be claimed.