Statistics: Box Plot (Box‐and‐Whiskers)

Box Plot (Box‐and‐Whiskers)

Box plots, sometimes called box-and-whiskers, take the stem-and-leaf one step further. A box plot will display a number of values of a distribution of numbers:

The median value
The lower quartile (Q₁)
The upper quartile (Q₃)
The interquartile range or IQR (distance between the lower and upper quartiles)
The symmetry of the distribution
The highest and lowest values

Use the set of values in Table 1 to examine each of the preceding items.

TABLE 1

Verbal SAT Scores of 20 High School Students

280, 340, 440, 490, 520, 540, 560, 560, 580, 580,

600, 610, 630, 650, 660, 680, 710, 730, 740, 740

The median (the middle value in a set that has been ordered lowest to highest) is the value above which half of the remaining values fall and below which the other half of the remaining values fall. Because there are an even number of scores in our example (20), the median score is the average of the two middle scores (10th and 11th)—580 and 600—or 590.

The lower quartile ( Q₁ or 25th percentile) is the median of the bottom half. The bottom half of this set consists of the first 10 numbers (ordered from low to high): 280, 340, 440, 490, 520, 540, 560, 560, 580, 580. The median of those 10 is the average of the 5th and 6th scores—520 and 540—or 530. The lower quartile score is 530.

The upper quartile ( Q₃ or 75th percentile) is the median score of the top half. The top half of this set consists of the last 10 numbers: 600, 610, 630, 650, 660, 680, 710, 730, 740,740. The median of these 10 is again the average of the 5th and 6th scores—in this case, 660 and 680—or 670. So 670 is the upper quartile score for this set of 20 numbers.

A box plot can now be constructed as follows: The left side of the box indicates the lower quartile; the right side of the box indicates the upper quartile; and the line inside the box indicates the median. A horizontal line is then drawn from the lowest value of the distribution through the box to the highest value of the distribution. (This horizontal line is the “whiskers.”)

Using the Verbal SAT scores in Table 1 , a box plot would look like Figure 1 .

Figure 1

A box plot of SAT scores displays median and quartiles.

Without reading the actual values, you can see by looking at the box plot in Figure 1 that the scores range from a low of 280 to a high of 740; that the lower quartile ( Q₁) is at 530; the median is at 590; and the upper quartile ( Q₃) is at 670. Because the median is slightly nearer the lower quartile than the upper quartile and the interquartile range is situated far to the right of the range of values, the distribution departs from symmetry.