Statistical Methods for Psychology

We earlier defined the median location of a set of Nscores as (N 1 1)/2. When the median

location is a whole number, as it will be when Nis odd, then the median is simply the value that

occupies that location in an ordered arrangement of data. When the median location is a frac-

tional number (i.e., when Nis even), the median is the average of the two values on each side of

that location. For the data in Table 2.8 the median location is (38 1 1)/2 5 19.5, and the median

is 3. To construct a boxplot, we are also going to take the first and third quartiles, defined ear-

lier. The easiest way to do this is to define the quartile location,which is defined as

If the median location is a fractional value, the fraction should be dropped from the nu-

merator when you compute the quartile location. The quartile location is to the quartiles what

the median location is to the median. It tells us where, in an ordered series, the quartile val-

ues^14 are to be found. For the data on hospital stay, the quartile location is (19 1 1)/2 5 10.

Thus, the quartiles are going to be the tenth scores from the bottom and from the top. These

values are 2 and 4, respectively. For data sets without tied scores, or for large samples, the

quartiles will bracket the middle 50% of the scores.

To complete the concepts required for understanding boxplots, we need to consider three

more terms: the interquartile range, inner fences, and adjacent values. As we saw earlier, the

interquartile range is simply the range between the first and third quartiles. For our data, the

interquartile range 4 22 5 2. An inner fenceis defined by Tukey as a point that falls 1.5

times the interquartile range below or above the appropriate quartile. Because the interquar-

tile range is 2 for our data, the inner fence is 2 3 1.5 5 3 points farther out than the quartiles.

Because our quartiles are the values 2 and 4, the inner fences will be at 2 23 52 1 and

4 13 5 7. Adjacent valuesare those actual values in the data that are no more extreme (no

farther from the median) than the inner fences. Because the smallest value we have is 1, that

is the closest value to the lower inner fence and is the lower adjacent value. The upper inner

fence is 7, and because we have a 7 in our data, that will be the higher adjacent value. The

calculations for all the terms we have just defined are shown in Table 2.8.

Quartile location=

Median location 11

2

Section 2.9 Boxplots: Graphical Representations of Dispersions and Extreme Scores 49

Table 2.8 Calculation and boxplots for data from Table 2.7

Median location 5 (N 1 1)/2 5 (38 1 1)/2 5 19.5

Median 5 3

Quartile location 5 (median location† 1 1)/2 5 (19 1 1)/2 5 10

Q 15 10th lowest score 5 2

Q 35 10th highest score 5 4

Interquartile range 5 4 22 5 2

Interquartile range * 1.5 5 2*1.5 5 3

Lower inner fence 5 Q 12 1.5 (interquartile range) 5 2 23 5 2 1

Upper inner fence 5 Q 31 1.5 (interquartile range) 5 4 13 57

Lower adjacent value 5 smallest value ≥lower fence 5 1

Upper adjacent value 5 largest value ≤upper fence 57

†Drop any fractional values.

0510

** * *

15 20 25 30 35

quartile location

inner fence

Adjacent values

(^14) Tukey referred to the quartiles in this situation as “hinges,” but little is lost by thinking of them as the quartiles.