Stem-And-Leaf Display

The stem-and-leaf display is an exploratory data analysis technique developed by John Tukey to summarize graphically the characteristics of a distribution. It is especially easy to produce by hand, and it is effective for examining distributions for small sample sizes. (All major statistical packages provide point- and-click routines that can be used readily to examine the distributions in the case of large sample sizes.) At the top of the next page are side-by-side stem-and-leaf displays for 100 randomly selected IQ scores and the subset of all scores greater than 99.

To construct a stem-and-leaf display, we place the data in ascending order and break each score into two parts, a stem (the leading digit[s], which contains the most salient information) and a leaf (the trailing digit[s], which contains less salient information). Two common line depths (i.e., “stem widths”) are created by forming stems that include the digits 0–4 and 5–9 (see Random IQ display) or by forming stems that include the digits 0–1, 2–3, 4–5, 6–7, and 8–9 (see IQ>99 display).

In the stem-and-leaf display for the variable Random IQ, the first line (1 6 4) indicates there is one score in the line (“line depth”), the IQ score 64, which is partitioned into a stem (6) and a leaf (4). For scores below the median, line depths are calculated by counting the total number of scores from the bottom (lowest score) of the distribution up to, and including, the scores in that line. For scores above the median, line [Page 967]depths are calculated by counting the total number of scores from the top (highest score) of the distribution down to, and including, the scores in that line. For the line that contains the median (IQ score 104), the depth of the line is reported in parentheses (20). Note that the center three depths (31, 20, 49) add to N = 100 since they account for all the scores within the distribution. Note also that some computer programs report leaf frequency instead of depths for all lines.

We can locate the center of the distribution (i.e., find the median) by eye or by counting in from either end until half of the observations are counted. We can examine the range of the data by locating the minimum and maximum values. We can locate the quartiles of the distribution either by eye or by counting. Turned on its side, the stem-and-leaf display forms a “digital histogram” that can be used to examine distribution shape, locate peaks and gaps, and identify unusual observations.

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