188 ◆AUTHORING A PHD
up or down until we reach the fourteenth observation (shown in
bold in the listing above). And we can find the quartiles in the
same way by partitioning in half the observations above and
below the median (the quartiles are the averages of the seventh
and eighth observations going from the top or from the
bottom). From the stem-and-leaf we can quickly generate a table
giving summary indices of central level and spread as follows:
Median 22
Top point 52 Bottom point 3
Range 49
Upper quartile 33 Lower quartile 19
Midspread 14With small amounts of data stem-and-leaf techniques are easily
applied using pen and paper. There is also a great deal to be said
for using them in this way because it keeps you in close touch
with your data (which might well be outputs from other statis-
tical packages, like frequency counts or charts). Where you get
a large number of data points (more than about 30) you can use
a PC package to do all the exploratory data-analysis techniques
set out here: for instance, SPSShas stem-and-leaf facilities.
Box-and-whisker plotsare a way of displaying the statistical
results of a number of stem-and-leaf analyses. They are like a
vertical bar chart, with a vertical axis showing the scale. The dif-
ference is that you draw in a box only from the upper to the
lower quartile points, and add a thick line to show the position
of the median, as shown in the right-hand bar of Figure 7.4.
To display the remaining data points, stretching away above
and below the middle mass, insert a single vertical line (the
whisker). The further away from the middle mass an observa-
tion lies, the more unusual it is. There may be a greater chance
that it is a fluke or a piece of bad data, or alternatively that it is
a significant extreme case, requiring detailed explanation.
Outlying observations (those lying a long way from the middle
box, specifically more than 1.5 times the midspread above the
upper quartile or below the lower quartile) are shown by blobs
on the whiskers (see the middle bar in Figure 7.4). Outliers are
often worth labelling individually with their name, to remind
you exactly which observations are highly unusual.