Authoring a PhD

HANDLING ATTENTION POINTS 7
Stem-and-leaf analysis goes a little bit further because it retains more of the information given in the original numbers.
Each number is divided into two parts, the larger stem part and the smaller leaf or unit part. We choose what to set as the stem in relation to the range of the data being analysed the variation from top to bottom score. Here we could set the stem as equal to s, just as in the frequency table above. But since we want to look a little deeper we could set the stem as fives instead, with (for instance) one stem running from 20 to, and another stem running from 25 to 29. On this basis the first number in the set is 25, which would separate into an uppers stem and a leaf of 5. The next number 46 would separate into an uppers stem and a leaf of 6. The next number would separate into a lowers stem and a leaf of 2, and soon. Working through the whole set of numbers above would give a stem-and leaf analysis as follows:
Stem
Leaf (1s)
(5s)
5 2
4 6
4 3
3 1 2 2 3 3 4 4 2
5 9 2
0 0 1 2 2 2 2 3 1
5 7 8 8 9 9 9 1
2 0
0 3
Upper outlier
Upper outlier
Upper quartile
⫽ 33
Median
⫽ 22
Lower quartile
⫽ 19
It is clear here that there is not just a single-peaked curve (one bell curve. Instead there is a main bulge of observations scoring from 15 to 23 (including 13 data points, and then another smaller bulge from 29 to 34 (including 7 data points. Since there are 27 observations we can find the median by counting

1 8 AUTHORING AP H D
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
⫽ Top point Bottom point Range Upper quartile
⫽ Lower quartile
⫽ 19
Midspread
⫽ With 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 statistical 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, SPSS has stem-and-leaf facilities.
Box-and-whisker plots are away 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 difference 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 observation lies, the more unusual it is. There maybe a greater chance that it is a fluke or apiece of bad data, or alternatively that it is a significant extreme case, requiring detailed explanation.
Outlying observations (those lying along 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 (seethe middle bar in Figure 7.4). Outliers are often worth labelling individually with their name, to remind you exactly which observations are highly unusual.

HANDLING ATTENTION POINTS It can be useful to look at a single box-and-whisker plot of the statistics from one stem-and-leaf. But these plots real value is in allowing you to compare the variation across different sets of data points, as shown in Figure 7.4. Here one can see at a glance:
◆
variations in the central level of the three different sets of observation, as shown by comparing the vertical position of the medians and of the middle boxes and
◆
variations in the spread of their data, shown by the vertical size of the shaded middle boxes, the vertical size of the boxes plus whiskers, and the presence or absence of outliers.
This is a sophisticated, multi-indicator comparison, yet accomplished in a very intuitive and accessible way. It can greatly assist your understanding of the data, and it can also convey a lot of information effectively to readers.
Smoothing data is another very useful data-reduction technique for any kind of information that is analysed using line graphs,
especially overtime movements of any kind of index. There are many cases where we acquire a large number of observations in a volatile data series, one that zigzags up and down a lot, such as the movements of stock markets, or commodity markets, or public opinion polls showing the popularity of a government.
The key difficulty here is to try and separate out the meaningless or temporary fluctuations from the underlying, long-run
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