St St Use the median to abolish those wretched outliers

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Put this one up against a reasonably untutored readership and you should get away with it.

Simply quote the median for a group of observations, but make no mention of the outliers and under no circumstances publish the full set of individual results. That way, your median will be dominated by the majority of the data points and the outliers should nicely disappear from view.

Calculating the median where there is an even number of observations In the example above (Table 2.2), the total number of timings is 15. With any odd number

Ranked time (s)

12 13

Middle pair

Middle pair

20 25

Average of 15 and 16 gives Median = 15.5 s

20 25

Figure 2.3 Calculation of a median with an even number of data points (timingsin seconds)

of data points, we can identify a single middle-ranking individual. However, with even numbers of observations there is no middle individual. In those cases, we identify the middle pair and then use the average of their values. An example of six timings is shown in Figure 2.3.

One slightly awkward consequence is that, although the timings were apparently made to the nearest whole number of seconds, we can end up with a median value that contains a fraction.

2.2.3 Mode - a global assessment variable for the response to an anti-inflammatory drug

The condition of a series of patients with arthritis is recorded using a global assessment variable. This is a composite measure that takes account of both objective measures of the degree of inflammation of a patient's joints and subjective measures of the restrictions on their quality of life. It is set up so that higher scores represent better condition. The patients are then switched to a new anti-inflammatory product for 3 months and re-assessed using the same global measure. We then calculate the change in score for each individual. A positive score indicates an improvement and a negative one a deterioration in the patient's condition. Sixty patients participated and the results are shown in Table 2.3.

A histogram of the above data (Figure 2.4) shows the difficulty we are going to have. Most of the patients have shown reduced joint inflammation, but there are two distinct sub-populations so far as side effects are concerned. Slightly under half the patients are relatively free of side effects, so their quality of life improves markedly, but for the remainder, side effects are of such severity that their lives are actually made considerably worse overall.

Mathematically, it is perfectly possible to calculate a mean or a median among these score changes and these are shown on Figure 2.4. However, neither indicator remotely encapsulates the situation. The mean (—0.77) is particularly unhelpful as it indicates a value that is very untypical - very few patients show changes close to zero. We need to describe the fact that, in this case, there are two distinct groups.

The first two sets of data we looked at (vaccine potencies and container opening timings) consisted ofvalues clustered around some single central point. Such data are referred to as 'unimodal'. The general term 'polymodal' is used for any case with

Table 2.3 Individual changes in a global assessment variable following treatment with an anti-inflammatory (61 patients)

Score changes

11

-9

-8

0

-9

2

-5

-15

-11

11

-13

-12

-13

-13

10

7

-18

-11

7

-13

9

-12

9

14

10

14

-9

-12

10

17

-10

-9

-14

6

11

-6

13

-11

13

-11

14

12

10

10

-6

-9

21

-9

9

6

2

8

-13

5

-12

-6

-7

10

-9

-12

1

Figure 2.4 Individual changes in a global assessment score. Neither mean nor median indicates a typical value with bimodal data. Only modes achieve this

Figure 2.4 Individual changes in a global assessment score. Neither mean nor median indicates a typical value with bimodal data. Only modes achieve this several clusterings. If we want to be more precise, we use terms such as bimodal or trimodal to describe the exact number of clusters. The arthritis data might be described generally as polymodal or more specifically as bimodal.

With polymodal data we need to indicate the central tendency of each cluster. This is achieved by quoting the most commonly occurring value (the 'mode') within each cluster. For the arthritis data, the two modes are score changes of —10 and + 10. We would therefore summarize the results as being bimodal with modes of —10 and +10.

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