Page 89 - Becoming Metric Wise

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Statistics

Figure 4.8 Illustration of the first procedure to obtain quantiles (second example).

we illustrate it also using this procedure as there is a slight difference. 50%
of 10 is 5. This value is exactly situated between two boxes above the
line. In that case one takes the average of the values in these two boxes.
Here this leads to (47 1 60)/2 5 53.5.
Consider now the somewhat more difficult situation where we have
13 observations: 1,1,3,4,4,4,5,8,10,15,25,50,90. These are illustrated in
Fig. 4.8.
Again we start with the first quartile: 25% of 135 3.25. This value is sit-
uated under the 4th box. This box happens to contain the value 4. Hence
the first quartile is 4. Similarly, 75% of 135 9.75. This value is situated under
the 10th box, which contains the number 15. Hence, 15 is the third quartile.
Finally, we check if we find the median: 50% of 135 6.5. This value is
under the 7th box, leading to a median5 second quartile equal to 5.
We point out that this method can also be used to determine deciles,
percentiles and any quantile in general. For example, the 16th percentile
of the data shown in Fig. 4.8 is 3. Indeed: 16% of 13 5 2.08; this value is
situated under the third box. Hence the 16th percentile is 3.

Procedure 2
This method uses interpolation. Values above the line are assumed to be
placed exactly in the middle of the box. We illustrate it for the situation
shown in Fig. 4.8.
What is the first quartile? The observation 3, placed in the middle
of the third box, corresponds to 2.5/13 5 0.1923; observation 4 placed in
the middle of the fourth box corresponds to 3.5/13 5 0.2692. We want
the value corresponding to 0.25. Using some simple mathematics,
we find that 0.25 corresponds to 3.750. Concretely, one calculates:
3 1 [(0.25 2 0.1923)/(0.2692 2 0.1923)].(4 2 3) 5 3.75. This is the first
quartile. See Fig. 4.9. Clearly, this value differs from the one obtained by
the first procedure.
The median or second quartile is clearly 5. Finally, we determine the
third quartile. The observation 15, placed in the middle of the tenth
box, corresponds to 9.5/13 5 0.7308; observation 25 placed in the middle
of the eleventh box corresponds to 10.5/13 5 0.8077. We want the