Why is the Standard Deviation Useful?
From the corn plaster trial data, the mean and standard deviation of the baseline corn size of the 200 trial patients are 3.8 and 1.8 mm respectively (two baseline sizes were missing). It turns out in many situations that about 95% of observations will be within two standard deviations of the mean. This is known as a reference interval or reference range and it is this characteristic of the standard deviation which makes it so useful. It holds for a large number of measurements commonly made in medicine. In particular it holds for data that follow a Normal distribution (see Chapter 4).
For example, the Association for Clinical Biochemistry and Laboratory Medicine gives a number of reference ranges in biochemistry such as for serum potassium of 3.5–5.3 mmol l−1 (labtestsonline 2019, https://labtestsonline.org.uk/articles/laboratory‐test‐reference‐ranges). This means in a normal, health population we would expect 19 out of 20 people to have serum potassium levels within these limits. For the corn plaster example, we would expect the majority of corns will be sized between 3.8–1.96 × 1.8 to 3.8 + 1.96 × 1.8 or 0.2 and 7.4 mm. Table 2.7 shows that there are 10 patients out of 200 (or 5%) who have a corn size above 7.4 mm and none below 1 mm; thus 95% of the observations in the data lie with two standard deviations of the mean.
Table 2.7 Frequency distribution the size of the corn, in mm, at baseline for 200 patients with corns who were recruited to a randomised control trial of the effectiveness of salicylic acid plasters compared with ‘usual’ scalpel debridement for the treatment of corns
(Source: data from Farndon et al. 2013).
| Size of corn at baseline (mm) | Frequency | Percentage | Cumulative percentage |
|---|---|---|---|
| 1 to <2 | 6 | 3.0 | 3.0 |
| 2 to <3 | 39 | 19.5 | 22.5 |
| 3 to <4 | 52 | 26.0 | 48.5 |
| 4 to <5 | 42 | 21.0 | 69.5 |
| 5 to <6 | 38 | 19.0 | 88.5 |
| 6 to <7 | 10 | 5.0 | 93.5 |
| 7 to <8 | 3 | 1.5 | 95.0 |
| 8 to <9 | 5 | 2.5 | 97.5 |
| 9 to <10 | 1 | 0.5 |
98.0
|