Medical Statistics. David Machin

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choice but for us to use its estimated value, , in the calculation. The consequence is that it is then better to divide by what are known as the degrees of freedom, which in this case is n−1, to obtain the SD.

Table 2.6 Calculation of the variance and standard deviation for 16 subjects from the corn size data.

	Corn			Square of
	size		Differences	differences
Subject	(mm)	Mean	from mean	from mean
(i)	(xi)	()	()	()2
1	1	3.625	−2.625	6.891
2	2	3.625	−1.625	2.641
3	2	3.625	−1.625	2.641
4	2	3.625	−1.625	2.641
5	2	3.625	−1.625	2.641
6	2	3.625	−1.625	2.641
7	3	3.625	−0.625	0.391
8	3	3.625	−0.625	0.391
9	3	3.625	−0.625	0.391
10	3	3.625	−0.625	0.391
11	4	3.625	0.375	0.141
12	4	3.625	0.375	0.141
13	5	3.625	1.375	1.891
14	6	3.625	2.375	5.641
15	6	3.625	2.375	5.641
16	10	3.625	6.375	40.641
Total	58		0.000	75.756
	n	Mean	df = n−1	Variance	SD
	16	3.625 mm	15	5.050 mm 2	2.247 mm

       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

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