Читать книгу Medical Statistics - David Machin - Страница 53

The calculations to work out the standard deviation for the 16 corn sizes are given in Table 2.6.

A convenient method of removing the negative signs is by squaring the deviations, which is given in the next column, which is then summed to get 75.756 mm². Note that the majority of this sum (54%) is contributed by one observation, the value of 10 mm from subject 16, which is the observation furthest from the mean. This illustrates that much of the value of an SD is derived from the outlying observations. (The standard deviation is vulnerable to outliers, so if the 10 was replaced by 100 we would get a very different result.) We now need to find the average squared deviation. Common sense would suggest dividing by n, but it turns out that this actually gives an estimate of the population variance, which is too small. This is because we use the estimated mean in the calculation in place of the true population mean. In fact, we seldom know the population mean so there is little 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