3.3 z-score / z-transformation

The -score is the result of transformation of data that converts a dataset of values, , that has a mean of and standard deviation to a set of values that has a mean of and a standard deviation of . It will be very useful when we need to compute probabilities associated with normal distributions. The -transformation is defined by

Example 3.12 : Find the -scores of the data given in the left column of the table below.

Data		-score,
18	324	(18-9.9)/6.2 = 1.3
15	225	(15-9.9)/6.2 = 0.8
12	144	(12-9.9)/6.2 = 0.3
6	36	(6-9.9)/6.2 = -0.6
8	64	(8-9.9)/6.2 = -0.3
2	4	(2-9.9)/6.2 = -1.3
3	9	(3-9.9)/6.2 = -1.1
5	25	(5-9.5)/6.2 = -0.8
20	400	(20-9.5)/6.2 = -1.7
10	100	(10-9.5)/6.2 = 0.1

The dataset size is . You need to compute the -score for each data value separately. To do the calculation, both and are needed. So in addition to the sum of the data, , we also need the sum of the values. The work of getting those sums is shown in the table above. With the and sums we get

and

and

Using these values for and in the third column of the table above, compute the -scores as shown. If we had computed the -scores more accurately, they would add up to zero, (the mean of the -scores is zero.)

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