3. Descriptive Statistics: Central Tendency and Dispersion
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 ![]() |
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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 |
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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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