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MORE ABOUT z-SCORES

As we saw also in chapter 6 on sampling, every real score in a distribution has a z-score, also called a standard score. A z-score tells you how far, in standard deviations, a real score is from the mean of the distribution. The formula for finding a z-score is:

To find the z-scores of the data on FEMILLIT in table 20.7, subtract 13.9 (the mean) from each raw score and divide the result by 14.94 (the sd). Table 20.11 shows these z-scores.

Table 20.11 z-Scores for the Data on FEMILLIT in Table 20.7

Country

Score - Mean x - x

z-Score

El Salvador

18.6 - 13.9 = 4.7

4.7/14.9 = 0.3154

Iran

20.7 - 13.9 = 6.8

6.8/14.9 = 0.4564

Latvia

0.2 - 13.9 = -13.7

-13.7/14.9 = -0.9195

Namibia

11.1 - 13.9 = -2.8

-2.8/14.9 = -0.1879

Panama

6.6 - 13.9 = -7.3

-7.3/14.9 = -0.4899

Slovenia

0.3 - 13.9 = -13.6

-13.6/14.9 = -0.9128

Suriname

7.4 - 13.9 = -6.5

-6.5/14.9 = -0.4362

Armenia

1.4 - 13.9 = -12.5

-12.5/14.9 = -0.8389

Chad

48.5 - 13.9 = 34.6

34.6/14.9 = 2.3221

Ghana

24.2 - 13.9 = 10.3

10.3/14.9 = 0.6913

Why Use Standard Scores?

There are several advantages to using standard scores rather than raw scores. First of all, although raw scores are always in specialized units (percentages of people, kilos of meat, hours of time, etc.), standard scores measure the difference, in standard deviations, between a raw score and the mean of the set of scores. A z-score close to 0 means that the raw score was close to the average. A z-score that is close to plus-or-minus 1 means that the raw score was about 1 sd from the mean, and so on.

What this means, in practice, is that when you standardize a set of scores, you create a scale that lets you make comparisons within chunks of your data.

For example, we see from table 20.11 that the female illiteracy rates for Ghana and

Chad are 24.2% and 48.5%, respectively. One of these raw numbers is about double the other. This tells us something, but the z-score tells us more. The percentage of female illiteracy in Ghana is almost twice the mean of 13.9 for these 10 countries but it’s only about .69 standard deviations above the mean. The percentage for Chad is about 3.5 times the mean, but it is 2.32 sd above the mean. Ghana’s score on this indicator of human development is within striking distance of the world mean, but Chad’s is way out on the right-hand tail of the distribution.

A second advantage of standard scores over raw measurements is that standard scores are independent of the units in which the original measurements are made. This means that you can compare the relative position of cases across different variables.

Medical anthropologists measure variables called ‘‘weight-for-length’’ and ‘‘length for age’’ in the study of nutritional status of infants across cultures. Linda Hodge and Darna Dufour (1991) studied the growth and development of Shipibo Indian children in Peru. They weighed and measured 149 infants, from newborns to 36 months in age.

By converting all measurements for height and weight to z-scores, they were able to compare their measurements of the Shipibo babies against standards set by the World Health Organization (Frisancho 1990) for healthy babies. The result: By the time Shipibo children are 12 months old, 77% of boys and 42% of girls have z-scores of —2 or more on length-for-age. In other words, by a year old, Shipibo babies are more than 2 sd under the mean for babies who, by this measure, are healthy.

By contrast, only around 10% of Shipibo babies (both sexes) have z-scores of —2 or worse on weight-for-length. By a year, then, most Shipibo babies are clinically ‘‘stunted’’ but they are not clinically ‘‘wasted.’’ This does not mean that Shipibo babies are small but healthy. Infant mortality is as high as 50% in some villages, and the z-scores on all three measures are similar to scores found in many developing countries where children suffer from malnutrition (box 20.5).

BOX 20.5

ON SCORING 1.96 ON THE SATS

z-scores do have one disadvantage: They can be downright unintuitive. Imagine trying to explain to people who have no background in statistics why you are proud of having scored a 1.96 on the SAT. Getting a z-score of 1.96 means that you scored almost 2 sd above the average and that only 2.5% of all the people who took the test scored higher than you did. If you got a z-score of — .50 on the SAT, that would mean that about a third of all test takers scored lower than you did. That's not too bad, but try explaining why you're not upset about getting a minus score on any test.

This is why T-scores (with a capital T—not to be confused with Student's t) were invented. The mean of a set of z-scores is always 0 and its standard deviation is always 1. T-scores are linear transformations of z-scores. For the SAT and GRE, the mean is set at 500 and the standard deviation is set at 100. A score of 400 on these tests, then, is one standard deviation below the mean; a score of 740 is 2.4 sd above the mean (Friedenberg 1995:85).

 
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