    Chapter 4: Analysing the Data Part II : Descriptive Statistics # Z-scores

Another useful transformation in statistics is standardisation. Sometimes called "converting to Z-scores" or "taking Z-scores" it has the effect of tranforming the original distribution to one in which the mean becomes zero and the standard devaition becomes 1. A Z-score quantifies the original score in terms of the number of standard deviations that that score is from the mean of the distribution. The formula for converting from an original or "raw" score to a Z-score is: The following data will be used as an example.

 id REASON CREATIVE ZCREATIV 1 15 12 -.23 2 10 13 .04 3 7 9 -1.05 4 18 18 1.41 5 5 7 -1.60 6 10 9 -1.05 7 7 14 .31 8 17 16 .86 9 15 10 -.78 10 9 12 -.23 11 8 7 -1.60 12 15 13 .04 13 11 14 .31 14 17 19 1.68 15 8 10 -.78 16 11 16 .86 17 12 12 -.23 18 13 16 .86 19 18 19 1.68 20 7 11 -.51

### Figure 4.8 Data for creativity and logical reasoning example.

This data will be used more extensively later in this chapter. To illustrate the calculation of the Z-score we will use the creativity data in the above table. SPSS Summarise and Descriptives has been used to generate the Z-scores and then converted to two decimal places only Ð see SPSS screens and outputs booklet).

The mean for Creativity is 12.85 and the sd = 3.66

Therefore the z-score for case #5 (a raw creativity score of 7) = The z-score for case #7 (a raw creativity score of 14) = A negative Z-score means that the original score was below the mean. A positive Z-score means that the original score was above the mean. The actual value corresponds to the number of standard deviations the score is from the mean in that direction. In the first example, a raw creativity score of 7 becomes a z-score of Ð1.60. This implies that the original score of 7 was 1.6 sd units below the mean. In the second example, a z-score of 0.31 implies that the raw score of 14 was 0.31 standard deviations above the mean.

The process of converting or transforming scores on a variable to Z-scores is called standardisation. There are other things in psychological science called "standardisation", so if someone says they "standardised" something, that doesnÕt necessarily mean they converted raw scores to standard scores. A distribution in standard form has a mean of 0 and a standard deviation of 1. However, it is important to note that a z-score transformation changes the central location of the distribution and the average variability of the distribution. It does not change the skewness or kurtosis.

Comparing Scores From Different Distributions. When scores are transformed to a Z-score, it is possible to use these new transformed scores to compare scores from different distributions. Suppose, for example, you took an introductory research methods unit and your friend studied English. You got a 76 (of 100) and your friend got 82 (also of 100). Intuitively, it might seem that your friend did better than you. But what if the class he took was easier than yours? Or what if students in his class varied less or more than students in your class in terms of final marks? In such situations, it is difficult to compare the scores. But if we knew the mean and standard deviations of the two distributions, we could compare these scores by comparing their Z-scores.

Suppose that the mean mark in your class was 54 and the standard deviation was 20 and the mean mark in your friendÕs class 72 and the standard deviation was 15. Your Z score is (76-54)/20 = 1.1. Your friendÕs Z score is (82-72)/15 = 0.67. So, using standard scores, you did better than your friend because your mark was more standard deviations above the class mean than your friends was above his own class mean.

In this example, the distributions were different (different means and standard deviations) but the unit of measurement was the same (% of 100). Using standard scores or percentiles, it is also possible to compare scores from different distributions where measurement was based on a different scale. For example, we could compare two scores from two different intelligence tests, even if the intelligence test scores were expressed in different units (eg, one as an intelligence quotient and one as a percent of answers given correctly). All we would need to know are the means and standard deviations of the corresponding distributions.

Alternatively, we could have used other measures of relative standing to compare across distributions. Scores on standardised tests are often expressed in terms of percentiles, for example. This way, you can know how you did in comparison to other people who took that test that year, in the last five years, or whatever the time period being used when the percentiles were constructed. For example, if your score was in the 65th percentile and your friendÕs was in the 40th percentile, you could justifiably claim that you did better than your friend because you outperformed a greater percent of students in your class than he did in his class.

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