Chapter 4: Analysing the Data |

## Z-scoresAnother 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 The following data will be used as an example.
## 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 Đ 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 Đ1.60. 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
Suppose that the mean mark in your class was 54 and the standard deviation was 20 and the mean mark in your friendŐs class 72 and the standard deviation was 15. Your Z score is (76-54)/20 = 1.1. Your friendŐs 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 |

© Copyright 2000 University of New England, Armidale, NSW, 2351. All rights reserved Maintained by Dr Ian Price |