280 Appendix: A Crash Course in Fundamental Statistical Concepts Table A.1 Partial z-Scores to Percentile Rank Table z-Score 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Percentile 84.13 86.43 88.49 90.32 91.92 93.32 94.52 95.54 For example, Table A.1 is a section from a normal table. There isn't an entry for 1.28 but we can see it falls somewhere between 88.49% and 90.32% (closer to 90.32%). Because the total area must add up to 100% under the curve, we can express a z-score of 1.28 as being higher than 90% of values or less than 10% of values (100% minus 90%). APPLYING THE NORMAL CURVE TO USER RESEARCH DATA The examples so far have been mostly about height, weight, and IQ scores--metrics that nicely follow a normal distribution. In our experience, user researchers rarely use these metrics, more typi- cally using measurements such as averages from rating scales and completion rates. Graphs of the distributions of these types of data are usually far from normal. For example, Figure A.8 shows 15 SUS scores from a usability test of the Budget.com website. It is hardly bell-shaped or even symmetrical. The average SUS score from this sample of 15 users is 80 with a standard deviation of 24. It's understandable to be a bit concerned about how much faith to put into this mean as a measure of central tendency because the data aren't symmetric. It is certainly even more of a concern about how we can use the normal curve to make inferences about this sort of data. It turns out this sample of 15 comes from a larger sample of 311 users, with all the values shown in Figure A.9. The mean SUS score of these data is 78. Again, the shape of this distribution makes you wonder if the normal curve is even relevant. However, if we take 1,000 random samples of 15 users from this large population of 311, then graph the 1,000 means, we get the graph shown in Figure A.10. Although the large sample of 311 SUS scores is not normal, the distribution of the random means shown in Figure A.10 does follow a normal distri- bution. The same principle applies if the population we draw randomly from is 311 or 311 million. CENTRAL LIMIT THEOREM Figure A.10 illustrates one of the most fundamental and important statistical concepts--the Central Limit Theorem. In short, this theorem states that as the sample size approaches infinity, the distribu- tion of sample means will follow a normal distribution regardless of what the parent population