    Chapter 4: Analysing the Data Part II : Descriptive Statistics # The standard normal distribution

Any distribution can be converted to a standardised distribution. However the symmetry of the original distribution remains unchanged. If the original distribution was skewed to start with, it will still be skewed after the z-score transformation. In the special case where the original distribution can be considered normal, standardising will result in what is known as the standard normal distribution. The advantage of this is that tables exist in any statistics textbook for the area under the curve for the standard normal distribution (or "normal curve"). From these tables you can estimate the answer to many questions about the original distribution.

Importantly, the percent (or proportion) of scores falling above or below any Z-score is known and tabled (in Howell, see Appendix Z, p. 695). So, consider the following example. Suppose the distribution of final exam marks in PESS202 is roughly normally distribution with a mean of 68 and a standard deviation of 10. What percent of students score above 73? To answer this question, you need to express 73 as a Z-score. Using the procedure already described, the Z-score is 0.5. In other words, 73 is one half of a standard deviation above the mean. So we want to know what percent of the scores are above one half of a standard deviation (that is, with a Z-score greater than 0.5). By using the table in Appendix Z of Howell, we see that the proportion of scores above a Z of 0.5 is .3085 if the distribution is normal. So expressed as a percentage, 30.85% or about 31% of students score above 73. Using the same procedure described earlier and a standard normal table, it is possible to entertain questions about percentages or proportions of scores greater than, less than, or between any score or set of scores.

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