The normal distribution is demonstrated by many phenomena -- for example, in the distribution of the weights of women in a population. Most will bunch up around the average (mean) weight, then fewer and fewer people will be found in the heaviest and lightest weight categories. When plotted, such data forms a bell-shaped curve, where the horizontal axis is weight and the vertical axis is the number of people of this weight. Using this general relationship, it is also possible to calculate proportions. In our example this could involve finding out what proportion (percentage) of women are under a certain weight.
Decide on the value, or values, that you want to use to define a group -- for example, the proportion of women below a certain weight, or between two weights. In our example, we wish to find the proportion of women below a certain value, which is given by the area under the normal curve to the left of the value.
Calculate the z-score for that value. This is given by the formula
where Z is the z-score, X is the value you are using, m is the population mean and s is the standard deviation of the population.
Consult a unit normal table to find the proportion of the area under the normal curve falling to the side of your value. The left-hand column gives the z-score to a single decimal place (0.0 to 3.0). Follow this down until you reach the correct row for your z-score. The top horizontal row gives the second decimal place for the z-score (0.00 to 0.09). Now follow your row horizontally until you reach the correct column.
Take the number obtained from the unit normal table and subtract this from 0.5. Now subtract the resulting number from 1. In our example, this gives the proportion of women below a certain weight. To obtain the percentage, we need to multiply this by 100.
- "Understanding Statistics"; Graham Upton and Ian Cook; 1997
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