Normal distribution (mu,sigma)

The ubiquitousness of the normal distribution is clearly not with mean 0 and standard deviation one; for example, many data such as heights and weights are never negative. But if data is normally distributed, it can be transformed to have mean 0 and standard deviation 1, and the transformed data will be standard normal for which the tables can be used. In particular, if some data {x} are normally distributed, the corresponding {z} will be normal with mean 0 and standard deviation 1 where the correspondence between the {x} and {z} is given by z = (x-mu)/sigma. Such z's are called z-scores. By subtracting mu, the mean has been shifted to 0, by dividing by sigma the standard deviation has been changed to 1.

Relative frequency of being in an interval

Assume that height is normally distributed with mean=57 and standard deviation = 5. What fraction of people are between the heights of 55 and 60 inches? The z-scores corresponding to 55 and 60 are (55-57)/5 = -.4 and (60-57)/5 = .6. At this stage the calculations are those for the standard normal distribution. From the table we get -.4 corresponds to .3446, .6 corresponds to .7257, hence the area between is .7257-.3446=.3811. The following figure may clarify what we have done.

Cutoff that relative frequency is below

If height has a mean of 67 and a standard deviation of 5, we can also ask what height 20% of the students will be shorter than. The first step is to convert the area (relative frequency) to a z-value using the table. Since we have specified the area to the left, we look for .2 in the body of the table, and find .2005 which corresponds to the z-value -.84.

Then we must rearrange the formula z = (x-mu)/sigma to x = mu + (z)(sigma) to get back to inches from standard deviation units. 67+(-.84)(5)=62.8. (Drawing a picture may help you remember when to add or subtract.)

Summary

If data is normally distributed, but the mean is not 0 and/or the standard deviation is not one, there are two stages to problems:
• If the interval is specified and you are looking for the relative frequency (probability): first convert from x-values to z-scores, then use the normal table to find the area (relative frequency, probability).
• If the relative frequency (probability) is specified, first use the normal table to find the z-values, then convert the z-values to x-values.

Applets: A good applet for showing the correspondence between raw data and z-scores by Gary McClelland is linked here (you need to hit enter after entering your values). An alternative version is also available.

Competencies: If height is normally distributed with mean 69 inches and standard deviation 4 inches (N(69, 4²)), what fraction of the people are between 60 and 70 inches in height?
If height is normally distributed with mean 69 inches and standard deviation 4 inches (N(69, 4²)), what height are 10% of the people taller than?