Normal Distributions (see section 1.3 in text)

These are bell-shaped curves that describe the distribution of values of many variables. If we know our data approximates a normal distribution, we can draw some conclusions on how frequent different data values occur.

Density curves

• Show the relative frequencies of a variable's values
• Normalized so that the area under the curve is equal to 1

Here's a link that shows the density curve of a normal distribution. It has an applet that allows you to change the mean and standard deviation.

Properties of normal distributions

• Bell-shaped
• Symmetric
• Unimodal
• Defined by the mean and the standard deviation

68% of the values are within 1 standard deviation of the mean. 95% of the values are within 2 standard deviations of the mean. 99.7% of the values are within 3 standard deviations of the mean.

Standard normal distribution

Defined as a normal distribution where:

• The mean is 0
• The standard deviation is 1

Describing a value in a normal distribution

We can take a value from any normal distribution and find its relative location on the standard normal distribution. This location is called its z-score. It is calculated by subtracting the mean and then dividing by the standard deviation.

Determining the cumulative proportion

If we know the z-score, we can determine what proportion of the data values are less than the z-score. This is the cumulative proportion. The tables in the front of the text show these values. Excel produces these values with the NORMSDIST function.

Normal quantile plot

These are useful for determining how well a distribution fits a normal distribution. Here are instructions for creating a normal quantile plot with Excel.

Log tranformations

Often data sets do not fit a normal distribution. In some cases (often for timing data), performing a log transformation on the data values produces a normal distribution.

The following Excel file demonstrates this:

• Excel worksheets