Shape of a probability density function
A probability density function (i.e. population histogram) can have any shape, though it is usually a fairly smooth curve. Indeed, we often have only rough information about its likely shape from a single sample histogram.
Normal distributions
One family of symmetric continuous probability density functions called normal distributions is particularly useful. Although normal distributions are only appropriate as population models for a small number of data sets, they are extremely important in statistics — their importance will be explained later in this chapter.
At this stage, we will use normal distributions to give a concrete example of a probability density function.
The shape of the normal distribution depends on two numerical values, called parameters, that can be adjusted to give a range of symmetric distributional shapes. The two normal parameters are called µ and σ and are the distribution's mean and standard deviation.
Shape of the normal family of distributions
Use the two sliders to adjust the normal parameters. Observe that the location and spread of the distribution are changed, but other aspects of its shape remain the same for all values of the parameters.
Note also that the total area under the probability density function remains the same (exactly 1.0) for all values of the parameters. This holds for all probability density functions.
For some data sets, a normal distribution does provide a reasonable model. The two parameters can be chosen to make the distribution's shape match that of a histogram of the data as closely as possible.
Job satisfaction ratings
Merchandise buyers are critically important to the success or failure of retail ventures, both through negotiations over pricing and decisions about the types of product that will be stocked. Because of high turnover of staff in this area (estimated to be over 25 percent per year), research was carried out to determine the factors related to turnover.
The diagram below shows job satisfaction ratings of a sample of 212 buyers on a scale of 0 (worst) to 30 (best), with a superimposed normal probability density function.
Use the sliders to adjust the normal parameters to obtain as close as possible a match to the histogram. This normal distribution can be used as an approximate model for how the data might have arisen.
We have used a subjective procedure of matching the shapes of the histogram and probability density 'by eye'. A more objective way to 'estimate' the normal parameters will be presented in the next chapter. Click the button Best fit to apply this objective method.
We will revisit normal distributions later in this chapter.