Uses in Practice 3

Uses in Practice
Adapted from the forthcoming CyberStats
introductory statistics course.

Page 3- From Sample to Model

Based on a sample of data, how can we decide if a normal curve is an appropriate model tor the population measurements?

Does the Sample Histogram Have a Bell Shape?

If the answer to this question is "yes," a normal curve model may be a useful description of the population. The sample mean and sample standard deviation can be used to estimate the mean and standard deviation of the population.

An Example

The normal curve model for how much college students sleep each night was based on a survey of 352 students at Penn State University.

Below you"ll see a histogram of the responses. The histogram looks more or less bell shaped so a normal curve model may be appropriate.

Histogram

Show Model

Model Only

By moving your mouse slowly over the links underneath the histogram, you can see three different graphs -

the sample histogram
the sample histogram and a normal curve [m = 7, s = 1. 7 ]
the normal curve model with m = 7 and s = 1. 7.

A bit of exploration should convince you that the normal curve is a reasonably good approximation of the histogram.

The Number of Bins Affects the Histogram's Appearance

In the activity below, use the slider labeled "Bin Width" to change the number of bars in the histogram.

As you do this consider these questions :

Does the number of histogram categories affect the apparent shape of the distribution?
Is it always obvious that a normal curve will approximate the histogram?

It's easy to see that the shape of the histogram is definitely affected by the number of categories. This makes it hard to determine that a normal curve model is a good approximation.

The overall impression for this example, however, is that the sample distribution has a bell shape.

How to Decide if a Normal Curve Describes a Sample

Think about whether the general features of the normal curve make sense for the variable in question. For instance, are measurements close to the mean more likely to occur than measurements more distant from the mean?
Examine a sample histogram of the data. You may need to try a few histograms, each with a different number of categories.
Look at how well the empirical rule works for the sample. It won't work well for skewed data.
For small samples, a more advanced plot, called a normal probability plot, can be used. When the sample size is small, a histogram is an unreliable estimate of the population model.