Small Sample Histograms - Shapes (Crunch-It)

In chapters 18-19, we can only use the techniques on small samples if we believe the underlying popluation distribution is normal. And sometimes the only information we have about the population distribution is the data in the sample.

That's not really adequate to say whether it's reasonable to assume the population is normally distributed. Sometimes it might be adequate to be clear that the population is NOT normally distributed.

In order to get a feeling for this, you simply need to look at a lot of graphs of small datasets from normal populations in order to see what shapes they can have.

In Crunch-It, you can investigate this in the following way:

In a blank Crunch-It worksheet (or one that already has some data - it doesn't matter) generate some random data from a Normal distribution. Choose Data > Simulation > Normal.
I chose to make samples of size 11.
I wanted ten of them, so I put them into columns 3 through 12.
I chose to use a mean of 3 and a standard deviation of 1.

After all that data appeared in the spreadsheet, I chose Graphics > Histogram
Then I entered all of the columns in the resulting box, so it would make ten histograms. When I viewed those histograms, I just have to hit Next each time to see the next histogram.

Notice that, even though these all came from a normal distribution, the shapes of the histograms of the small samples don't all look alike. This shows there can be quite a bit of variability in these shapes.

To go even further, I repeated this for a very skewed population distribution, which I chose to be Chi-Squared with parameter 1.

In this case, many of the samples of size 11 did show a high outlier. So this illustrates that even a small sample is likely to show an outlier if the population is very skewed.