Chapter 8 Statistics
8.6 The Normal Distribution
Learning Objectives
By the end of this section, you will be able to:
- Describe the characteristics of the normal distribution
- Apply the 68-95-99.7 percent groups to normal distribution datasets
- Use the normal distribution to calculate a [latex]z[/latex]-score
- Find and interpret percentiles and quartiles
Many datasets that result from natural phenomena tend to have histograms that are symmetric and bell-shaped. Imagine finding yourself with a whole lot of time on your hands and nothing to keep you entertained but a coin, a pencil, and paper. You decide to flip that coin 100 times and record the number of heads. With nothing else to do, you repeat the experiment 10 times total. Using a computer to simulate this series of experiments, here’s a sample for the number of heads in each trial:
54, 51, 40, 42, 53, 50, 52, 52, 47, 54
It makes sense that we’d get somewhere around 50 heads when we flip the coin 100 times, and it makes sense that the result won’t always be exactly 50 heads. In our results, we can see numbers that were generally near 50 and not always 50, like we thought.
Moving toward Normality
Let’s take a look at a histogram for the dataset in our section opener:
This is interesting, but the data seem pretty sparse. There were no trials where you saw between 43 and 47 heads, for example. Those results don’t seem impossible; we just didn’t flip enough times to give them a chance to pop up. So let’s do it again, but this time we’ll perform 100 coin flips 100 times. Rather than review all 100 results, which could be overwhelming, let’s instead visualize the resulting histogram.
From the histogram, we see that most of the trials resulted in between, say, 44 and 56 heads. There were some more unusual results: one trial resulted in 70 heads, which seems really unlikely (though still possible!). But we’re starting to maybe get a sense of the distribution. More data would help, though. Let’s simulate another 900 trials and add them to the histogram!
We can still see that 70 is a really unusual observation, though we came close in another trial (one that had 68 heads). Now the distribution is coming more into focus: it looks quite symmetric and bell-shaped. Let’s just go ahead and take this thought experiment to an extreme conclusion: 10,000 trials.
The distribution is pretty clear now. Distributions that are symmetric and bell-shaped like this pop up in all sorts of natural phenomena, such as the heights of people in a population, the circumferences of eggs of a particular bird species, and the numbers of leaves on mature trees of a particular species. All of these have bell-shaped distributions. Additionally, the results of many types of repeated experiments generally follow this same pattern, as we saw with the coin-flipping example; this fact is the basis for much of the work done by statisticians. It’s a fact that’s important enough to have its own name: the Central Limit Theorem.
The Norm