# Statistics

## Applications of the Normal Distribution


### Learning Objectives

After completing this section, you should be able to:
1. Apply the normal distribution to real-world scenarios.
As we saw in The Normal Distribution, the word “standardized” is closely associated with the normal distribution. This is why tests like college entrance exams, state achievement tests for K–12 students, and Advanced Placement tests are often called “standardized tests”: scores are assigned in a way that forces them to follow a normal distribution, with a mean and standard deviation that are consistent from year to year. Standardization also allows people like college admissions officers to directly compare an applicant who took the ACT (a college entrance exam) to an applicant who instead chose to take the SAT (a different college entrance exam). Standardization allows us to compare individuals from different groups; this is among the most important applications of the normal distribution. We’ll explore this and other real-world uses of the normal distribution in this section.

### College Entrance Exams

There are two good ways to compare two data values from different groups: using -scores and using percentiles. The two methods will always give consistent results (meaning that we won’t find, for example, that the first value is better using -scores but the second value is better using percentiles), so use whichever method is more comfortable for you.

### Coin flipping

In the opening of The Normal Distribution, we saw that the number of heads we get when we flip a coin 100 times is distributed normally. It can be shown that if  is the number of flips, then the mean of that distribution is  and the standard deviation is  (as long as ). So, for 100 flips, the mean of the distribution is 50 and the standard deviation is 5.
          In that opening example, one of our early runs gave us 70 heads in 100 flips, which we noted seemed unusual. Using the normal distribution, we can identify exactly how unusual that really is. Using Google Sheets, the formula “=NORM.DIST(70, 50, 5, TRUE)” gives us 0.999968, which is the 99.997th percentile! How is that useful? Suppose you need to test whether a coin is fair, and so you flip it 100 times. While we might be suspicious if we get 70 heads out of the 100 flips, we now have a numerical measure for how unusual that is: If the coin were fair, we would expect to see 70 heads (or more) only  of the time. That’s really unlikely! Analysis like this is related to hypothesis testing, an important application of statistics in the sciences and social sciences.

### Analyzing Data That Are Normally Distributed

Whenever we’re working with a dataset that has a distribution that looks symmetric and bell-shaped, we can use techniques associated with the normal distribution to analyze the data.

### Check Your Understanding


### Key Concepts

1. We can use -scores to compare data values from different datasets.