1. MAT 1000 Mathematics in Today's World

2. Last Time We looked at the standard deviation, a measurement of the spread of a distribution. We introduced a special type of distribution, the normal distribution. These highly symmetric distributions are very common. We saw how, using only the mean and standard deviation, we can find the first and third quartiles of a normal distribution.

3. Today Using the mean and standard deviation, we can find out much more about a normal distribution. In particular, we will be able to easily find any of the percentiles of the distribution. To do so, we need to find standard scores, or z-scores. First, we address the question of why normal distributions are so common.

4. Today Note: Today’s material is not in the textbook.

5. Example of normal distributions Physical characteristics like height or weight. The annual returns on the S&P 500 over the last 50 years. Cars in the parking lot of a mall. How long it takes a kernel of popcorn to pop in the microwave.

6. Why are normal distributions so common? Normal distributions are “bell” shaped, so most of the data is close to the center (the mean), and only rarely are there numbers far from the mean. We expect this distribution whenever there are many conflicting forces that tend to cancel each other out. This is the case in lots of situations.

7. Why are normal distributions so common? Example What forces can affect stock returns during a year? Usually lots of little things: new products, bad publicity, changing government regulations, even the weather. If we combine the returns of 500 companies, then all of these small factors tend to cancel out. This means the S&P 500 return will usually be close to the average.

8. Why are normal distributions so common? Example What forces determine a person’s height? Lots of reasons. There are genetic factors, but things like childhood nutrition or illness also play a role. With lots of small forces that tend to conflict, it’s no surprise that most people tend to be close to average height.

9. Why are normal distributions so common? Isn’t it true that in any data set most of the data will be close to the mean? Absolutely not! Suppose 10 people take a test. Five score 0, and five score 100. The mean is 50, but nobody is close to that. Why do people believe that most of the data in a distribution should be close to the mean? Precisely because normal distributions are so common.

10. Percentiles The median and the first and third quartiles are examples of what are called percentiles. For any number P between 0 and 100, we can find the P th percentile of a distribution. By definition, P percent of the data is less than the P th percentile For example, Q 1 could also be called the 25 th percentile—25% of the data is less than Q 1

11. Percentiles

15. Standard scores

17. Finding percentiles

18. Table of percentiles

19. Finding percentiles

21. Table of percentiles

22. Finding percentiles

24. Comparing percentiles Using the mean and standard deviation of a normal distribution, we can find the percentile of any data value from that distribution. Percentiles are also very useful for comparing data values from different distributions. Is a 600 on the SAT math test better or worse than a 26 on the ACT math test? We can’t compare the numbers—the SAT is out of 800 and the ACT is out of 36.

25. Comparing percentiles

26. From the table, a standard score of 1 is the 84 th percentile, while a standard score of 1.3 is the 90 th percentile. So a 26 on the ACT is better than a 600 on the SAT. In what sense is it a better score? Percentiles describe these scores relative to all the other test takers. Scoring higher than 90% of the people who took a test is better than scoring higher than 84%.

27. Another normal distribution One of the most important examples of a normal distribution is the sampling distribution of statistics In a sample survey, we choose a sample and compute a statistic. A different sample would have given a different statistic. If we consider every possible sample, we would have a distribution of statistics (which numbers occur, and how often they occur).

28. Another normal distribution It turns out that if our sample size is large enough, the distribution of statistics will be normal. What is a large enough sample? A general rule of thumb is a sample size of 30.

29. Another normal distribution Example In 2012 Barack Obama won the presidential election with 51.1% of the vote. In the run up to the election, there were many polls of likely voters. These polls were producing statistics to estimate a parameter: the proportion of all voters who were going to vote for Obama. Now, we know the value of this parameter to be 51.1%

30. Another normal distribution Example Suppose a polling company sampled 100 voters before the election. It turns out that the distribution of statistics for a sample size of 100 is normal with mean 51.1% and standard deviation 5%. (We’ll see the formulas for these later in the course.) As decimals these are 0.511 and 0.05.

31. Another normal distribution

33. Example So, in about 58% of the possible samples of 100 voters, we would have seen more than 50% of the sample supporting Obama.

34. Another normal distribution