Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 16 Mathematics of Normal Distributions 16.1Approximately Normal Distributions of Data 16.2Normal Curves and Normal Distributions 16.3Standardizing Normal Data 16.4The Rule 16.5Normal Curves as Models of Real- Life Data Sets 16.6Distribution of Random Events 16.7Statistical Inference

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. This table is a frequency table giving the heights of 430 NBA players listed on team rosters at the start of the 2008–2009 season. Example 16.1Distribution of Heights of NBA Players

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The bar graph for this data set is shown. Example 16.1Distribution of Heights of NBA Players

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. We can see that the bar graph fits roughly the pattern of a somewhat skewed (off-center) bell-shaped curve (the orange curve). Example 16.1Distribution of Heights of NBA Players

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. An idealized bell-shaped curve for this data (the red curve) is shown for comparison purposes. The data would be even more bell- shaped if it weren’t for all the 6’7” to 7’ players. Example 16.1Distribution of Heights of NBA Players

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. This is not a quirk of nature but rather a reflection of the way NBA teams draft players. Example 16.1Distribution of Heights of NBA Players

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The table on the next slide shows the scores of N = 1,494,531 college-bound seniors on the mathematics section of the 2007 SAT. (Scores range from 200 to 800 and are grouped in class intervals of 50 points.) The table shows the score distribution and the percentage of test takers in each class interval. Example SAT Math Scores

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Here is a bar graph of the data. Example SAT Math Scores

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The orange bell-shaped curve traces the pattern of the data in the bar graph. If the data followed a perfect bell curve, it would follow the red curve shown in the figure. Example SAT Math Scores

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Unlike the curves in Fig.16-3, here the orange and red curves are very close. Example SAT Math Scores

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The two very different data sets discussed in Examples 16.1 and 16.2 have one thing in common–both can be described as having bar graphs that roughly fit a bell- shaped pattern. In Example 16.1, the fit is crude; in Example 16.2, it is very good. In either case, we say that the data set has an approximately normal distribution. Approximately Normal Distribution

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The word normal in this context is to be interpreted as meaning that the data fits into a special type of bell-shaped curve; the word approximately is a reflection of the fact that with real-world data we should not expect an absolutely perfect fit. A distribution of data that has a perfect bell shape is called a normal distribution. Normal Distribution

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Perfect bell-shaped curves are called normal curves. Every approximately normal data set can be idealized mathematically by a corresponding normal curve (the red curves in Examples 16.1 and 16.2). This is important because we can then use the mathematical properties of the normal curve to analyze and draw conclusions about the data. Normal Curves

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The tighter the fit between the approximately normal distribution and the normal curve, the better our analysis and conclusions are going to be. Thus, to understand real-world data sets that have an approximately normal distribution, we first need to understand some of the mathematical properties of normal curves. Normal Curves

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. As usual, we will use the letter N to represent the size of the data set. In real- life applications, data sets can range in size from reasonably small (a dozen or so data points) to very large (hundreds of millions of data points), and the larger the data set is, the more we need a good way to describe and summarize it. Data Set

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Example 14.1Stat 101 Test Scores

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Like students everywhere, the students in the Stat 101 class have one question foremost on their mind when they look at the results: How did I do? Each student can answer this question directly from the table. It’s the next question that is statistically much more interesting. How did the class as a whole do? To answer this last question, we will have to find a way to package the results into a compact, organized, and intelligible whole. Example 14.1Stat 101 Test Scores

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The first step in summarizing the information in Table 14-1 is to organize the scores in a frequency table such as Table In this table, the number below each score gives the frequency of the score–that is, the number of students getting that particular score. Example 14.2Stat 101 Test Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. We can readily see from Table 14-2 that there was one student with a score of 1, one with a score of 6, two with a score of 7, six with a score of 8, and so on. Notice that the scores with a frequency of zero are not listed in the table. Example 14.2Stat 101 Test Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. We can do even better. Figure 14-1 (next slide) shows the same information in a much more visual way called a bar graph, with the test scores listed in increasing order on a horizontal axis and the frequency of each test score displayed by the height of the column above that test score. Notice that in the bar graph, even the test scores with a frequency of zero show up–there simply is no column above these scores. Example 14.2Stat 101 Test Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Figure 14-1 Example 14.2Stat 101 Test Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Bar graphs are easy to read, and they are a nice way to present a good general picture of the data. With a bar graph, for example, it is easy to detect outliers–extreme data points that do not fit into the overall pattern of the data. In this example there are two obvious outliers–the score of 24 (head and shoulders above the rest of the class) and the score of 1 (lagging way behind the pack). Example 14.2Stat 101 Test Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Sometimes it is more convenient to express the bar graph in terms of relative frequencies –that is, the frequencies given in terms of percentages of the total population. Figure 14-2 shows a relative frequency bar graph for the Stat 101 data set. Notice that we indicated on the graph that we are dealing with percentages rather than total counts and that the size of the data set is N = 75. Example 14.2Stat 101 Test Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Figure 14-2 Example 14.2Stat 101 Test Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. This allows anyone who wishes to do so to compute the actual frequencies. For example, Fig indicates that 12% of the 75 students scored a 12 on the exam, so the actual frequency is given by 75  0.12 = 9 students. The change from actual frequencies to percentages (or vice versa) does not change the shape of the graph–it is basically a change of scale. Example 14.2Stat 101 Test Scores: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Frequency charts that use icons or pictures instead of bars to show the frequencies are commonly referred to as pictograms. The point of a pictogram is that a graph is often used not only to inform but also to impress and persuade, and, in such cases, a well- chosen icon or picture can be a more effective tool than just a bar. Here’s a pictogram displaying the same data as in figure Bar Graph versus Pictogram

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Figure 14-3 Bar Graph versus Pictogram

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. This figure is a pictogram showing the growth in yearly sales of the XYZ Corporation between 2001 and It’s a good picture to Example 14.3Selling the XYZ Corporation show at a shareholders meeting, but the picture is actually quite misleading.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. This figure shows a pictogram for exactly the same data with a much more accurate and sobering picture of how well the XYZ Example 14.3Selling the XYZ Corporation Corporation had been doing.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The difference between the two pictograms can be attributed to a couple of standard tricks of the trade: (1) stretching the scale of the vertical axis and (2) “cheating” on the choice of starting value on the vertical axis. As an educated consumer, you should always be on the lookout for these tricks. In graphical descriptions of data, a fine line separates objectivity from propaganda. Example 14.3Selling the XYZ Corporation