1. Excursions in Modern Mathematics, 7e: 16.1 - 1Copyright © 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 68-95-99.7 Rule 16.5Normal Curves as Models of Real- Life Data Sets 16.6Distribution of Random Events 16.7Statistical Inference

2. Excursions in Modern Mathematics, 7e: 16.1 - 2Copyright © 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

3. Excursions in Modern Mathematics, 7e: 16.1 - 3Copyright © 2010 Pearson Education, Inc. The bar graph for this data set is shown. Example 16.1Distribution of Heights of NBA Players

4. Excursions in Modern Mathematics, 7e: 16.1 - 4Copyright © 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 16.22007 SAT Math Scores

5. Excursions in Modern Mathematics, 7e: 16.1 - 5Copyright © 2010 Pearson Education, Inc.

6. Excursions in Modern Mathematics, 7e: 16.1 - 6Copyright © 2010 Pearson Education, Inc. Here is a bar graph of the data. Example 16.22007 SAT Math Scores

7. Excursions in Modern Mathematics, 7e: 16.1 - 7Copyright © 2010 Pearson Education, Inc. The two very different data sets have one thing in common–both can be described as having bar graphs that roughly fit a bell- shaped pattern. These data sets have an approximately normal distribution. Approximately Normal Distribution

8. Excursions in Modern Mathematics, 7e: 16.1 - 8Copyright © 2010 Pearson Education, Inc. A distribution of data that has a perfect bell shape is called a normal distribution. Normal Distribution

9. Excursions in Modern Mathematics, 7e: 16.1 - 9Copyright © 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 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

10. Excursions in Modern Mathematics, 7e: 16.1 - 10Copyright © 2010 Pearson Education, Inc. Some bells are short and squat,others are tall and skinny, and others fall somewhere in between. Normal Curves

11. Excursions in Modern Mathematics, 7e: 16.1 - 11Copyright © 2010 Pearson Education, Inc. Symmetry. Every normal curve has a vertical axis of symmetry, splitting the bell-shaped region outlined by the curve into two identical halves. This is the only line of symmetry of a normal curve. Essential Facts About Normal Curves

12. Excursions in Modern Mathematics, 7e: 16.1 - 12Copyright © 2010 Pearson Education, Inc. Median / mean. We will call the point of intersection of the Essential Facts About Normal Curves horizontal axis and the line of symmetry of the curve the center of the distribution.

13. Excursions in Modern Mathematics, 7e: 16.1 - 13Copyright © 2010 Pearson Education, Inc. Median / mean. The center represents both the median M and the mean (average)  of the data. In a normal distribution, M = . The fact that in a normal distribution the median equals the mean implies that 50% of the data are less than or equal to the mean and 50% of the data are greater than or equal to the mean. Essential Facts About Normal Curves

14. Excursions in Modern Mathematics, 7e: 16.1 - 14Copyright © 2010 Pearson Education, Inc. In a normal distribution, M = . (If the distribution is approximately normal, then M ≈ .) MEDIAN AND MEAN OF A NORMAL DISTRIBUTION

15. Excursions in Modern Mathematics, 7e: 16.1 - 15Copyright © 2010 Pearson Education, Inc. Standard Deviation. The standard deviation is an important measure of spread, and it is particularly useful when dealing with normal (or approximately normal) distributions.– Denoted by the Greek letter  called sigma Essential Facts About Normal Curves

16. Excursions in Modern Mathematics, 7e: 16.1 - 16Copyright © 2010 Pearson Education, Inc. Standard Deviation. If you were to bend a piece of wire into a bell- shaped normal curve, at the very top you would be bending the wire downward. Essential Facts About Normal Curves

17. Excursions in Modern Mathematics, 7e: 16.1 - 17Copyright © 2010 Pearson Education, Inc. Standard Deviation. But, at the bottom you would be bending the wire upward. Essential Facts About Normal Curves

18. Excursions in Modern Mathematics, 7e: 16.1 - 18Copyright © 2010 Pearson Education, Inc. Standard Deviation. As you move your hands down the wire, the curvature gradually changes, and there is one point on each side of the curve where Essential Facts About Normal Curves the transition from being bent downward to being bent upward takes place. Such a point is called a point of inflection of the curve.

19. Excursions in Modern Mathematics, 7e: 16.1 - 19Copyright © 2010 Pearson Education, Inc. Standard Deviation. The standard deviation of a normal distribution is the horizontal distance Essential Facts About Normal Curves between the line of symmetry of the curve and one of the two points of inflection, P´ or P in the figure.

20. Excursions in Modern Mathematics, 7e: 16.1 - 20Copyright © 2010 Pearson Education, Inc. In a normal distribution, the standard deviation  equals the distance between a point of inflection and the line of symmetry of the curve. STANDARD DEVIATION OF A NORMAL DISTRIBUTION

21. Excursions in Modern Mathematics, 7e: 16.1 - 21Copyright © 2010 Pearson Education, Inc. Quartiles. We learned in Chapter 14 how to find the quartiles of a data set. When the data set has a normal distribution, the first and third quartiles can be approximated using the mean  and the standard deviation . Multiplying the standard deviation by 0.675 tells us how far to go to the right or left of the mean to locate the quartiles. Essential Facts About Normal Curves

22. Excursions in Modern Mathematics, 7e: 16.1 - 22Copyright © 2010 Pearson Education, Inc. In a normal distribution, Q 3 ≈  + (0.675)  and Q 1 ≈  – (0.675) . QUARTILES OF A NORMAL DISTRIBUTION Pg. 604 # 6

23. Excursions in Modern Mathematics, 7e: 16.1 - 23Copyright © 2010 Pearson Education, Inc. We have seen that normal curves don’t all look alike, but this is only a matter of perception. All normal distributions tell the same underlying story but use slightly different dialects to do it. One way to understand the story of any given normal distribution is to rephrase it in a simple language that uses the mean  and the standard deviation  as its only vocabulary. This process is called standardizing the data. Standardizing the Data

24. Excursions in Modern Mathematics, 7e: 16.1 - 24Copyright © 2010 Pearson Education, Inc. To standardize a data value x, we measure how far x has strayed from the mean  using the standard deviation  as the unit of measurement. A standardized data value is often referred to as a z-value. The best way to illustrate the process of standardizing normal data is by means of a few examples. z-value

25. Excursions in Modern Mathematics, 7e: 16.1 - 25Copyright © 2010 Pearson Education, Inc. Let’s consider a normally distributed data set with mean  = 45 ft and standard deviation  = 10 ft. We will standardize several data values, starting with a couple of easy cases. Example 16.4Standardizing Normal Data

26. Excursions in Modern Mathematics, 7e: 16.1 - 26Copyright © 2010 Pearson Education, Inc. ■ x 4 = 21.58 is... uh, this is a slightly more complicated case. How do we handle this one? First, we find the signed distance between the data value and the mean by taking their difference (x 4 –  ). In this case we get 21.58 ft – 45 ft = –23.42 ft. (Notice that for data values smaller than the mean this difference will be negative.) Example 16.4Standardizing Normal Data

27. Excursions in Modern Mathematics, 7e: 16.1 - 27Copyright © 2010 Pearson Education, Inc. ■ If we divide this difference by  = 10 ft, we get the standardized value z 4 = –2.342. This tells us the data point x 4 is –2.342 standard deviations from the mean  = 45 ft (D in the figure). Example 16.4Standardizing Normal Data

28. Excursions in Modern Mathematics, 7e: 16.1 - 28Copyright © 2010 Pearson Education, Inc. In a normal distribution with mean  and standard deviation , the standardized value of a data point x is z = (x –  )/ . STANDARDING RULE Pg. 605 # 18a,b,

29. Excursions in Modern Mathematics, 7e: 16.1 - 29Copyright © 2010 Pearson Education, Inc. One important point to note is that while the original data is given with units, there are no units given for the z-value. The units for the z-value are standard deviations, and this is implicit in the very fact that it is a z-value. Standardizing Normal Data

30. Excursions in Modern Mathematics, 7e: 16.1 - 30Copyright © 2010 Pearson Education, Inc. The process of standardizing data can also be reversed, and given a z-value we can go back and find the corresponding x-value. To do this take the formula z = (x –  )/  and solve for x in terms of z. x =  +  z Pg. 605 # 22a,c Groups Pg. 604 #4d, 8, 18c,d, 22b,d, 25 Finding the Value of a Data Point

31. Excursions in Modern Mathematics, 7e: 16.1 - 31Copyright © 2010 Pearson Education, Inc. In a typical bell-shaped distribution, most of the data are concentrated near the center. As we move away from the center the heights of the columns drop rather fast, and if we move far enough away from the center, there are essentially no data to be found. The 68-95-99.7 Rule

32. Excursions in Modern Mathematics, 7e: 16.1 - 32Copyright © 2010 Pearson Education, Inc. These are all rather informal observations, but there is a more formal way to phrase this, called the 68-95-99.7 rule. This useful rule is obtained by using one, two, and three standard deviations above and below the mean as special landmarks. In effect, the 68-95-99.7 rule is three separate rules in one. The 68-95-99.7 Rule

33. Excursions in Modern Mathematics, 7e: 16.1 - 33Copyright © 2010 Pearson Education, Inc. 1.In every normal distribution, about 68% of all the data values fall within one standard deviation above and below the mean. In other words, 68% of all the data have standardized values between z = –1 and z = 1. THE 68-95-99.7 RULE

34. Excursions in Modern Mathematics, 7e: 16.1 - 34Copyright © 2010 Pearson Education, Inc. THE 68-95-99.7 RULE

35. Excursions in Modern Mathematics, 7e: 16.1 - 35Copyright © 2010 Pearson Education, Inc. 2. In every normal distribution, about 95% of all the data values fall within two standard deviations above and below the mean. In other words, 95% of all the data have standardized values between z = –2 and z = 2. THE 68-95-99.7 RULE

36. Excursions in Modern Mathematics, 7e: 16.1 - 36Copyright © 2010 Pearson Education, Inc. THE 68-95-99.7 RULE

37. Excursions in Modern Mathematics, 7e: 16.1 - 37Copyright © 2010 Pearson Education, Inc. 3. In every normal distribution, about 99.7% (i.e., practically 100%) of all the data values fall within three standard deviations above and below the mean. In other words, 99.7% of all the data have standardized values between z = –3 and z = 3. THE 68-95-99.7 RULE

38. Excursions in Modern Mathematics, 7e: 16.1 - 38Copyright © 2010 Pearson Education, Inc. THE 68-95-99.7 RULE

39. Excursions in Modern Mathematics, 7e: 16.1 - 39Copyright © 2010 Pearson Education, Inc. Earlier in the text, we defined the range R of a data set (R = Max – Min) and, in the case of an approximately normal distribution, we can conclude that the range is about six standard deviations. This is true as long as we can assume that there are no outliers. Pg. 606 # 32a, 34, 42, 52c Groups # 34, 44a,d, 52a,b,d,e Practical Implications