1. Chapter 6 Measures of Location Normal Distributions 1 Larson/Farber 4th ed

2. Objectives Define & Understand Percentiles & their Graphs Define & interpret z-scores Define and Understand the Normal Curve Interpret and Apply the Empirical Rule Interpret graphs of normal probability distributions Find areas under the standard normal curve 2 Larson/Farber 4th ed

3. + Describing Location in a Distribution Measuring Position: Percentiles One way to describe the location of a value in a distributionis to tell what percent of observations are less than it. Definition: The p th percentile of a distribution is the value with p percent of the observations less than it. 6 7 7 2334 7 5777899 8 00123334 8 569 9 03 Jenny earned a score of 86 on her test. How did she perform relative to the rest of the class? Example, p. 85 Her score was greater than 21 of the 25 observations. Since 21 of the 25, or 84%, of the scores are below hers, Jenny is at the 84 th percentile in the class’s test score distribution. 6 7 7 2334 7 5777899 8 00123334 8 569 9 03

4. + Cumulative Relative Frequency Graphs A cumulative relative frequency graph (or ogive ) displays the cumulative relative frequency of eachclass of a frequency distribution. Describing Location in a Distribution Age of First 44 Presidents When They Were Inaugurated AgeFrequencyRelative frequency Cumulative frequency Cumulative relative frequency 40- 44 22/44 = 4.5% 22/44 = 4.5% 45- 49 77/44 = 15.9% 99/44 = 20.5% 50- 54 1313/44 = 29.5% 2222/44 = 50.0% 55- 59 1212/44 = 34% 3434/44 = 77.3% 60- 64 77/44 = 15.9% 4141/44 = 93.2% 65- 69 33/44 = 6.8% 4444/44 = 100%

5. + Describing Location in a Distribution Was Barack Obama, who was inaugurated at age 47,unusually young? Estimate and interpret the 65 th percentile of the distribution Cumulative Relative Frequency GraphsA cumulative relative frequency graph (or ogive ) displays the cumulative relative frequency of each class of a frequencydistribution. 47 11 65 58

6. 6 Comparing data sets How do we compare results when they are measured on two completely different scales? One solution might be to look at percentiles What might you say about a woman that is in the 50 th percentile and a man in the 15 th percentile?

7. 7 Another way of comparing Another way of comparing: Look at whether the data point is above or below the mean, and by how much. Example: Compare an SAT score of 1080 to an ACT score of 28 given the mean SAT is 896 with standard deviation of 174 & the mean ACT is 20.6 with a standard deviation of 5.2

8. Slide 6- 8 The Standard Deviation as a Ruler The trick in comparing very different-looking values is to use standard deviations as our rulers. The standard deviation tells us how the whole collection of values varies, so it’s a natural ruler for comparing an individual to a group. As the most common measure of variation, the standard deviation plays a crucial role in how we look at data.

9. Slide 6- 9 Standardizing with z-scores We compare individual data values to their mean, relative to their standard deviation using the following formula: We call the resulting values standardized values, denoted as z. They can also be called z-scores.

10. Slide 6- 10 Standardizing with z-scores (cont.) Standardized values have no units. z-scores measure the distance of each data value from the mean in standard deviations. A negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean.

11. Slide 6- 11 Benefits of Standardizing Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean. Thus, we can compare values that are measured on different scales, with different units, or from different populations.

12. Slide 6- 12 Back to z-scores As the formula indicates, standardizing data into z- scores shifts the data by subtracting the mean and rescales the values by dividing by their standard deviation.  Standardizing into z-scores does not change the shape of the distribution.  Standardizing into z-scores changes the center by making the mean 0.  Standardizing into z-scores changes the spread by making the standard deviation 1.

13. Slide 6- 13 When Is a z-score Big? A z-score gives us an indication of how unusual a value is because it tells us how far it is from the mean. Remember that a negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean. The larger a z-score is (negative or positive), the more unusual it is.

14. EXAMPLE Enter the following data into L1 on your calculator and construct a histogram. Also run the 1-Vars Stats and record the mean and sample standard deviation. Amount of precipitation in Newark over the course of a year: 3.542.884.164.24.094.024.763.73.82 3.63.653.81

15. Slide 6- 15 When Is a z-score Big? (cont.) There is no universal standard for z-scores, but there is a model that shows up over and over in Statistics. This model is called the Normal model (You may have heard of “bell-shaped curves.”). Normal models are appropriate for distributions whose shapes are unimodal and roughly symmetric. These distributions provide a measure of how extreme a z-score is.

16. Slide 6- 16 When Is a z-score Big? (cont.) There is a Normal model for every possible combination of mean and standard deviation obtained by rescaling the data to z-scores.  We write N(μ,σ) to represent a Normal model with a mean of μ and a standard deviation of σ.

17. Properties of Normal Distributions Normal distribution A continuous probability distribution for a random variable, x. The most important continuous probability distribution in statistics. The graph of a normal distribution is called the normal curve. x 17 Larson/Farber 4th ed

18. Properties of Normal Distributions 1.The mean, median, and mode are equal. 2.The normal curve is bell-shaped and symmetric about the mean. 3.The total area under the curve is equal to one. 4.The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the mean. x Total area = 1 μ 18 Larson/Farber 4th ed

19. Properties of Normal Distributions 5.Between μ – σ and μ + σ (in the center of the curve), the graph curves downward. The graph curves upward to the left of μ – σ and to the right of μ + σ. The points at which the curve changes from curving upward to curving downward are called the inflection points. μ  3σ μ + σ μ  2σμ  σμ  σ μμ + 2σμ + 3σ x Inflection points 19 Larson/Farber 4th ed

20. Means and Standard Deviations A normal distribution can have any mean and any positive standard deviation. The mean gives the location of the line of symmetry. The standard deviation describes the spread of the data. μ = 3.5 σ = 1.5 μ = 3.5 σ = 0.7 μ = 1.5 σ = 0.7 20 Larson/Farber 4th ed

21. Example: Understanding Mean and Standard Deviation 1.Which curve has the greater mean? Solution: Curve A has the greater mean (The line of symmetry of curve A occurs at x = 15. The line of symmetry of curve B occurs at x = 12.) 21 Larson/Farber 4th ed

22. Example: Understanding Mean and Standard Deviation 2.Which curve has the greater standard deviation? Solution: Curve B has the greater standard deviation (Curve B is more spread out than curve A.) 22 Larson/Farber 4th ed

23. Example: Interpreting Graphs The heights of fully grown white oak trees are normally distributed. The curve represents the distribution. What is the mean height of a fully grown white oak tree? μ = 90 (A normal curve is symmetric about the mean) σ = 3.5 (The inflection points are one standard deviation away from the mean) Solution: 23 Larson/Farber 4th ed

24. The Standard Normal Distribution Standard normal distribution A normal distribution with a mean of 0 and a standard deviation of 1. 33 1 22 11 023 z Area = 1 Any x-value can be transformed into a z-score by using the formula 24 Larson/Farber 4th ed

25. The Standard Normal Distribution If each data value of a normally distributed random variable x is transformed into a z-score, the result will be the standard normal distribution. Normal Distribution x     z Standard Normal Distribution 25 Larson/Farber 4th ed

26. Slide 6- 26 When Is a z-score Big? (cont.) When we use the Normal model, we are assuming the distribution is Normal. We cannot check this assumption in practice, so we check the following condition:  Nearly Normal Condition: The shape of the data’s distribution is unimodal and symmetric.  This condition can be checked with a histogram or a Normal probability plot (to be explained later).

27. Slide 6- 27 The First Three Rules for Working with Normal Models Make a picture. And, when we have data, make a histogram to check the Nearly Normal Condition to make sure we can use the Normal model to model the distribution.

28. Slide 6- 28 The 68-95-99.7 Rule Normal models give us an idea of how extreme a value is by telling us how likely it is to find one that far from the mean. We can find these numbers precisely, but until then we will use a simple rule that tells us a lot about the Normal model…

29. Slide 6- 29 The 68-95-99.7 Rule (cont.) It turns out that in a Normal model:  about 68% of the values fall within one standard deviation of the mean;  about 95% of the values fall within two standard deviations of the mean; and,  about 99.7% (almost all!) of the values fall within three standard deviations of the mean.

30. Slide 6- 30 The 68-95-99.7 Rule (cont.) The following shows what the 68-95-99.7 Rule tells us:

31. The Empirical Rule

32. Normal Distributions The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7 th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N (6.84, 1.55). a) What is the mean & standard deviation of this NormalDistribution? b) Sketch the Normal density curve for this distribution. c) What percent of ITBS vocabulary scores are less than 3.74? d) What percent of the scores are between 5.29 and 9.94? AN EXAMPLE: 68+13.5 = 81.5%

33. Example – Using the Normal Distribution (Empirical Rule) IQ scores are normally distributed with mean 100 and standard deviation 15. Find the proportion of the population with IQ scores in the given interval. Also find the probability that a randomly selected individual has an IQ score in the given interval. (a) Between 85 and 115 (b) At least 130 (c) At most 130

34. Slide 6- 34 Finding Normal Percentiles by Hand When a data value doesn’t fall exactly 1, 2, or 3 standard deviations from the mean, we can look it up in a table of Normal percentiles. Table Z in Appendix G provides us with normal percentiles, but many calculators and statistics computer packages provide these as well.

35. Slide 6- 35 Finding Normal Percentiles by Hand (cont.) Table Z is the standard Normal table. We have to convert our data to z-scores before using the table. Figure 6.5 shows us how to find the area to the left when we have a z-score of 1.80:

36. Normal Distributions Normal Distribution Calculations When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards? Using Table A, we can find the area to the left of z=2.63 and the area to the left of z=0.13. 0.9957 – 0.5517 = 0.4440. About 44% of Tiger’s drives travel between 305 and 325 yards.

37. AP Statistics, Section 2.2, Part 1 37 Cautions We should only use the z-table when the distributions are normal, and data has been standardized The z-table only gives the amount of data found below the z-score, THAT IS THE AREA TO THE LEFT OF THE z-score! If you want to find the portion found above the z-score, subtract the probability found on the table from 1.

38. EXAMPLE: Find the proportion of observations from the standard Normal distribution the is greater than.81

39. Will my calculator do any of this normal stuff? ONLY Normalpdf – use for graphing ONLY Normalcdf – will find proportion of area from lower bound to upper bound in z-scores or data values Invnorm (inverse normal) – will find z-score or data value for a given proportion of area THESE COMMANDS ARE FOUND IN THE DISTRIBUTION MENU of VARS key

40. Finding Normal Percentiles using the calculator  Go to the Distribution key on your calculator  Find NORMCDF  Use the key stroke: NORMCDF(min z,max z)  Without using z-scores: NORMCDF(min value, max value, Mean, St. Dev)

41. AP Statistics, Section 2.2, Part 1 41 Example Using Data Values Men’s heights are Normally distributed according to N(69,2.5). a)What proportion of men are between 68 and 70 inches tall? b) What percent of men are taller than 68 inches? c) What percent of men are shorter than 68 inches?

42. Application of the Normal Curve The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. If you order a pizza, find the probability that the delivery time will be: a.between 25 and 27 minutes.a. ___________ b.less than 30 minutes.b. __________ c.less than 22.7 minutes.c. __________.3413.9938.1251

43. Example: Finding Probabilities for Normal Distributions A survey indicates that for each trip to the supermarket, a shopper spends an average of 45 minutes with a standard deviation of 12 minutes in the store. The length of time spent in the store is normally distributed and is represented by the variable x. A shopper enters the store. Find the probability that the shopper will be in the store for between 24 and 54 minutes. 43 Larson/Farber 4th ed

44. Example: Finding Probabilities for Normal Distributions Find the probability that the shopper will be in the store more than 39 minutes. (Recall μ = 45 minutes and σ = 12 minutes) 44 Larson/Farber 4th ed

45. Example: Finding Probabilities for Normal Distributions If 200 shoppers enter the store, how many shoppers would you expect to be in the store more than 39 minutes? Solution: Recall P(x > 39) = 0.6915 200(0.6915) =138.3 (or about 138) shoppers 45 Larson/Farber 4th ed

46. Example: Finding Probabilities for Normal Distributions A survey indicates that people use their computers an average of 2.4 years before upgrading to a new machine. The standard deviation is 0.5 year. A computer owner is selected at random. Find the probability that he or she will use it for fewer than 2 years before upgrading. Assume that the variable x is normally distributed. 1.Draw a picture 2.Use your calculator to find the proportion of Area under the normal model less than 2 46 Larson/Farber 4th ed

47. Example: Using Technology to find Normal Probabilities Assume that cholesterol levels of men in the United States are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the United States. What is the probability that his cholesterol level is less than 175? 47 Larson/Farber 4th ed

48. Slide 6- 48 From Percentiles to Scores: z in Reverse Sometimes we start with areas and need to find the corresponding z-score or even the original data value. Example: What z-score represents the first quartile (25% mark) in a Normal model?

49. Slide 6- 49 From Percentiles to Scores: z in Reverse (cont.) Look in Table Z for an area of 0.2500. The exact area is not there, but 0.2514 is pretty close. This figure is associated with z = -0.67, so the first quartile is 0.67 standard deviations below the mean.

50. INVERSE NORM We can also use the calculator to also find the z-score for a particular area: INVNORM(prop of area to the left) So, to solve the previous problem, keystroke: INVNORM(.25) Note: when using the calculator, entering μ,σ will “un-standardize” the data, that is return a data value and not a z-score

51. AP Statistics, Section 2.2, Part 1 51 Working backwards How tall must a man be in order to be in the 90 th percentile? (Recall men’s model was N(69, 2.5)

52. AP Statistics, Section 2.2, Part 1 52 Working backwards How tall must a woman be in order to be in the top 15% of all women if heights follow a Normal model N(60, 2.3)?

53. Scores for a civil service exam are normally distributed, with a mean of 75 and a standard deviation of 6.5. To be eligible for civil service employment, you must score in the top 5%. What is the lowest score you can earn and still be eligible for employment? ? 0 z 5% ? 75 x Solution: 1 – 0.05 = 0.95 An exam score in the top 5% is any score above the 95 th percentile. Find the score that corresponds to a cumulative area of 0.95. 53 Larson/Farber 4th ed

54. Solution: Finding a Specific Data Value USE INVNORM(1-.05,75,6.5) 85.69 5% 75 54 Larson/Farber 4th ed The lowest score you can earn and still be eligible for employment is 86

55. The Verbal SAT test has a mean score of 500 and a standard deviation of 100. Scores are normally distributed. A major university determines that it will accept only students whose Verbal SAT scores are in the top 4%. What is the minimum score that a student must earn to be accepted?

56. ...students whose Verbal SAT scores are in the top 4%. Mean = 500, standard deviation = 100 =.04.9600 X= ?? Using INVNORM (.9600, 500, 100 ) =

57. Mean = 500, standard deviation = 100 =.04.9600 X= The cut-off score is 675. Check your answer using NORMCDF(675,5000,500,100) = HW: P. 133 31 THRU 35, & 45

58. AP Statistics, Section 2.2, Part 1 58 REMINDER: CAUTION!!! Whether using the calculator or Table, we should only use the z-table when the distributions are normal, and data has been standardized

59. Slide 6- 59 Are You Normal? How Can You Tell? When you actually have your own data, you must check to see whether a Normal model is reasonable. Looking at a histogram of the data is a good way to check that the underlying distribution is roughly unimodal and symmetric.

60. Slide 6- 60 Are You Normal? How Can You Tell? (cont.) A more specialized graphical display that can help you decide whether a Normal model is appropriate is the Normal probability plot. If the distribution of the data is roughly Normal, the Normal probability plot approximates a diagonal straight line. Deviations from a straight line indicate that the distribution is not Normal.

61. Slide 6- 61 Are You Normal? How Can You Tell? (cont.) Nearly Normal data have a histogram and a Normal probability plot that look somewhat like this example:

62. Slide 6- 62 Are You Normal? How Can You Tell? (cont.) A skewed distribution might have a histogram and Normal probability plot like this:

63. Slide 6- 63 What Can Go Wrong? Don’t use a Normal model when the distribution is not unimodal and symmetric.