1. 1 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Chapter 5 The Standard Deviation as a Ruler and the Normal Model

2. 2 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 2 Objectives: The student will be able to: 19. Compare values from two different distributions using their z- scores. 20. Use Normal models (when appropriate) and the 68-95-99.7 Rule to estimate the percentage of observations falling within one, two, or three standard deviations of the mean. 21. Determine the percentages of observations that satisfy certain conditions by using the Normal model and determine “extraordinary” values. 22. Determine whether a variable satisfies the Nearly Normal condition by making a normal probability plot or histogram. 23. Determine the z-score that corresponds to a given percentage of observations.

3. 3 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 5.1 Standardizing with z-Scores

4. 4 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Comparing Athletes Dobrynska took the gold in the Olympics with a long jump of 6.63 m for the women’s heptathlon, 0.5 m higher than average. Fountain won the 200 m run with a time of 23.21 s, 1.5 s faster than average. Whose performance was more impressive?

5. 5 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How Many Standard Deviations Above? Long Jump200 m Run Mean6.11 m24.71 s SD0.24 m0.70 s Individual6.63 m23.21 s The standard deviation helps us compare. Long Jump: 1 SD above: 6.11 + 0.24 = 6.35 2 SD above: 6.11 + (2)(0.24) = 6.59 Just over 2 standard deviations above

6. 6 Copyright © 2014, 2012, 2009 Pearson Education, Inc. The z-Score In general, to find the distance between the value and the mean in standard deviations: 1. Subtract the mean from the value. 2. Divide by the standard deviation. This is called the z-score.

7. 7 Copyright © 2014, 2012, 2009 Pearson Education, Inc. The z-score The z-score measures the distance of the value from the mean in standard deviations. A positive z-score indicates the value is above the mean. A negative z-score indicates the value is below the mean. A small z-score indicates the value is close to the mean when compared to the rest of the data values. A large z-score indicates the value is far from the mean when compared to the rest of the data values.

8. 8 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How Many Standard Deviations Above? Long Jump200 m Run Mean6.11 m24.71 s SD0.24 m0.70 s Individual6.63 m23.21 s Long Jump:200 m Run: Dobrynska’s long jump was a little more impressive than Fountain’s 200 m run. Standard Deviations from the Mean

9. 9 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Practice Assume verbal SAT scores have a mean of 500 and a std. dev. of 100. What is the z-score of somebody who scores 500 on the verbal portion of the SAT? Assume IQ scores have a mean of 100 and a std. dev. of 16. Albert Einstein reportedly had an IQ of 160. What is the z-score of his IQ? Slide 1- 9

10. 10 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 10 Examples Suppose your stats professor reports your test grade as a z-score and you received a score of 2.20. What does that mean? If the mean was 80 and the standard deviation was 7, what was your grade?

11. 11 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 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. Example – Which student performed better? Student A received a 85 on a 100 point quiz with a mean of 90 and standard deviation of 5. Student B received a 35 on a 50 point quiz with a mean of 37 and a standard deviation of 3. We must compare z-scores!

12. 12 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Practice Recall: Suppose a basketball player scored the following number of points in his last 15 games: 4, 4, 3, 4, 7, 16, 12, 15, 6, 8, 5, 9, 8, 25, 11 Use these scores to calculate the z-scores for the 3 lowest-scoring and 3 highest-scoring games. Slide 1- 12

13. 13 Copyright © 2014, 2012, 2009 Pearson Education, Inc. A college student received a score of 78 on her Math exam and a score of 86 on her French exam. The overall results on the French exam had a mean of 82 and a standard deviation of 8, while the math exam had a mean of 54 and a standard deviation 12. On which exam did she do relatively better? Math z-score: 2.0 French z-score: 0.5 She did relatively better on her math exam. Slide 1- 13

14. 14 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Who is relatively taller: A non-basketball playing man who is 75 inches tall (assume non-basketball playing men have a mean height of 71.5 inches tall and a standard deviation of 2.1 inches). A male basketball player who is 85 inches tall (assume male basketball players have a mean height of 80 inches and a standard deviation of 3.3) Slide 1- 14

15. 15 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 5.2 Shifting and Scaling

16. 16 Copyright © 2014, 2012, 2009 Pearson Education, Inc. National Health and Examination Survey Who? What? When? Where? Why? How? 80 male participants between 19 and 24 who measured between 68 and 70 inches tall Their weights in kilograms 2001 – 2002 United States To study nutrition and health issues and trends National survey

17. 17 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Shifting Weights Mean: 82.36 kg Maximum Healthy Weight: 74 kg How are shape, center, and spread affected when 74 is subtracted from all values? Shape and spread are unaffected. Center is shifted by 74.

18. 18 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Rules for Shifting If the same number is subtracted or added to all data values, then: The measures of the spread – standard deviation, range, and IQR – are all unaffected. The measures of position – mean, median, and mode – are all changed by that number.

19. 19 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Rescaling If we multiply all data values by the same number, what happens to the position and spread? To go from kg to lbs, multiply by 2.2. The mean and spread are also multiplied by 2.2.

20. 20 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How Rescaling Affects the Center and Spread When we multiply (or divide) all the data values by a constant, all measures of position and all measures of spread are multiplied (or divided) by that same constant.

21. 21 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Example: Rescaling Combined Times in the Olympics The mean and standard deviation in the men’s combined event at the Olympics were 168.93 seconds and 2.90 seconds, respectively. If the times are measured in minutes, what will be the new mean and standard deviation? Mean: 168.93 / 60 = 2.816 minutes Standard Deviation: 2.90 / 60 = 0.048 minute

22. 22 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Shifting, Scaling, and z-Scores Converting to z-scores: Subtract the mean Divide by the standard deviation The shape of the distribution does not change. Changes the center by making the mean 0 Changes the spread by making the standard deviation 1

23. 23 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 5.3 Normal Models

24. 24 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Models “ All models are wrong, but some are useful. ” George Box, statistician −1 < z < 1: Not uncommon z = ±3: Rare z = 6: Shouts out for attention!

25. 25 Copyright © 2014, 2012, 2009 Pearson Education, Inc. The Normal Model Bell Shaped: unimodal, symmetric A Normal model for every mean and standard deviation.  (read “mew”) represents the population mean.  (read “sigma”) represents the population standard deviation. N( ,  ) represents a Normal model with mean  and standard deviation .

26. 26 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Parameters and Statistics Parameters: Numbers that help specify the model  Statistics: Numbers that summarize the data, s, median, mode N(0, 1) is called the standard Normal model, or the standard Normal distribution. The Normal model should only be used if the data is approximately symmetric and unimodal.

27. 27 Copyright © 2014, 2012, 2009 Pearson Education, Inc. The 68-95-99.7 Rule 68% of the values fall within 1 standard deviation of the mean. 95% of the values fall within 2 standard deviations of the mean. 99.7% of the values fall within 3 standard deviations of the mean.

28. 28 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Working With the 68-95-99.7 Rule Each part of the SAT has a mean of 500 and a standard deviation of 100. Assume the data is symmetric and unimodal. If you earned a 700 on one part of the SAT how do you stand among all others who took the SAT? Think → Plan: The variable is quantitative and the distribution is symmetric and unimodal. Use the Normal model N(500, 100).

29. 29 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Show and Tell Show → Mechanics: Make a picture. 700 is 2 standard deviations above the mean. Tell → Conclusion: 95% lies within 2 standard deviations of the mean. 100% - 95% = 5% are outside of 2 standard deviations of the mean. Above 2 standard deviations is half of that. 5% / 2 = 2.5% Your score is higher than 2.5% of all scores on this test.

30. 30 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Practice Assume IQ scores have a normal model distribution with a mean of 100 and a std. dev. of 16. Find the following percentages using the 68-95-99.7% rule a. % of people with 84<=IQ<=116 b. % of people with IQ>=100 c. % of people 68<=IQ<=132 d. % of people who are “geniuses” (a genius is someone with an IQ>=132) e. % of people 84<=IQ<=132 Slide 1- 30

31. 31 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Three Rules For Using the Normal Model 1. Make a picture. 2. Make a picture. 3. Make a picture. When data is provided, first make a histogram to make sure that the distribution is symmetric and unimodal. Then sketch the Normal model.

32. 32 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 5.4 Finding Normal Percentiles

33. 33 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What if z is not −3, −2, −1, 0, 1, 2, or 3? If the data value we are trying to find using the Normal model does not have such a nice z-score, we will use a computer. Example: Where do you stand if your SAT math score was 680?  = 500,  = 100 Note that the z -score is not an integer:

34. 34 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 34 Finding Normal Percentiles 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 D provides us with normal percentiles, but many calculators and statistics computer packages provide these as well. You will do these using your TI-83

35. 35 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 35 Finding Normal Percentiles using Technology – TI-83 To find what percentage of a standard Normal model is found in the region a < z < b (note, for infinity use any large number or 1E99!) use the DISTR function normalcdf(a,b,μ,σ) If your a and b are z-scores then μ = 0 and σ = 1

36. 36 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 36 Finding Normal Percentiles Example: Where do you stand if your SAT math score was 680?  = 500,  = 100 z= 1.8 The figure shows us how to find the area to the left when we have a z-score of 1.80: Normalcdf(-1E99, 1.80, 0, 1) =.964 Alternatively we can compute Normalcdf(-1E99, 680, 500, 100)=.964

37. 37 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Using StatCrunch for the Normal Model What percent of all SAT scores are below 680?  = 500,  = 100 Stat → Calculators → Normal Fill in info, hit Compute

38. 38 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Using StatCrunch for the Normal Model What percent of all SAT scores are below 680?  = 500,  = 100 Stat → Calculators → Normal Fill in info, hit Compute 96.4% of SAT scores are below 680.

39. 39 Copyright © 2014, 2012, 2009 Pearson Education, Inc. A Probability Involving “Between” What is the proportion of SAT scores that fall between 450 and 600?  = 500,  = 100 Think → Plan: Probability that x is between 450 and 600 = Probability that x < 600 – Probability that x < 450 Variable: We are told that the Normal model works. N(500, 100)

40. 40 Copyright © 2014, 2012, 2009 Pearson Education, Inc. A Probability Involving “Between” What is the proportion of SAT scores that fall between 450 and 600?  = 500,  = 100 Show → Mechanics: Use StatCrunch to find each of the probabilities. Probability that x is between 450 and 600 = Probability that x < 600 – Probability that x < 450 = 0.8413 – 0.3085 = 0.5328

41. 41 Copyright © 2014, 2012, 2009 Pearson Education, Inc. A Probability Involving “Between” What is the proportion of SAT scores that fall between 450 and 600?  = 500,  = 100 Probability that x is between 450 and 600 = Probability that x < 600 – Probability that x < 450 = 0.8413 – 0.3085 = 0.5328 Conclusion: The Normal model estimates that about 53.28% of SAT scores fall between 450 and 600.

42. 42 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Underweight Cereal Boxes Based on experience, a manufacturer makes cereal boxes that fit the Normal model with mean 16.3 ounces and standard deviation 0.2 ounces, but the label reads 16.0 ounces. What fraction will be underweight? Think → Plan: Find Probability that x < 16.0 Variable: N(16.3, 0.2)

43. 43 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Underweight Cereal Boxes What fraction of the cereal boxes will be underweight (less than 16.0)?  = 16.3,  = 0.2 Probability x < 16.0 = 0.0668 Conclusion: I estimate that approximately 6.7% of the boxes will contain less than 16.0 ounces of cereal.

44. 44 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 44 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 in a Normal model? invNorm(.25, 0, 1)

45. 45 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 45 Finding the z-score given percentile using technology– TI-83 To find the score with a given tail probability, use the DISTR function invNorm(p,μ,σ). NOTE: invNorm always considers p to be the percentage in the left (negative) tail To find a right tail or center probability, you will have to do some work to find the right p to use with invNorm. If you want a z-score (not a raw score, then you can leave off μ and σ (or write 0, 1)

46. 46 Copyright © 2014, 2012, 2009 Pearson Education, Inc. From Percentiles to Scores: z in Reverse Suppose a college admits only people with SAT scores in the top 10%. How high a score does it take to be eligible?  = 500,  = 100 Think → Plan: We are given the probability and want to go backwards to find x. Variable: N(500, 100) Solve this using InvNorm(.9, 500, 100)

47. 47 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 47 Percentiles and Z-scores What percent of a standard Normal model is found in each region? Draw a picture for each a) z > -2.05 b) z < -0.33 c) 1.2 < z < 1.8 d) |z| < 1.28 In a standard Normal model, what value(s) of z cut(s) off the region described? Draw a picture first! a) The highest 20% b) The highest 75% c) The lowest 3% d) The middle 90%

48. 48 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 48 More Percentiles and Z-scores What percent of a standard Normal model is found in each region? Draw a picture for each a) z > -1.05 b) z < -0.40 c) 1.3 < z < 2.0 In a standard Normal model, what value(s) of z cut(s) off the region described? Draw a picture first! a) The highest 20% b) The highest 60% c) The lowest 6% d) The middle 75%

49. 49 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 49 Additional exercises Some IQ tests are standardized to a normal model with a mean of 100 and a standard deviation of 16. A) Draw the model for these IQ scores clearly labeling showing what the 68-95-99.7 Rule predicts about the scores B) In what interval would you expect to find the central 95% of IQ scores to be found? C) About what percent of people should have IQ scores above 116? D) About what percent of people should have IQ scores between 68 and 84? E) About what percent of people would have IQ scores above 132?

50. 50 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 44) Based on the Normal model N(100, 16) describing IQ scores, what percent of people’s IQ scores would you expect to be – Over 80? – Under 90? – Between 112 and 132? 46) In the same model, what cutoff value bounds – The highest 5% of all IQs? – The lowest 30% of the IQs? – The middle 80% of the IQs? Slide 1- 50

51. 51 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Section 5.5 Normal Probability Plots

52. 52 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Checking if the Normal Model Applies A histogram will work, but there is an alternative method. Instead use a Normal Probability Plot. Plots each value against the z-score that would be expected had the distribution been perfectly normal. If the plot shows a line or is nearly straight, then the Normal model works. If the plot strays from being a line, then the Normal model is not a good model.

53. 53 Copyright © 2014, 2012, 2009 Pearson Education, Inc. The Normal Model Applies The Normal probability plot is nearly straight, so the Normal model applies. Note that the histogram is unimodal and somewhat symmetric.

54. 54 Copyright © 2014, 2012, 2009 Pearson Education, Inc. The Normal Model Does Not Apply The Normal probability plot is not straight, so the Normal model does not apply applies. Note that the histogram is skewed right.

55. 55 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What Can Go Wrong Don’t use the Normal model when the distribution is not unimodal and symmetric. Always look at the picture first. Don’t use the mean and standard deviation when outliers are present. Check by making a picture. Don’t round your results in the middle of the calculation. Always wait until the end to round. Don’t worry about minor differences in results. Different rounding can produce slightly different results.