2. 1 Chapter 6 Part 1 Using the Mean and Standard Deviation Together z-scores 68-95-99.7 rule Changing units (shifting and rescaling data)

3. 2 Z-scores: Standardized Data Values Measures the distance of a number from the mean in units of the standard deviation

4. 3 z-score corresponding to y

5. 4 n Exam 1: y 1 = 88, s 1 = 6; exam 1 score: 91 Exam 2: y 2 = 88, s 2 = 10; exam 2 score: 92 Which score is better?

6. 5 Comparing SAT and ACT Scores n SAT Math: Eleanor’s score 680 SAT mean =500 sd=100 n ACT Math: Gerald’s score 27 ACT mean=18 sd=6 n Eleanor’s z-score: z=(680-500)/100=1.8 n Gerald’s z-score: z=(27-18)/6=1.5 n Eleanor’s score is better.

7. 6 Z-scores add to zero Student/Institutional Support to Athletic Depts For the 9 Public ACC Schools: 2013 ($ millions) SchoolSupporty - ybarZ-score Maryland15.56.41.79 UVA13.14.01.12 Louisville10.91.80.50 UNC9.20.10.03 VaTech7.9-1.2-0.34 FSU7.9-1.2-0.34 GaTech7.1-2.0-0.56 NCSU6.5-2.6-0.73 Clemson3.8-5.3-1.47 Mean=9.1000, s=3.5697 Sum = 0

8. 7 In a recent year the mean tuition at 4-yr public colleges/universities in the U.S. was $6185 with a standard deviation of $1804. In NC the mean tuition was $4320. What is NC’s z-score? 1. 1.03 2. -1.03 3. 2.39 4. 1865 5. -1865

9. Changing Units of Measurement How shifting and rescaling data affect data summaries

10. Shifting and rescaling: linear transformations zOriginal data x 1, x 2,... x n zLinear transformation: x * = a + bx, (intercept a, slope b) x x*x* 0 a Shifts data by a Changes scale

11. Linear Transformations x* = a+ b x Examples: Changing 1.from feet (x) to inches (x*): x*=12x 2.from dollars (x) to cents (x*): x*=100x 3.from degrees celsius (x) to degrees fahrenheit (x*): x* = 32 + (9/5)x 4.from ACT (x) to SAT (x*): x*=150+40x 5.from inches (x) to centimeters (x*): x* = 2.54x 0 12 0 100 32 9/5 150 40 0 2.54

12. Shifting data only: b = 1 x* = a + x  Adding the same value a to each value in the data set:  changes the mean, median, Q 1 and Q 3 by a  The standard deviation, IQR and variance are NOT CHANGED. yEverything shifts together. ySpread of the items does not change.

13. Shifting data only: b = 1 x* = a + x (cont.) zweights of 80 men age 19 to 24 of average height (5'8" to 5'10") x = 82.36 kg z NIH recommends maximum healthy weight of 74 kg. To compare their weights to the recommended maximum, subtract 74 kg from each weight; x* = x – 74 (a=-74, b=1) z x* = x – 74 = 8.36 kg 1.No change in shape 2.No change in spread 3.Shift by 74

14. Shifting and Rescaling data: x* = a + bx, b > 0 Original x data: x 1, x 2, x 3,..., x n Summary statistics: mean x median m 1 st quartile Q 1 3 rd quartile Q 3 stand dev s variance s 2 IQR x* data: x* = a + bx x 1 *, x 2 *, x 3 *,..., x n * Summary statistics: new mean x* = a + bx new median m* = a+bm new 1 st quart Q 1 *= a+bQ 1 new 3 rd quart Q 3 * = a+bQ 3 new stand dev s* = b  s new variance s* 2 = b 2  s 2 new IQR* = b  IQR

15. Rescaling data: x* = a + bx, b > 0 (cont.) zweights of 80 men age 19 to 24, of average height (5'8" to 5'10") zx = 82.36 kg zmin=54.30 kg zmax=161.50 kg zrange=107.20 kg zs = 18.35 kg z Change from kilograms to pounds: x* = 2.2x (a = 0, b = 2.2) z x* = 2.2(82.36)=181.19 pounds z min* = 2.2(54.30)=119.46 pounds z max* = 2.2(161.50)=355.3 pounds z range*= 2.2(107.20)=235.84 pounds z s* = 18.35 * 2.2 = 40.37 pounds

16. Example of x* = a + bx 4 student heights in inches (x data) 62, 64, 74, 72 x = 68 inches s = 5.89 inches Suppose we want centimeters instead: x * = 2.54x (a = 0, b = 2.54) 4 student heights in centimeters: 157.48 = 2.54(62) 162.56 = 2.54(64) 187.96 = 2.54(74) 182.88 = 2.54(72) x * = 172.72 centimeters s * = 14.9606 centimeters Note that x * = 2.54x = 2.54(68)=172.2 s * = 2.54s = 2.54(5.89)=14.9606 not necessary! UNC method Go directly to this. NCSU method

17. Example of x* = a + bx x data: Percent returns from 4 investments during 2003: 5%, 4%, 3%, 6% x = 4.5% s = 1.29% Inflation during 2003: 2% x* data: Inflation-adjusted returns. x* = x – 2% (a=-2, b=1) x* data: 3% = 5% - 2% 2% = 4% - 2% 1% = 3% - 2% 4% = 6% - 2% x* = 10%/4 = 2.5% s* = s = 1.29% x* = x – 2% = 4.5% –2% s* = s = 1.29% (note! that s* ≠ s – 2%) !! not necessary! Go directly to this

18. Example zOriginal data x: Jim Bob’s jumbo watermelons from his garden have the following weights (lbs): 23, 34, 38, 44, 48, 55, 55, 68, 72, 75 s = 17.12; Q 1 =37, Q 3 =69; IQR = 69 – 37 = 32 zMelons over 50 lbs are priced differently; the amount each melon is over (or under) 50 lbs is: zx* = x  50 (x* = a + bx, a=-50, b=1) -27, -16, -12, -6, -2, 5, 5, 18, 22, 25 s* = 17.12; Q* 1 = 37 - 50 =-13, Q* 3 = 69 - 50 = 19 IQR* = 19 – (-13) = 32 NOTE: s* = s, IQR*= IQR

19. Z-scores: a special linear transformation a + bx Example. At a community college, if a student takes x credit hours the tuition is x* = $250 + $35x. The credit hours taken by students in an Intro Stats class have mean x = 15.7 hrs and standard deviation s = 2.7 hrs. Question 1. A student’s tuition charge is $941.25. What is the z-score of this tuition? x* = $250+$35(15.7) = $799.50; s* = $35(2.7) = $94.50

20. Z-scores: a special linear transformation a + bx (cont.) Example. At a community college, if a student takes x credit hours the tuition is x* = $250 + $35x. The credit hours taken by students in an Intro Stats class have mean x = 15.7 hrs and standard deviation s = 2.7 hrs. Question 2. Roger is a student in the Intro Stats class who has a course load of x = 13 credit hours. The z-score is z = (13 – 15.7)/2.7 = -2.7/2.7 = -1. What is the z-score of Roger’s tuition? Roger’s tuition is x* = $250 + $35(13) = $705 Since x* = $250+$35(15.7) = $799.50; s* = $35(2.7) = $94.50 This is why z-scores are so useful!!

21. SUMMARY: Linear Transformations x* = a + bx z Linear transformations do not affect the shape of the distribution of the data -for example, if the original data is right- skewed, the transformed data is right-skewed

22. SUMMARY: Shifting and Rescaling data, x* = a + bx, b > 0

24. 23 68-95-99.7 rule Mean and Standard Deviation (numerical) Histogram (graphical) 68-95-99.7 rule

25. 24 The 68-95-99.7 rule; applies only to mound-shaped data

26. 25 68-95-99.7 rule: 68% within 1 stan. dev. of the mean 68% 34% y-s y y+s

27. 26 68-95-99.7 rule: 95% within 2 stan. dev. of the mean 95% 47.5% y-2s y y+2s

28. 27 Example: textbook costs 286291307308315316327 328340342346347348348 349354355355360361364 367369371373377380381 382385385387390390397 398409409410418422424 425426428433434437440 480

29. 28 Example: textbook costs (cont.) 286291307308315316327328 340342346347348348349354 355355360361364367369371 373377380381382385385387 390390397398409409410418 422424425426428433434437 440480

30. 29 Example: textbook costs (cont.) 286291307308315316327328 340342346347348348349354 355355360361364367369371 373377380381382385385387 390390397398409409410418 422424425426428433434437 440480

31. 30 Example: textbook costs (cont.) 286291307308315316327328 340342346347348348349354 355355360361364367369371 373377380381382385385387 390390397398409409410418 422424425426428433434437 440480

32. 31 The best estimate of the standard deviation of the men’s weights displayed in this dotplot is 1. 10 2. 15 3. 20 4. 40

33. 32 End of Chapter 6 Part 1. Next: Part 2 Normal Models

34. 33 Z-scores add to zero Student/Institutional Support to Athletic Depts For the 9 Public ACC Schools: 2013 ($ millions) SchoolSupporty - ybarZ-score Maryland15.56.41.79 UVA13.14.01.12 Louisville10.91.80.50 UNC9.20.10.03 VaTech7.9-1.2-0.34 FSU7.9-1.2-0.34 GaTech7.1-2.0-0.56 NCSU6.5-2.6-0.73 Clemson3.8-5.3-1.47 Mean=9.1000, s=3.5697 Sum = 0