1 Chapter 5 Part 1 Using the Mean and Standard Deviation Together z-scores rule Changing units (shifting and rescaling data)

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

3 z-score corresponding to y

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?

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=( )/100=1.8 n Gerald’s z-score: z=(27-18)/6=1.5 n Eleanor’s score is better.

6 Z-scores add to zero Student/Institutional Support to Athletic Depts For the 9 Public ACC Schools: 2013 ($ millions) SchoolSupporty - ybarZ-score Maryland UVA Louisville UNC VaTech FSU GaTech NCSU Clemson Mean=9.1000, s= Sum = 0

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 tuition was $4320. What is NC’s z-score?

rule Mean and Standard Deviation (numerical) Histogram (graphical) rule

10 The rule; applies only to mound-shaped data

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

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

13 Example: textbook costs

14 Example: textbook costs (cont.)

15 Example: textbook costs (cont.)

16 Example: textbook costs (cont.)

17 The best estimate of the standard deviation of the men’s weights displayed in this dotplot is

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

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

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 /

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.

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 = 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

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

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 = kg zmin=54.30 kg zmax= kg zrange= kg zs = kg z Change from kilograms to pounds: x* = 2.2x (a = 0, b = 2.2) z x* = 2.2(82.36)= pounds z min* = 2.2(54.30)= pounds z max* = 2.2(161.50)=355.3 pounds z range*= 2.2(107.20)= pounds z s* = * 2.2 = pounds

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: = 2.54(62) = 2.54(64) = 2.54(74) = 2.54(72) x * = centimeters s * = centimeters Note that x * = 2.54x = 2.54(68)=172.2 s * = 2.54s = 2.54(5.89)= not necessary! UNC method Go directly to this. NCSU method

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

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 =38, Q 3 =68; IQR = 68 – 38 = 30 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 = =-12, Q* 3 = = 18 IQR* = 18 – (-12) = 30 NOTE: s* = s, IQR*= IQR

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 $ What is the z-score of this tuition? x* = $250+$35(15.7) = $799.50; s* = $35(2.7) = $94.50

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 The z-score does not depend on the unit of measurement. This is why z-scores are so useful!!

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

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

32 End of Chapter 5 Part 1. Next: Part 2 Normal Models