2.4 (cont.) 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 =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 = =-13, Q* 3 = = 19 IQR* = 19 – (-13) = 32 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