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2.4 (cont.) Changing Units of Measurement How shifting and rescaling data affect data summaries Shifting and rescaling: linear transformations Original data x1, x2, . . . xn Linear transformation: x* = a + bx, (intercept a, slope b) Shifts data by a Changes scale x* a 0 x Linear Transformations 2.54 32 12 40 100 00 0a+ 9/5 x x* = 150 b 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, Q1 and Q3 by a The standard deviation, IQR and variance are NOT CHANGED. Everything shifts together. Spread of the items does not change. Shifting data only: b = 1 x* = a + x (cont.) weights of 80 men age 19 to 24 of average height (5'8" to 5'10") x = 82.36 kg 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) 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: x1, x2, x3, . . ., xn Summary statistics: mean x median m 1st quartile Q1 3rd quartile Q3 stand dev s variance s2 IQR x* data: x* = a + bx x1*, x2*, x3*, . . ., xn* Summary statistics: new mean x* = a + bx new median m* = a+bm new 1st quart Q1*= a+bQ1 new 3rd quart Q3* = a+bQ3 new stand dev s* = b s new variance s*2 = b2 s2 new IQR* = b IQR Rescaling data: x* = a + bx, b > 0 (cont.) weights of 80 men age 19 to 24, of average height (5'8" to 5'10") x = 82.36 kg min=54.30 kg max=161.50 kg range=107.20 kg s = 18.35 kg Change from kilograms to pounds: x* = 2.2x (a = 0, b = 2.2) x* = 2.2(82.36)=181.19 pounds min* = 2.2(54.30)=119.46 pounds max* = 2.2(161.50)=355.3 pounds range*= 2.2(107.20)=235.84 pounds s* = 18.35 * 2.2 = 40.37 pounds Example of x* = a + bx 4 student heights in inches (x data) not 62, 64, 74, 72 necessary! UNC x = 68 inches method s = 5.89 inches Suppose we want centimeters instead: Go directly to x* = 2.54x this. NCSU (a = 0, b = 2.54) method 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 Example of x* = a + bx x data: Percent returns from 4 investments during 2003: 5%, 4%, 3%, 6% not x = 4.5% necessary! s = 1.29% Inflation during 2003: 2% x* data: Inflation-adjusted returns. Go directly to this 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%) !! Example Original 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; Q1=37, Q3 =69; IQR = 69 – 37 = 32 Melons over 50 lbs are priced differently; the amount each melon is over (or under) 50 lbs is: x* = 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 Z-scores: a special linear transformation a + bx z xx s x s 1 s x a bx where a x s ,b 1 s 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 z 941.25 799.50 141.75 1.5 94.50 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 705 - 799.50 -94.50 on the unit of measurement. z= = =-1 94.50 94.50 This is why z-scores are so useful!! SUMMARY: Linear Transformations x* = a + bx Assembly Time (seconds) Assembly Time (minutes) 30 20 15 10 5 0 Frequency Frequency 25 30 20 10 0 Linear transformations do not affect the shape of the distribution of the data -for example, if the original data is rightskewed, the transformed data is right-skewed SUMMARY: Shifting and Rescaling data, x* = a + bx, b > 0 original data x1 , x2 , x3 ,... transformed data x1* , x2* , x3* ,... summary statistics mean x median m summary statistics new mean x * a bx new median m* a bm 1st Q1 new Q1* a bQ1 3rd Q3 new Q3* a bQ3 st dev s new st dev s* bs var. s 2 new var. s *2 b 2 s 2 IQR new IQR* bIQR