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2.4 (cont.) Changing Units of Measurement

2.4 (cont.) Changing Units of Measurement. How shifting and rescaling data affect data summaries. x *. Shifts data by a. a. Changes scale. 0. x. Shifting and rescaling: linear transformations. Original data x 1 , x 2 , . . . x n Linear transformation:

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2.4 (cont.) Changing Units of Measurement

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  1. 2.4 (cont.)Changing Units of Measurement How shifting and rescaling data affect data summaries

  2. x* Shifts data by a a Changes scale 0 x Shifting and rescaling: linear transformations • Original data x1, x2, . . . xn • Linear transformation: x* = a + bx, (intercept a, slope b)

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

  4. Shifting data only: b = 1x* = 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.

  5. 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 Shifting data only: b = 1x* = a + x (cont.) • No change in shape • No change in spread • Shift by 74

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

  7. 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 Rescaling data: x* = a + bx, b > 0 (cont.)

  8. 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 Example of x* = a + bx not necessary!UNC method Go directly to this. NCSU method

  9. 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%) !! Example of x* = a + bx not necessary! Go directly to this

  10. 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

  11. 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

  12. 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!!

  13. SUMMARY: Linear Transformations x* = a + bx • 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

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

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