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)

x

x*

0

aShifts data by a

Changes scale

Linear Transformationsx* = a+ b x

Examples: Changing1. from feet (x) to inches (x*): x*=12x2. from dollars (x) to cents (x*):

x*=100x3. from degrees celsius (x) to degrees

fahrenheit (x*): x* = 32 + (9/5)x 4. from ACT (x) to SAT (x*):

x*=150+40x5. from inches (x) to centimeters (x*):

x* = 2.54x

0 120 10032 9/5150 400 2.54

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.

Shifting data only: b = 1x* = 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 kg1. 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 xmedian m1st quartile Q1

3rd quartile Q3

stand dev svariance s2

IQR

x* data: x* = a + bxx1*, x2*, x3*, . . ., xn*

Summary statistics:new mean x* = a +

bxnew median m* =

a+bmnew 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)62, 64, 74, 72x = 68 inchess = 5.89 inches

Suppose we wantcentimeters 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 centimeterss* = 14.9606 centimeters

Note thatx* = 2.54x = 2.54(68)=172.2s* = 2.54s = 2.54(5.89)=14.9606

not necessary!UNC method

Go directly to this. NCSU method

Example of x* = a + bxx data:Percent returns from 4investments during2003:5%, 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!

thats* ≠ s – 2%) !!

not necessary!

Go directly to this

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, 75s = 17.12; Q1=37, Q3 =69; IQR = 69 – 37 =

32Melons 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

1 1where ,

x x x xz x a bx a b

s s s s 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

941.25 799.50 141.75 1.594.50 94.50

z

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 isz = (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

705-799.50 -94.50z= = =-194.50 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

Assembly Time (seconds)

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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

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

* * *1 2 3 1 2 3

*

*

*1 1 1

*3 3 3

original data , , ,... transformed data , , ,...

summary statistics summary statistics

mean new mean

median new median

1st new

3rd new

st dev

x x x x x x

x x a bx

m m a bm

Q Q a bQ

Q Q a bQ

*

2 2 2 2

new st dev

var. new var. *

new *

s s bs

s s b s

IQR IQR bIQR