Linear TransformationsLinear Transformations2.1b 2.1b

Target Goal: I can transform linear data and Target Goal: I can transform linear data and determine its effect on center and spread.determine its effect on center and spread.

HW: pg 107: 19-21, 23HW: pg 107: 19-21, 23

Changing the Unit of Changing the Unit of MeasurementMeasurement

Linear TransformationsLinear Transformations

CChanges the original variable x into the hanges the original variable x into the new variable xnew variable xnewnew given by an equation of given by an equation of

the form:the form:

xxnewnew = = aa + + bbx x

Linear TransformationsLinear Transformations xxnewnew = = aa + + bbxx

Adding the constant Adding the constant aa shifts all values shifts all values of x of x upward or downward by the same amount.upward or downward by the same amount.

MultiplyingMultiplying by the positive constant by the positive constant bb changes changes the the size of the unit size of the unit of measurement.of measurement.

AddingAdding a constant to each observation a constant to each observation does not does not change the spreadchange the spread..

Linear transformations Linear transformations do not change the shape do not change the shape (symmetric, skewed right or left) of a distribution. (symmetric, skewed right or left) of a distribution.

Effects of Linear TransformationsEffects of Linear Transformations

Linear transformations: miles/kilometersLinear transformations: miles/kilometers Fahrenheit/Celsius Fahrenheit/Celsius

MultiplyingMultiplying each observation each observation byby a positive a positive number number bb multiplies both measures of center multiplies both measures of center ((mean and medianmean and median) and measures of spread ) and measures of spread ((standard deviation and IQRstandard deviation and IQR) by ) by bb..

AddingAdding the same number the same number aa (either positive or (either positive or negative)negative) to each observation: to each observation: adds adds aa to to measures of measures of centercenter and to and to quartilesquartiles but but does does not change measures of spread.not change measures of spread.

Ex: Los Angeles Laker’s Salaries (2000Ex: Los Angeles Laker’s Salaries (2000))

= $4.14 mil, M = $2.6 mil, s = 4.76 mil= $4.14 mil, M = $2.6 mil, s = 4.76 milx

Ex. Los Angeles Laker’s SalariesEx. Los Angeles Laker’s Salaries

Plot (a) stems = millions, Describe distribution:Plot (a) stems = millions, Describe distribution:

Right skewed with two outliers.Right skewed with two outliers.

Five-number summary is:Five-number summary is:$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1

(a) Suppose that each member of the team (a) Suppose that each member of the team receives a $100,000 bonus for winning the receives a $100,000 bonus for winning the NBA championship.NBA championship.

How will this affect the shape, center and How will this affect the shape, center and spread?spread?

$100,000 =$100,000 =$0.1 mill

Five-number summary was:Five-number summary was:$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1

The linear transformation isThe linear transformation is

xxnewnew = = aa + + bbxx ; x; xnewnew = =

[each value in (a) and the mean and median will all [each value in (a) and the mean and median will all increase by 0.1: plot (b)]increase by 0.1: plot (b)]

xxnew new = 0.1 + 1x = 0.1 + 1x

Five-number summary was:Five-number summary was:$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1

We added 0.1; We added 0.1; Old: mean = $4.14 mil, M = $2.6 mil, s = 4.76 Old: mean = $4.14 mil, M = $2.6 mil, s = 4.76

milmil meanmeannewnew = =

$4.24 mil$4.24 mil medianmediannewnew = =

$2.7 mil $2.7 mil ssnewnew = =

no changeno change

Five-number summary was:Five-number summary was:$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1

The new five-number summary is:The new five-number summary is:$0.4 mil, $1.1 mil, $2.7 mil, $4.7mil, $17.2$0.4 mil, $1.1 mil, $2.7 mil, $4.7mil, $17.2

Does the spread change? Does the spread change?

NoNo

Ex. L.A. Lakers cont.Ex. L.A. Lakers cont.

(b) Each player is offered a 10% increase (b) Each player is offered a 10% increase base salary.base salary.

The linear transformation isThe linear transformation is

xxnewnew = = aa + + bbxx

xxnew new = =0 + 1.10x0 + 1.10x

Calculate the following:Calculate the following: mean = $4.14(1.10) = mean = $4.14(1.10) =

median = $2.6(1.10) =median = $2.6(1.10) =

Spread and IQR also increased by 10%.Spread and IQR also increased by 10%.

$4.55

$2.86

Five-number summary was:Five-number summary was:$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1$0.3 mil, $1.0 mil, $2.6 mil, $4.6 mil, $17.1

The new five-number summary is:The new five-number summary is:[see graph (c)] [see graph (c)]

xxnew new = 0 + 1.10x = 0 + 1.10x

$0.33mil, $1.10 mil, $2.86 mil, $5.06 mil, $18.81mil$0.33mil, $1.10 mil, $2.86 mil, $5.06 mil, $18.81mil

Example: CockroachesExample: Cockroaches Maria measures the lengths of 5 cockroaches that she Maria measures the lengths of 5 cockroaches that she

finds at school. Here are her results (finds at school. Here are her results (in inchesin inches):):

1.41.4 2.22.2 1.11.1 1.61.61.21.2

a.a. Find the mean and standard deviation of Maria’s Find the mean and standard deviation of Maria’s measurements. (use calc)measurements. (use calc)

Enter data into L1,Enter data into L1,

STAT:CALC:1-VAR Stats: enter: L1 or,STAT:CALC:1-VAR Stats: enter: L1 or, 22ndnd STAT(LIST):MATH:stdDev(7): L1 STAT(LIST):MATH:stdDev(7): L1

Mean = 7.5/5 = 1.5Mean = 7.5/5 = 1.5

s = .436s = .436

Example: CockroachesExample: Cockroaches

b.b. Maria’s science teacher is furious to discover Maria’s science teacher is furious to discover that she has measured the cockroach lengths that she has measured the cockroach lengths in inches rather than centimeters (There are in inches rather than centimeters (There are 2.54 cm in 1 inch). She gives Maria two 2.54 cm in 1 inch). She gives Maria two minutes to report the mean and standard minutes to report the mean and standard deviation of the 5 cockroaches in deviation of the 5 cockroaches in centimeterscentimeters. . Maria succeeded. Will you? Maria succeeded. Will you?

xxnewnew = = aa + + bbxx

To obtain mean and s in centimeters, To obtain mean and s in centimeters, multiply the results in inches by 2.54:multiply the results in inches by 2.54:

New Mean = 1.5 x2.54 =New Mean = 1.5 x2.54 =

3.81cm3.81cm

New s = .436 x 2.54 =New s = .436 x 2.54 =

1.107 cm1.107 cm

c. Considering the 5 cockroaches that Maria c. Considering the 5 cockroaches that Maria found as a small sample from the population of found as a small sample from the population of all cockroaches at her school, all cockroaches at her school, what would you what would you estimate as the estimate as the average lengthaverage length of the of the population of cockroaches?population of cockroaches? How sure of the How sure of the estimate are you? estimate are you?

The average cockroach length can be est. by The average cockroach length can be est. by the mean length of the sample ; 1.5 inches.the mean length of the sample ; 1.5 inches.

This is a questionable estimate because our This is a questionable estimate because our sample is so small.sample is so small.

Example: mean and Example: mean and s s are not enoughare not enough

The mean x bar and standard deviation s The mean x bar and standard deviation s measure center and spread but are not a measure center and spread but are not a complete description of a distribution. complete description of a distribution.

Data sets with different shapes can have Data sets with different shapes can have the same mean and standard deviation. the same mean and standard deviation.

To demonstrate this fact, To demonstrate this fact, use your use your calculatorcalculator to find the to find the mean mean and and ss for the for the following two small data sets:following two small data sets:

Set A: Set A:

mean = 7.501, s = 2.032mean = 7.501, s = 2.032

Set B: Set B:

mean = 7.501, s = 2.031mean = 7.501, s = 2.031

Then make a stem-plot of each and Then make a stem-plot of each and comment on the comment on the shapeshape of each of each distribution.distribution.

Set A is skewed to the left.Set A is skewed to the left.

Set B has a high outlier.Set B has a high outlier.