1

Chapter 6 Part 1Using the Mean and Standard

Deviation Together

z-scores

68-95-99.7 rule

Changing units (shifting and rescaling data)

2

Z-scores: Standardized Data Values

Measures the distance of a number from the mean in units of

the standard deviation

3

z-score corresponding to y

where

original data value

the sample mean

s the sample standard deviation

the z-score corresponding to

y yz

s

y

y

z y

4

Exam 1: y1 = 88, s1 = 6; exam 1 score: 91

Exam 2: y2 = 88, s2 = 10; exam 2 score: 92

Which score is better?

1

2

91 88 3z .5

6 692 88 4

z .410 10

91 on exam 1 is better than 92 on exam 2

If data has mean and standard deviation ,

then standardizing a particular value of

indicates how many standard deviations

is above or below the mean .

y s

y

y

y

5

Comparing SAT and ACT Scores

SAT Math: Eleanor’s score 680

SAT mean =500 sd=100 ACT Math: Gerald’s score 27

ACT mean=18 sd=6 Eleanor’s z-score: z=(680-500)/100=1.8 Gerald’s z-score: z=(27-18)/6=1.5 Eleanor’s score is better.

6

Z-scores add to zeroStudent/Institutional Support to Athletic Depts For the 9 Public ACC

Schools: 2013 ($ millions)

School Support y - ybar Z-score

Maryland 15.5 6.4 1.79

UVA 13.1 4.0 1.12

Louisville 10.9 1.8 0.50

UNC 9.2 0.1 0.03

VaTech 7.9 -1.2 -0.34

FSU 7.9 -1.2 -0.34

GaTech 7.1 -2.0 -0.56

NCSU 6.5 -2.6 -0.73

Clemson 3.8 -5.3 -1.47

Mean=9.1000, s=3.5697

Sum = 0 Sum = 0

7

In a recent year the mean tuition at 4-yr public colleges/universities in the U.S. was $6185 with a standard deviation of $1804. In NC the mean tuition was $4320. What is NC’s z-score?

1. 2. 3. 4 5

37%

55%

0%2%7%

1. 1.03

2. -1.03

3. 2.39

4. 1865

5. -1865

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 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 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 snew 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 = 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

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

This is why z-scores are so useful!!

SUMMARY: Linear Transformations x* = a + bx

Assembly Time (seconds)

0

5

10

15

20

25

30

Fre

qu

en

cy

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

Assembly Time (minutes)

0

10

20

30

Fre

qu

en

cy

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

23

68-95-99.7 rule

Mean andStandard Deviation

(numerical)

Histogram(graphical)

68-95-99.7 rule

24

The 68-95-99.7 rule; applies only to mound-shaped data

approximately 68% of the measurements

are within 1 standard deviation of the mean,

that is, in ( , )

approx. 95% of the measurements are within

2 stand. dev. of the mean, i.e., in ( 2 , 2 )

almos

y s y s

y s y s

t all the measurements are within 3 stan.

dev of the mean, i.e., in ( 3 , 3 )y s y s

25

68-95-99.7 rule: 68% within 1 stan. dev. of the mean

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-5

-4.5 -4

-3.5 -3

-2.5 -2

-1.5 -1

-0.5 0

0.5 1

1.5 2

2.5 3

3.5 4

4.5 5

68%

34%34%

y-s y y+s

26

68-95-99.7 rule: 95% within 2 stan. dev. of the mean

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-5

-4.5 -4

-3.5 -3

-2.5 -2

-1.5 -1

-0.5 0

0.5 1

1.5 2

2.5 3

3.5 4

4.5 5

95%

47.5% 47.5%

y-2s y y+2s

27

Example: textbook costs

375.48

42.72

50

y

s

n

286 291 307 308 315 316 327 328340 342 346 347 348 348 349 354355 355 360 361 364 367 369 371373 377 380 381 382 385 385 387390 390 397 398 409 409 410 418422 424 425 426 428 433 434 437440 480

28

Example: textbook costs (cont.)286 291 307 308 315 316 327 328340 342 346 347 348 348 349 354355 355 360 361 364 367 369 371373 377 380 381 382 385 385 387390 390 397 398 409 409 410 418422 424 425 426 428 433 434 437440 480

375.48 42.72

( , ) (332.76, 418.20)

32percentage of data values in this interval 64%;

5068-95-99.7 rule: 68%

y s

y s y s

1 standard deviation interval about the mean

29

Example: textbook costs (cont.)286 291 307 308 315 316 327 328340 342 346 347 348 348 349 354355 355 360 361 364 367 369 371373 377 380 381 382 385 385 387390 390 397 398 409 409 410 418422 424 425 426 428 433 434 437440 480

375.48 42.72

( 2 , 2 ) (290.04, 460.92)

48percentage of data values in this interval 96%;

5068-95-99.7 rule: 95%

y s

y s y s

2 standard deviation interval about the mean

30

Example: textbook costs (cont.)286 291 307 308 315 316 327 328340 342 346 347 348 348 349 354355 355 360 361 364 367 369 371373 377 380 381 382 385 385 387390 390 397 398 409 409 410 418422 424 425 426 428 433 434 437440 480

375.48 42.72

( 3 , 3 ) (247.32, 503.64)

50percentage of data values in this interval 100%;

5068-95-99.7 rule: 99.7%

y s

y s y s

3 standard deviation interval about the mean

31

The best estimate of the standard deviation of the men’s weights

displayed in this dotplot is

1 2 3 4

4%9%

71%

16%

1. 10

2. 15

3. 20

4. 40

32

End of Chapter 6 Part 1.Next: Part 2 Normal Models

33

Z-scores add to zeroStudent/Institutional Support to Athletic Depts For the 9 Public ACC

Schools: 2013 ($ millions)

School Support y - ybar Z-score

Maryland 15.5 6.4 1.79

UVA 13.1 4.0 1.12

Louisville 10.9 1.8 0.50

UNC 9.2 0.1 0.03

VaTech 7.9 -1.2 -0.34

FSU 7.9 -1.2 -0.34

GaTech 7.1 -2.0 -0.56

NCSU 6.5 -2.6 -0.73

Clemson 3.8 -5.3 -1.47

Mean=9.1000, s=3.5697

Sum = 0 Sum = 0