1. Chapter 6 Part 1 Using the Mean and Standard Deviation Together
z-scores
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

4. Exam 1: y1 = 88, s1 = 6; exam 1 score: 91
Which score is better?

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

6. Z-scores add to zero
Student/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
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

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 $ In NC the tuition was $ What is NC’s z-score?
1.03
-1.03
2.39
1865
-1865

8. Changing Units of Measurement
How shifting and rescaling data affect data summaries

9. 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 x* a

10. Linear Transformations x* = a+ b x
2.54 12
150 40
100 32
9/5
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

11. 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.

12. 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 = 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
No change in shape
No change in spread
Shift by 74

13. 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

14. 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 = kg
min=54.30 kg
max= kg
range= kg
s = kg
Change from kilograms to pounds:
x* = 2.2x (a = 0, b = 2.2)
x* = 2.2(82.36)= pounds
min* = 2.2(54.30)= pounds
max* = 2.2(161.50)=355.3 pounds
range*= 2.2(107.20)= pounds
s* = * 2.2 = pounds

15. Example of x* = a + bx 4 student heights in inches (x data)
4 student heights in centimeters:
= 2.54(62)
= 2.54(64)
= 2.54(74)
= 2.54(72)
x* = centimeters
s* = centimeters
Note that
x* = 2.54x = 2.54(68)=172.2
s* = 2.54s = 2.54(5.89)=
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)
not necessary!UNC method
Go directly to this. NCSU method

16. Example of x* = a + bx 3% = 5% - 2% 2% = 4% - 2% 1% = 3% - 2%
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%) !!
not necessary!
Go directly to this

17. 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=38, Q3 =68; IQR = 68 – 38 = 30
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 = =-12, Q*3 = = 18
IQR* = 18 – (-12) = 30
NOTE: s* = s, IQR*= IQR

18. 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 $ What is the z-score of this tuition?
x* = $250+$35(15.7) = $799.50; s* = $35(2.7) = $94.50

19. 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!!

20. 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

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

23. 68-95-99.7 rule Mean and Histogram Standard Deviation (graphical)
(numerical)
Histogram
(graphical)
rule

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

25. 68-95-99.7 rule: 68% within 1 stan. dev. of the mean
34%
34%
y-s y y+s

26. 68-95-99.7 rule: 95% within 2 stan. dev. of the mean
47.5%
47.5%
y-2s y y+2s

27. Example: textbook costs

28. Example: textbook costs (cont.)

29. Example: textbook costs (cont.)

30. Example: textbook costs (cont.)

31. The best estimate of the standard deviation of the men’s weights displayed in this dotplot is10 15 20 40

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