Download Standard deviation and the Normal Model

Scaling, Shifting, Combining
Variables
Chapters 5 and 6
Let’s have a look at our test scores…
(the top 3 and bottom 3 have been deleted)
Here are the summary statistics:
Statistic
measures
of center
or
position
Value
Mean 86
Median 87
Q1 81
Q3 92
Std Dev 9.1
measures
of spread
IQR 11
(Q3 – Q1)
Range 40
(max-min)
Let’s pretend that I add… 50 points to EVERYONE’s score.
(don’t worry, I won’t. that’s just… ridiculous)
What would happen to these
statistics?
Statistic
Value
after
+50?
Mean 86
136
Median 87
137
Q1 81
131
Q3 92
142
Std Dev 9.1
9.1
IQR 11
11
Range 40
40
Shifting data (adding)
Adding (or subtracting) a constant to every data value…
• adds (or subtracts) that constant to measures of
position/center…
• …but does NOT change measures of spread.
Statistic
Value
after
+50?
Mean 86
136
Median 87
137
Q1 81
131
Q3 92
142
Std Dev 9.1
9.1
IQR 11
11
Range 40
40
Let’s say that I multiply EVERYONE’s score by 2.
(perish the thought.)
Now what would happen?
Statistic
Value
after
x2?
Mean 86
172
Median 87
174
Q1 81
162
Q3 92
184
Std Dev 9.1
18.2
IQR 11
22
Range 40
80
Scaling data (multiplying)
Multiplying every data value by a constant…
• causes measures of position/center
and
measures of spread
to be multiplied by that constant.
Statistic
Value
after
x2?
Mean 86
172
Median 87
174
Q1 81
162
Q3 92
184
Std Dev 9.1
18.2
IQR 11
22
Range 40
80
combining scaling and shifting!
wheeeee!!!
We have the following “test” scores:
46
49
52
min = 46
Q1 = 50.5
median = 55
Q3 = 61.5
max = 68
54
56
59
64
68
mean = 56
range = 22
IQR = 11
std dev = 7.426
Suppose the teacher first multiplies all test scores by 2,
then subtracts 30 from each…
46
92
62
49
98
68
52 54 56 59 64 68 x2
104 108 112 118 128 136 -30
74 78 82 88 98 106
x2 then -30
mean = 56
x2 then -30
Q1 = 50.5 x2 then -30
range = 22
x2
IQR = 11
x2
min = 46
median = 55
x2 then -30
Q3 = 61.5 x2 then -30
max = 68
x2 then -30
std dev = 7.426 x2
Practice with scaling and shifting
1
2
4
7
7
9
12
Multiply each value by 5 and add 4.
Find these new measures of center and spread.
Measures of center and spread:
Mean
6
34
Median
7
39
Range
11
55
IQR
7
St Dev
3.9
35
19.5
BACK TO THE BACONATOR TRIPLE…
(THIS ONE’S NOT ON YOUR PRINTOUT…)
81810
23 29 33 39 40 45 52 58 63 65 ____
5 number summary: IF THE “81” WERE MISTYPED AS…
Would it affect the…
Min:
23
Q1:
36
b) Median?
Q2:
45
c) IQR?
Q3:
60.5
Max: 81
a) Mean?
(YES)
(NO)
(NO… since Q3 and Q1 would not
change)
d) SD?
(YES, since the mean would
change)
e) Range?
changed)
(YES, since the maximum has
combining two variables!!!
When adding or subtracting two
independent variables, “x” and “y”:
 X Y   X  Y
 X Y    
2
X
2
Y
ALWAYS ADD VARIANCES!
 X Y   X  Y
Only possible if “x” and “y”
are INDEPENDENT
Natalie and Michelle are roommates that both work as waitresses
in two different restaurants. The amount of money that Michelle
earns in a week is a random variable with a mean of $500 and a
standard deviation of $75. The amount of money that Natalie
earns in a week is a random variable with a mean of $600 and a
standard deviation of $100. The two of them work in different
parts of Austin, so we will assume that the two ladies’ earnings are
independent of one another. We will also assume that the
distributions for each ladies’ weekly earnings are approximately
normally distributed.
M = 500
M = 75
N = 600
N = 100
a) What are the mean and standard deviation for Michelle and Natalie’s combined
earnings (M + N) for one week?
M+N = $1100
M+N = $125
b) What are the mean and standard deviation for the difference in (N – M),
Natalie’s earnings and Michelle’s earnings for one week?
N-M = $100
N-M = $125
STOP!!!
(be sure to check for updated HW)