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3 Summarizing Exam Scores Exam 1 – Score: 67 – Mean: – Standard Deviation: Exam 2 – Score: 64 – Mean: – Standard Deviation:
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1
The Standard Deviation as a Ruler
A student got a 67/75 on the first exam and a 64/75 on the second exam. She was disappointed that she did not score as well on the second exam.
To her surprise, the professor said she actually did better on the second exam, relative to the rest of the class.
2
The Standard Deviation as a Ruler
How can this be?Both exams exhibit variation in the
scores.However, that variation may be
different from one exam to the next.The standard deviation provides a ruler
for comparing the two exam scores.
3
Summarizing Exam ScoresExam 1
– Score: 67– Mean:
– Standard Deviation:
Exam 2– Score: 64– Mean:
– Standard Deviation:
5.59y
61.8s
1.50y
86.11s
4
Standardizing
syyz
Look at the number of standard deviations thescore is from the mean.
5
Standardized Exam ScoresExam 1
– Score: 67
Exam 2– Score: 64
87.061.8
5.5967
z
z
17.186.11
1.5064
z
z
6
Standardized Exam ScoresOn exam 1, the 67 was 0.87
standard deviations better than the mean.
On exam 2, the 64 was 1.17 standard deviations better than the mean.
7
StandardizingShifts the distribution by
subtracting off the mean.Rescales the distribution by
dividing by the standard deviation.
8
Distribution of Low Temps
5
10
15
20
Cou
nt
-10 0 10 20 30 40 50Low Temperature (o F)
9
Shifting the Distribution
5
10
15
20
Cou
nt
-40 -30 -20 -10 0 10 20Low Temperature – 32 (o F)
10
ShiftingTemperature (o F)
– Median: 24.0o F– Mean: 24.4o F
– IQR: 16.0o F– Std Dev: 11.22o F
Temp – 32 (o F)– Median: –8o F– Mean: –7.6o F
– IQR: 16.0o F– Std Dev: 11.22o F
11
ShiftingWhen adding (or subtracting) a
constant:– Measures of position and center
increase (or decrease) by that constant.
– Measures of spread do not change.
12
Rescaling
5
10
15
Cou
nt
-20 -15 -10 -5 0 5 10Low Temperature (o C)
13
RescalingTemp – 32 (o F)
– Median: –8o F– Mean: –7.6o F
– IQR: 16.0o F– Std Dev: 11.22o F
Temperature (o C)– Median: –4.4o F– Mean: –4.2o F
– IQR: 8.9o F– Std Dev: 6.24o F
14
RescalingWhen multiplying (or dividing) by
a constant:– All measures of position, center and
spread are multiplied (or divided) by that constant.
15
StandardizingStandardizing does not change the
shape of the distribution.Standardizing changes the center by
making the mean 0.Standardizing changes the spread by
making the standard deviation 1.
16
Normal ModelsOur conceptualization of what the
distribution of an entire population of values would look like.
Characterized by population parameters: μ and σ.
17
40 45 50 55 60 65 70 75 80
0
10
20
30
Height
Per
cent
18
Describe the sampleShape is symmetric and mounded
in the middle.Centered at 60 inches.Spread between 45 and 75 inches.30% of the sample is between 60
and 65 inches.
19
Normal ModelsOur conceptualization of what the
distribution of an entire population of values would look like.
Characterized by a bell shaped curve with population parameters– Population mean = μ– Population standard deviation = σ.
20
40 45 50 55 60 65 70 75 80
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Height
Den
sity
Sample Data
21
40 45 50 55 60 65 70 75 80
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Height (inches)
Den
sity
Normal Model
22
Population – all items of interest.
Example: All children age 5 to 19.
Variable: Height
Sample – afew items from the population.Example: 550
children.
Normal Model
40 45 50 55 60 65 70 75 80
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Height (inches)
Den
sity
40 45 50 55 60 65 70 75 80
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Height
Den
sity
23
Normal ModelHeightCenter:
– Population mean, μ = 60 in.Spread:
– Population standard deviation, σ = 6 in.
24
68-95-99.7 RuleFor Normal Models
– 68% of the values fall within 1 standard deviation of the mean.
– 95% of the values fall within 2 standard deviations of the mean.
– 99.7% of the values fall within 3 standard deviations of the mean.
25
Normal Model - Height68% of the values fall between
60 – 6 = 54 and 60 + 6 = 66.95% of the values fall between
60 – 12 = 48 and 60 + 12 = 72.99.7% of the values fall between
60 – 18 = 42 and 60 + 18 = 78.
26
From Heights to Percentages
What percentage of heights fall above 70 inches?
Draw a picture.How far away from the mean is 70
in terms of number of standard deviations?
27
807570656055504540
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
Height (inches)
Den
sity
Normal Model
Shaded area?
28
Standardizing
67166070 .z
yz
29
Standard Normal Model Table Z: Areas under the standard
Normal curve in the back of your text.
On line:http://davidmlane.com/hyperstat/z_table.html
30
From Percentages to Heights
What height corresponds to the 75th percentile?
Draw a picture.The 75th percentile is how many
standard deviations away from the mean?
31
807570656055504540
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
Height (inches)
Den
sity
Normal Model
25%
50%
25%
32
Standard Normal Model Table Z: Areas under the standard
Normal curve in the back of your text.
On line:http://davidmlane.com/hyperstat/z_table.html
33
Reverse Standardizing
0264606706660670
..*y
y.
yz
34
Do Data Come from a Normal Model?
The histogram should be mounded in the middle and symmetric.
The data plotted on a normal probability (quantile) plot should follow a diagonal line.– The normal quantile plot is an option in
JMP: Analyze – Distribution.
35
Do Data Come from a Normal Model?
Octane ratings – 40 gallons of gasoline taken from randomly selected gas stations.
Amplifier gain – the amount (decibels) an amplifier increases the signal.
Height – 550 children age 5 to 19.
36
.01
.05
.10
.25
.50
.75
.90
.95
.99
-3
-2
-1
0
1
2
3
Nor
mal
Qua
ntile
Plo
t
2
4
6
8
Cou
nt
85 90 95Octane Rating
37
.01
.05
.10
.25
.50
.75
.90
.95
.99
-3
-2
-1
0
1
2
3
Nor
mal
Qua
ntile
Plo
t
5
10
15
20
25
Cou
nt
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12Amplifier Gain (dB)
38
.01
.05
.10
.25
.50
.75
.90
.95
.99
-3
-2
-1
0
1
2
3
Nor
mal
Qua
ntile
Plo
t
50
100
150
Cou
nt
45 50 55 60 65 70 75
39
Nearly normal?Is the histogram basically
symmetric and mounded in the middle?
Do the points on the Normal Quantile plot fall close to the red diagonal (Normal model) line?