Transcript Ch 3 - csusm

Chapter 3
Data Characterization
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Today: Mean and Variance
 Mean:
Exercise:
 also called “average”
 Formula:
mean=
compute the mean weight for the
Chargers’ offense players and
defense players.
Sum of data
Number of data
 Characterize the center of the
data distribution
 The most commonly used
measure
Which mean should be higher?
Why?
Are they population mean or sample
mean?
 Sample mean x
 The average derived from sample
Ways to compute the mean:
1. Use calculator.
 Population mean 
 The average derived from the
population
2. Use Excel. (function: average)
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Sensitivity to outliers
Compute the mean for the following 2 groups of data
 Household income in
community a: (Unit =10000$)
 Household income in
community b: (Unit =10000$)
#1
#2
#3
#4
#5
#6
#7
#8
#1
#2
#3
#4
#5
#6
#7
#8
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3
5
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5
4
100
If the mayor decide to provide more public facilities to poor
communities, and the decision is made based on whether
the mean income in the community is below $50,000 per year.
Does such a decision make sense?
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Compute the mean from
frequency table
Below is a frequency table showing
Create a histogram using the data on
the number of days the teams finish
the left, locate the mean on the graph.
their projects
 How to describe the shape of the
histogram?
Relative
Days to
Complete
Frequency
5
4
?
6
12
?
7
8
?
8
6
?
9
4
?
10
2
?
Frequency
 What is the relationship between
the mean and peak?
Use relative frequency to find out the
mean.
How many days on average does a team
finish one project?
mean=
total days
total teams
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Compute the mean from Histogram
Histogram
Frequency
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6
6
Histogram conveys the same
information as the frequency table
5
5
4
4
3
3
2
2
1
0
0
55
mean=
0
15
15
25
25
35
36
45
45
55
55
More
total data value 15  3  25  6  35  5  45  4  55  2

 33
data size
3 65 4 2
Mathematical Expression:
x=33 if sample,  =33 if population
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Weighted Mean
 The mean assumes that each piece of information
equally.
 E.g. average score of the students.
 Sometimes, different data should be put in different
weight.
 One may be more important than the other.
• E.g. some instructor assign 60% on the homework score, and 40%
on the final exam. If a student’s homework score is 84, and got 70
in the exam, compute the student’s final score. (weighted mean of
homework score and exam score)
-- this teacher thinks homework reveals more comprehensive
information about a student’s knowledge, and hence put more
weight.
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When to use weighted mean?
 Some other examples of weighted mean:
 A student’s GPA. A course with more credit takes more weight.
 An economic growth indicator. (some industries affects the economy more than
others)
 Crush time leader: a player who perform the best in the last few minutes of the
game. – can reveal the person’s performance under pressure.
 Expectation – you will see in chapter 4
• E.g. in a gambling game, if with 60% chance you lose one dollar, and with 40%
chance you gain one dollar, the expectation is
60%x$(-1)+40%x$1=-$0.2
 Other examples? (average Cal State Tuition)
Always think whether you should use weighted mean or simple mean.
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Break
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Variance

A measure of data spread.

Also called “the average of squared deviations from the mean”
The larger the variance, the fat the histogram
-- sample variance
N
n
s 
2
 (x
i 1
-- population variance
i
 x)
2
σ2 
2
(x

μ)
 i
i 1
n -1
N
Note the difference!
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Steps to compute the variance
1.
Identify whether the data are of a population or sample
(the formulae are different.)
2.
Use the following table to compute the deviation:
a)
Data
list
5
4
Distance from
the mean
Square
the distance
=5-mean=1.167
=(1.67)2=1.36
Find out the mean:
mean=
5 4 453 2
 3.833
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b)
Find out the distance
(fill out the 2nd column)
c)
Find out the squared distance
4
(the 3rd column)
5
d)
Add up the 3rd column
3
e)
divided by
2
i.
population size; or
ii.
sample size -1
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Comparing variance vs. histogram
Find the variance for the following groups of sample data:
Compare the mean and variance.
Create the histogram to compare the distribution.
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14
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15
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15
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17
20
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What does variance mean?
Variance indicate variation:
 The larger the variance, the more spread out the
data.
 Indicates unpredictability.
 E.g.
• Weather data: weather changes dramatically, hard to predict
tomorrow’s temperature
(If look at temperature data: which has larger variance,
Chicago or San Diego?)
• Stock: more risk on returns.
• A person’s performance: consistency. emotional…
• Other examples?
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Use frequency table to compute the
population variance:
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15
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Data value
Frequency
Relative
Frequency
15
14
1
0.125
16
15
3
0.375
16
16
3
0.375
16
17
1
0.125
17
Data
distance
square
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15
15
15
16
16
Data
distance
square
14
15
16
Compute the
weighted average
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16
17
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Standard Deviation
 Square root of variance.
 An indicator of data deviation, can be directly
compared to the mean.
s= s
2
OR
Sample variance
Sample standard deviation
= 
2
Exercise: compute the
standard deviation from
the histogram on slide no.
5 and locate it on the
histogram.
Population variance
Population standard deviation
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Empirical Rule
 If the data is bell shaped
(most of the time), then
95%
68%
99.7%
 68% of all data will fall in
the range of
μ
μσ
μ  2σ
 
 95% of all data will fall in
the range of
  2
 99.7% of all data will fall in
the range of
  3
μ  3σ
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