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# Lecture 3

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```									   Lecture 3

Stats 2B03

September 15th , 2010

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Recall...

y Recall...
In lecture 2 we discussed:
y Goals for Lec. 3
y Notation
q Types of Sampling
Mode
SRS (SRSWR & SRSWOR)
The Mean
Stratiﬁed
Median
Systematic
Cluster
q Working with ordinal data
Frequency charts
Histograms
Stem and Leaf Plots

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Goals for Lec. 3

y Recall...
q   Measures of Central Tendency
y Goals for Lec. 3
y Notation                  Mean
Mode                        Median
The Mean                    Mode
Median

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Notation

y Recall...
q N represents the population size (if it’s ﬁnite)
y Goals for Lec. 3
y Notation           q n represents the sample size
Mode

The Mean

Median

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Def. 1

y Recall...
Sample Mode
y Goals for Lec. 3
y Notation
The value (or bin) which occurs with highest
Mode
y Def. 1                 frequency
y Sample/Discrete
y Continuous         Population Mode
The Mean

Median                   The value with the highest probability/likelihood

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Def. 1

y Recall...
Sample Mode
y Goals for Lec. 3
y Notation
The value (or bin) which occurs with highest
Mode
y Def. 1                 frequency
y Sample/Discrete
y Continuous         Population Mode
The Mean

Median                   The value with the highest probability/likelihood

May not be unique

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Sample/Discrete

y Recall...
y Goals for Lec. 3
y Notation

Mode
y Def. 1
y Sample/Discrete
y Continuous

The Mean

Median

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Continuous

y Recall...

0.25
y Goals for Lec. 3
y Notation

Mode

0.20
y Def. 1
y Sample/Discrete
y Continuous                     0.15

The Mean
density

Median
0.10
0.05
0.00

2   4   6   8   10

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Def. 2

y Recall...
Sample Mean - x
¯
y Goals for Lec. 3
y Notation
The arithmetic average of a sample
Mode
n
x1 + x2 + . . . + x n       i=1 xi
The Mean
y Def. 2                 x=
¯                          =
y µ and x
¯                             n                   n
y Example 1.
Population Mean - µ
y With bins

Median
The average of the population
N
x1 + x2 + . . . + x N     i=1 xi
µ=                          =
N                  N
(This is correct when N is ﬁnite)

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µ and x
¯

y Recall...
We wish to make inference about µ, the population average,
y Goals for Lec. 3
y Notation           and with do this with the x
¯
Mode

The Mean
y Def. 2
True value    Estimate
y µ and x
¯                      (Population)   (Sample)
y Example 1.
Center        µ             x
¯
y With bins

Median

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µ and x
¯

y Recall...
We wish to make inference about µ, the population average,
y Goals for Lec. 3
y Notation           and with do this with the x
¯
Mode

The Mean
y Def. 2
True value    Estimate
y µ and x
¯                      (Population)   (Sample)
y Example 1.
Center        µ             x
¯
y With bins

Median

Use the mean when the data seems to be normally
distributed (i.e., tails not too heavy)

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Example 1.

y Recall...
Consider the data from the previous lecture
y Goals for Lec. 3
y Notation
11,11,15,17,19,20,20,21,23,28,28,28,32,35,59
Mode

The Mean
What is the mean?
y Def. 2
y µ and x
¯            What is the mean if we remove the largest observation?
y Example 1.
y With bins

Median

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With bins

y Recall...
Treat the middle of the bins as the value
y Goals for Lec. 3
y Notation

Mode
Freq   Count    CumCnt     Percent    CumPct
The Mean
10-19       5         5       33.33      33.33
y Def. 2              20-29       7        12       46.67      80.00
y µ and x
¯
y Example 1.
30-39       2        14       13.33      93.33
y With bins           50-59       1        15        6.67     100.00
Median                  N=       15

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Def. 2

y Recall...
Sample Median
y Goals for Lec. 3
y Notation
The point which half of a sample is below and half
Mode
above
The Mean

Median
Population Median
y Def. 2
y Sample Median
y Example 2.             The point of the population at which half is below
y Comparison             and half above

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Sample Median

y Recall...
If n is odd choose the middle most item (n = 5)
y Goals for Lec. 3
y Notation
e.g., 7,8,9,12,14
Mode

The Mean

Median
y Def. 2
If n is even take the average of the two middle most items
y Sample Median      (n = 6)
y Example 2.
y Comparison              e.g., 7,8,9,12,13,14

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Sample Median

y Recall...
If n is odd choose the middle most item (n = 5)
y Goals for Lec. 3
y Notation
e.g., 7,8,9,12,14
Mode

The Mean

Median
y Def. 2
If n is even take the average of the two middle most items
y Sample Median      (n = 6)
y Example 2.
y Comparison              e.g., 7,8,9,12,13,14

P.S., Don’t forget to sort the data!

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Example 2.

y Recall...
Consider the data from the previous lecture
y Goals for Lec. 3
y Notation
11,11,15,17,19,20,20,21,23,28,28,28,32,35,59
Mode

The Mean
What is the median?
Median
What is the median if we remove the largest observation?
y Def. 2
y Sample Median
y Example 2.
y Comparison

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Comparison

y Recall...
y Goals for Lec. 3
All Data Minus Largest
y Notation                  Mean
Mode

The Mean                   Median
Median
y Def. 2
y Sample Median
y Example 2.
y Comparison

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Comparison

y Recall...
y Goals for Lec. 3
All Data Minus Largest
y Notation                  Mean
Mode

The Mean                   Median
Median
y Def. 2
y Sample Median
y Example 2.
y Comparison
The median is more robust
(i.e., it changes less due to outliers)

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