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Lecture 3

VIEWS: 3 PAGES: 19

  • pg 1
									   Lecture 3

    Stats 2B03

September 15th , 2010




                        1 / 15
                     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




                                                   2 / 15
                     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




                                                        3 / 15
                     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




                                                                          4 / 15
                     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




                                                                             5 / 15
                     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




                                                                             5 / 15
                     Sample/Discrete

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

Mode
y Def. 1
y Sample/Discrete
y Continuous

The Mean

Median




                                       6 / 15
                     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




                                                             7 / 15
                     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)




                                                                   8 / 15
                     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




                                                                                  9 / 15
                     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)




                                                                                  9 / 15
                     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




                                                                              10 / 15
                     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




                                                                       11 / 15
                     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




                                                                              12 / 15
                     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




                                                                                  13 / 15
                     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!




                                                                                  13 / 15
                     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




                                                                                14 / 15
                     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




                                                             15 / 15
                     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)




                                                                     15 / 15

								
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