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DESCRIPTIVE MEASURES

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DESCRIPTIVE MEASURES Powered By Docstoc
					    Perform Descriptive
    Measures Statistical
        Procedures




FINANCE SCHOOL       805A-44A-1033-VG-1
            Terminal Learning Objective

    Action: Perform descriptive measures statistical
            procedures by computing the measures of central
            tendency and dispersion; constructing a frequency
            distribution (absolute, relative, and cumulative);
            plotting a histogram, frequency polygon, and
            cumulative frequency distribution; and identify the
            skewness of the direction.

Conditions: Given the Formulas provided in the Summary
            Sheet and a data set (population or sample).

 Standard: IAW the Descriptive Measures Programmed Text
           and Accepted Statistical Techniques.



                    FINANCE SCHOOL                    805A-44A-1033-VG-2
               What is (or are) Statistics?


“There are Three Types of Lies: LIES, DAMNED LIES,
and STATISTICS.” -- DISRAELI. Other quotes are:

There was a Man who Drowned in a River only 2 Feet Deep
--- On Average, That Is.

People use Statistics as a Drunk uses a Lamppost
--- More for Support than for Illumination.

 Statistics are like a Razor
--- Vital, Variable, And Vicious!

Figures Don’t Lie, But Liars Figure!


                        FINANCE SCHOOL              805A-44A-1033-VG-3
               Descriptive Terms


• Population (N): is the whole thing.

• Sample (n): is a part of the whole.

• Parameter.

• Statistic.




                FINANCE SCHOOL          805A-44A-1033-VG-4
         Sampling Procedures



              Select a Sample


Population                       Sample
Parameter                        Statistic
             Make an Inference




              FINANCE SCHOOL         805A-44A-1033-VG-5
  Measures of
Central Tendency


    Mean.

   Median.

    Mode.



 FINANCE SCHOOL    805A-44A-1033-VG-6
                          MEAN


•   Population Mean ():  = X       Where:

                            X = observations
                                       = arithmetic
                                         population mean
                                      N = # observations



                                      Where:
•   Sample Mean (X): X = X
                                      X = observations
                     n
                                      X = arithmetic
                                          population mean
                                      n = # observations

                     FINANCE SCHOOL              805A-44A-1033-VG-7
                  Population




 = X = 7 + 9 + 10 + 10 + 11 + 13 = 60 = 10
N            6                6




                FINANCE SCHOOL                 805A-44A-1033-VG-8
                Extreme Values




 = 7 + 9 + 10 + 10 + 11 + 13 + 31 = 91 = 13
        7                     7




                FINANCE SCHOOL                 805A-44A-1033-VG-9
                   Characteristics
                    of the Mean

• Familiar to most people.

• Always exists.

• Always unique.

• Lends itself to statistical manipulation.

• Accounts for each individual item.

• Reliable estimator.


                   FINANCE SCHOOL             805A-44A-1033-VG-10
                    MEDIAN



The middle item (or mean of the two middle items)
found within a set of data, when the data is
arranged in order of magnitude.




                 FINANCE SCHOOL             805A-44A-1033-VG-11
                   Characteristics
                    of the Median


• Always exists.

• Always unique.

• Ordered array required.

• Less reliable than the mean as an estimator.




                   FINANCE SCHOOL            805A-44A-1033-VG-12
                     MODE


The most frequently occurring observation or
observations in the data.

          X            X             X
          7            7             7
          9            9             9
         10           10             9
                                      ]
         13           10
                         ]          10
         16           13            13
         17           15            13
                                      ]
       No Mode      Unimodal      Bimodal


                 FINANCE SCHOOL             805A-44A-1033-VG-13
                Characteristics
                 of the Mode


• May not always exist.

• Requires no calculation.

• Are not unique.




                FINANCE SCHOOL    805A-44A-1033-VG-14
         Arithmetic Mean



Population Mean = 

Sample Mean = X




          FINANCE SCHOOL   805A-44A-1033-VG-15
Measures of Dispersion



       Range.

      Variance.

 Standard Deviation.




    FINANCE SCHOOL       805A-44A-1033-VG-16
                     RANGE



The difference between the two extreme values of a
set of a data.




 RANGE = LARGEST VALUE - SMALLEST VALUE

                  Rg = Xmax - Xmin




                FINANCE SCHOOL              805A-44A-1033-VG-17
                Average Deviation



Each data point of the data set differs from the mean by
a measurable amount. We can measure this in a
general sense by the formula:
                               di = Xi - 
where di is the difference between any specific data
point, Xi , and the mean.




                  FINANCE SCHOOL                 805A-44A-1033-VG-18
                  Original Data Set



Let’s consider our original data set:

7, 9, 10, 10, 11, 13

the mean is:

 = 7 + 9 + 10 + 10 + 11 + 13 = 60 = 10
                 6              6




                   FINANCE SCHOOL         805A-44A-1033-VG-19
                        Data Point

Now we find the deviation of each data point from the mean:
          X        X-  = d
          7        7 - 10 = -3
          9        9 - 10 = -1
         10       10 - 10 = 0
         10       10 - 10 = 0
         11       11 - 10 = +1
         13       13 - 10 = +3
                 (X - ) = 0
                    N

For this and any data set, the average of the deviations
about the mean is Zero.
                     FINANCE SCHOOL                 805A-44A-1033-VG-20
                     VARIANCE


The mean of the squares of the variations from the
mean of a frequency distribution.
                                            2
                                      x)
                            2
                                  x - N
                                     2

Population Variance:      
                                      N
                                            2
                                      x)
                            2
                                  x - n
                                    2

Sample Variance:          S
                                     n-1

                  FINANCE SCHOOL                805A-44A-1033-VG-21
                Standard Deviation


The square root of the mean of the standard deviations.

                                                  2
Population Std Deviation:                 x)
                                      x - N
                                         2

                       2
                            
                                          N

Sample Std Deviation:                            2
                                          x)
                              2
                                      x - n
                                         2

                  S   S
                                         n-1

                  FINANCE SCHOOL              805A-44A-1033-VG-22
           Squared Deviations


    X          X-  = d       (X - )2
    7          7 - 10 = -3        9
    9          9 - 10 = -1        1
   10         10 - 10 = 0         0
   10         10 - 10 = 0         0
   11         11 - 10 = +1        1
   13         13 - 10 = +3        9
 X = 60    (X - ) = 0        20
                N




             FINANCE SCHOOL              805A-44A-1033-VG-23
         Mean Squared Deviation



The average or mean squared deviation is found
by dividing the sum of the squared deviations by
the number of observations:

    (X - )2 = 20 = 3 1/3
       N        6
The mean squared deviation is also called the
variance, and is denoted by the lower case Greek
letter (sigma) squared: 2




              FINANCE SCHOOL              805A-44A-1033-VG-24
               Alternative Formulation



An alternative formulation for both the variance and
standard deviation simplifies the work involved. These
are:
                                              (X)2
          Variance: 2 = (X - )2 = X2 - N
                                N            N

                                                (X)2
          Standard Deviation = =       X2 - N
                                             N


                    FINANCE SCHOOL                805A-44A-1033-VG-25
                          Worksheet A

Worksheet A for 2 =  (X - )2
                                  N

            X          X-  = d          (X - )2
            7          7 - 10 = -3           9
            9          9 - 10 = -1           1
           10         10 - 10 = 0            0
           10         10 - 10 = 0            0
           11         11 - 10 = +1           1
           13         13 - 10 = +3           9
        X = 60      (X - ) = 0          20
                         N

 = 60 = 10        2 = 20 = 3.3333    = 3.3333 = 1.8257
    6                   6

                        FINANCE SCHOOL               805A-44A-1033-VG-26
            Worksheet B
                         (X)2
Worksheet B for 2 = X2 - N
                      N
               X                   X2
               7                   49
               9                   81
              10                  100
              10                  100
              11                  121
              13                  169
X    =       60     X2     =    620

            (60)2
2 = 620 - 6 = 620 - 600 = 20 = 3.3333
        6         6         6

= 3.3333 = 1.8257

          FINANCE SCHOOL                 805A-44A-1033-VG-27
              Coefficient of Variation


C = 100(/)

- standard deviation is a measure of absolute variability.

- coefficient of variation is a measure of relative
variability.

              Group A                Group B
           = $6.61                   $5.82
          =   .57                    .51
          C = 8.62%                    8.76%


                   FINANCE SCHOOL                     805A-44A-1033-VG-28
                                          Federalist Papers


                        0.4
                                                                                      Hamilton
                       0.35
                                                                                      Madison
                        0.3
Proportion of Papers




                                                                                      Disputed
                       0.25

                        0.2

                       0.15

                        0.1

                       0.05

                         0
                              0   1   3   5     7    9   11    13   15   17   19

                                               Rate of Usage




                                              FINANCE SCHOOL                       805A-44A-1033-VG-29
           Pure Chemical Company


Output of 10 curing VATs 19 Jan **.

           VAT         Gallons Produced
            A                65
            B                67
            C                66
            D                68
            E                66
            F                67
            G                66
            H                65
             I               64
            J                68
                 FINANCE SCHOOL           805A-44A-1033-VG-30
            Frequency Distribution
          (Pure Chemical Company)

VAT output 19 Jan **.
                            Frequency
      Output Gallons    Absolute   Relative
           64              1      1/10 = 0.1

            65             2       2.10 = 0.2

            66             3       3.10 = 0.3

            67             2       2/10 = 0.2

           68              2       2/10 = 0.2
      TOTALS              10              1.0
                 FINANCE SCHOOL           805A-44A-1033-VG-31
                                    Histogram
                             (Pure Chemical Company)

Curing VAT output 19 Jan **.
                     3                                                         0.3




                                                                                     Relative Frequency
Absolute Frequency




                     2                                                         0.2



                     1                                                         0.1



                     0                                                         0
                         0    64    65         66           67   68   0
                                         Output (Gallons)



                                   FINANCE SCHOOL                         805A-44A-1033-VG-32
                                  Frequency Polygon
                               (Pure Chemical Company)

                     Curing VAT output 19 Jan **.
                     3                                                        0.3




                                                                                    Relative Frequency
Absolute Frequency




                     2                                                        0.2


                                                                              0.1
                     1


                     0                                                        0

                          0     64     65         66           67   68    0
                                            Output (Gallons)



                                     FINANCE SCHOOL                      805A-44A-1033-VG-33
         Population Test
             N = 34

ROSTER            ROSTER
NUMBER   GRADE    NUMBER   GRADE
  1        98       18       99
  2        84       19       78
  3        86       20       99
  4       100       21       86
  5        94       22       95
  6        97       23      100
  7        81       24       93
  8        86       25       97
  9        99       26       97


          FINANCE SCHOOL      805A-44A-1033-VG-34
         Population Test
          N = 34 (Cont)

ROSTER            ROSTER
NUMBER   GRADE    NUMBER   GRADE
  10       87       27       79
  11       86       28       91
  12       90       29       88
  13       89       30       93
  14       96       31       94
  15       94       32       84
  16       98       33       89
  17       90       34       91



          FINANCE SCHOOL      805A-44A-1033-VG-35
                  Frequency Distribution
                      for Test Scores
                                              FREQUENCY          .
MIDPOINT   CLASS INTERVAL   TALLY   ABSOLUTE RELATIVE CUMULATIVE
   80         78-81                    3       .088       .088
   84         82-85                    2       .059       .147
   88         86-89                    8       .235       .382
   92         90-93                    6       .176       .558
   96         94-97                    8       .235       .793
  100         98-100                   7       .206       .999
                                       34      .999*
   * NOT EXACT DUE TO ROUNDING.




                        FINANCE SCHOOL                    805A-44A-1033-VG-36
                              Test Score Histogram



                     8
Absolute Frequency




                     6

                     4

                     2

                     0
                         78   82    86    90     94   98   102
                                         Grade



                                   FINANCE SCHOOL                805A-44A-1033-VG-37
                              Test Score Frequency Polygon

                                                                           Cumulative
                                                                           Frequency
                                                                           Distribution
                        1
Cumulative Frequency




                       0.8

                       0.6

                       0.4

                       0.2

                        0
                             76   80     84         88           92   96      100
                                              Grade (Midpoint)



                                       FINANCE SCHOOL                         805A-44A-1033-VG-38
            Symmetrical Frequency
                Distribution



Frequency


                  50%   50%

                    Mean            X
                    Median
                    Mode



               FINANCE SCHOOL           805A-44A-1033-VG-39
             Skewed Frequency
               Distributions
    MODE

           MEDIAN
F
              MEAN

                           X
    (1) SKEWED RIGHT                    MODE


                       F       MEDIAN
                               MEAN

                                                       X
                               (2) SKEWED LEFT
               FINANCE SCHOOL               805A-44A-1033-VG-40
              Descriptive Measures

                    SUMMARY
• Compute and explain the uses of the following
  measures of central tendency and dispersion:

      - Mean             - Range.
      - Median           - Variance.
      - Mode             - Standard Deviation.


• Construct and plot a frequency distribution and
  identify the skewness of the distribution.


                  FINANCE SCHOOL              805A-44A-1033-VG-41
              Terminal Learning Objective

    Action: Perform descriptive measures statistical procedures
            by computing the measures of central tendency and
            dispersion; constructing a frequency distribution
            (absolute, relative, and cumulative); plotting a
            histogram, frequency polygon, and cumulative
            frequency distribution; and identify the skewness of
            the direction.

Conditions: Given the Formulas provided in the Summary Sheet
            and a data set (population or sample).

 Standard: IAW the Descriptive Measures Programmed Text and
           Accepted Statistical Techniques



                      FINANCE SCHOOL                   805A-44A-1033-VG-42

				
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