mode.ppt

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					                       THE MODE:

        The mode is defined as that value which occurs most
frequently in a set of data i.e. it indicates the most common
result.


                        EXAMPLE:

Suppose that the marks of eight students in a particular test
                         are as follows:
     2, 7, 9, 5, 8, 9, 10, 9

Obviously, the most common mark is 9. In other words,
mode = 9.
           MODE IN CASE OF RAW DATA
      PERTAINING TO A CONTINUOUS VARIABLE
       In case of a set of values (pertaining to a continuous
variable) that have not been grouped into a frequency
distribution (i.e. in case of raw data pertaining to a
continuous variable), the mode is obtained by counting the
number of times each value occurs.
       Let us consider an example. Suppose that the
government of a country collected data regarding the
percentages of revenues spent on Research & Development
by 49 different companies, and obtained the following
figures:
               EXAMPLE
       Percentage of Revenues Spent on
         Research and Development
Company    Percentage   Company   Percentage
  1          13.5         14         9.5
  2          8.4          15         8.1
  3          10.5         16        13.5
  4          9.0          17         9.9
  5          9.2          18         6.9
  6          9.7          19         7.5
  7          6.6          20        11.1
  8          10.6         21         8.2
  9          10.1         22         8.0
  10         7.1          23         7.7
  11         8.0          24         7.4
  12         7.9          25         6.5
  13         6.8          26         9.5
   Percentage of Revenues Spent on
     Research and Development
Company   Percentage   Company   Percentage
  27         8.2         39         6.5
  28         6.9         40         7.5
  29         7.2         41         7.1
  30         8.2         42        13.2
  31         9.6         43         7.7
  32         7.2         44         5.9
  33         8.8         45         5.2
  34        11.3         46         5.6
  35         8.5         47        11.7
  36         9.4         48         6.0
  37        10.5         49         7.8
  38         6.9
                          DOT PLOT

          The horizontal axis of a dot plot contains a scale for
 the quantitative variable that we are wanting to represent.
          The numerical value of each measurement in the data
 set is located on the horizontal scale by a dot.     When data
 values repeat, the dots are placed above one another,
 forming a pile at that particular numerical location.




                                                               R&D
4.5      6        7.5       9        10.5      12       13.5
                        Dot Plot
As is obvious from the above diagram, the value 6.9 occurs 3
times whereas all the other values are occurring either once
                          or twice.
                   Hence the modal value is 6.9.




                                                         R&D
 4.5      6       7.5      9      10.5    12      13.5



          ˆ
         X= 6.9
      Also, this dot plot shows that almost all of the R&D
percentages are falling between 6% and 12%, most of the
percentages are falling between 7% and 9%.
We will be interested to note that
 mode is such a measure that can be
 computed even in case of nominal
 and ordinal levels of measurements.
            For example
The marital status of an adult can be
  classified into one of the following
  five mutually exclusive categories:
Single, married, divorced, separated
  and widowed.
Nominal scale is that where a certain
 order exists between the groupings.
For example:
Speaking of human height, an adult
 can be regarded as tall, medium or
 short.
A company has developed five
 different bath oils, and, in order to
 determine consumer-preference, the
 company conducts a market survey.
Number of Respondents favouring
       various bath-oils
                         400
                                          Bath oils
    No. of Respondents



                         300


                         200


                         100


                          0
                               I    II    III         IV   V

                                   Mode
The largest number of respondents
 favaoured bath-oil NO.II, as
 evidenced by the bar-chart.
Thus, we can say that Bath-oil No.II is
 the mode.
    THE MODE IN CASE OF A DISCRETE FREQUENCY
                  DISTRIBUTION:

        In case of a discrete frequency distribution,
identification of the mode is immediate; one simply finds that
value which has the highest frequency.
           Example:                No. of Passengers   No. of Flights
         An airline found the              X                  f
following      numbers        of          28                1
passengers in fifty flights of a          33                1
forty-seater plane.                       34                2
Highest Frequency fm = 13                 35                3
occurs against the X value 13.            36                5
                                          37                7
Hence:
                                          38                10
         Mode =       ˆ
                      X     = 39          39                13
                                          40                8
                                         Total              50
       THE MODE IN CASE OF THE FREQUENCY
     DISTRIBUTION OF A CONTINUOUS VARIABLE:

      In case of grouped data, the modal group is easily
recognizable (the one that has the highest frequency).
At what point within the modal group does the mode lie?
                       Mode:
            ˆ                   f m  f1
            X l                                   xh
                        f m  f1    f m  f 2 
where
   l = lower class boundary of the modal class,
   fm = frequency of the modal class,
   f1 = frequency of the class preceding the
        modal class,
   f2 = frequency of the class following modal
        class, and
   h = length of class interval of the modal class
EPA MILEAGE RATINGS

  Class          Class
                              Frequency
  Limit        Boundaries
30.0 – 32.9   29.95 – 32.95      2
33.0 – 35.9   32.95 – 35.95      4
36.0 – 38.9   35.95 – 38.95      14
39.0 – 41.9   38.95 – 41.95      8
42.0 – 44.9   41.95 – 44.95      2
                      Total      30
EPA MILEAGE RATINGS

      Class           Class       No. of
      Limits       Boundaries      Cars
    30.0 – 32.9   29.95 – 32.95      2
    33.0 – 35.9   32.95 – 35.95    4 = f1
    36.0 – 38.9   35.95 – 38.95   14 = fm
    39.0 – 41.9   38.95 – 41.95    8 = f2
    42.0 – 44.9   41.95 – 44.95      2
      It is evident that the third class is the modal class.
The mode lies somewhere between 35.95 and 38.95.

       In order to apply the formula for the mode, we
note that fm = 14, f1 = 4 and f2 = 8.

       Hence we obtain:
Hence, we obtained:

ˆ                     14  4
X      35.95                      3
                14  4  14  8
                  10
       35.95         3
                10  6
       35.95  1.875
       37.825
                               Number of Cars




                            0
                            2
                            4
                            6
                            8
                           10
                           12
                           14
                           16
                                                Y


                   29
                     .9
                       5

                   32
                     .9
                       5

                   35
                     .9
                       5

                   38
                     .9
                       5

Miles per gallon   41
                     .9
                       5

                   44
                     .9
                       5
                           X
                      The frequency polygon of the same distribution was:

                  Y
                 16
                 14
Number of Cars




                 12
                 10
                  8
                  6
                  4
                  2
                  0                                                            X
                           5


                                   5


                                           5


                                                   5


                                                           5


                                                                   5


                                                                           5
                         .4


                                 .4


                                         .4


                                                 .4


                                                         .4


                                                                 .4


                                                                         .4
                       28


                               31


                                       34


                                               37


                                                       40


                                                               43


                                                                       46
                                       Miles per gallon
Frequency curve was as indicated by the dotted line in the following figure:
                          Y
                     16
                     14
    Number of Cars

                     12
                     10
                      8
                      6
                      4
                      2
                      0                                                          X
                             5


                                     5


                                             5


                                                     5


                                                             5


                                                                     5


                                                                             5
                          .4


                                  .4


                                          .4


                                                  .4


                                                          .4


                                                                  .4


                                                                          .4
                      28


                                 31


                                         34


                                                 37


                                                         40


                                                                 43


                                                                         46
                                                                  Miles per gallon
In this example, the mode is 37.825, and if we locate this value on the X-axis,
we obtain the following picture:

                             Y
                        16
                        14
       Number of Cars



                        12
                        10
                         8
                         6
                         4
                         2
                         0                                                           X
                                5


                                        5


                                                5


                                                        5


                                                                5


                                                                         5


                                                                                 5
                             .4


                                     .4


                                             .4


                                                     .4


                                                             .4


                                                                      .4


                                                                              .4
                         28


                                    31


                                            34


                                                    37


                                                            40


                                                                     43


                                                                             46
                                                                      Miles per gallon
                                                            ˆ
                                                            X       = 37.825
              Example
The following table contains the ages
 of 50 managers of child-care centers
 in five cities of a developed country.
   Ages of a sample of managers
    of Urban child-care centers
   42         26        32         34         57
   30         58        37         50         30
   53         40        30         47         49
   50         40        32         31         40
   52         28        23         35         25
   30         36        32         26         50
   55         30        58         64         52
   49         33        43         46         32
   61         31        30         40         60
   74         37        29         43         54
Convert this data into Frequency Distribution and
find the modal age.
 Frequency Distribution of
 Child-Care Managers Age
Class Interval    Frequency
   20 – 29           6
   30 – 39           18
   40 – 49           11
   50 – 59           11
   60 – 69           3
   70 – 79           1
    Total            50
                   Mode:
        ˆ                f m  f1
        X l                                xh
                 f m  f1    f m  f 2 
where
 l = lower class boundary of the modal
   class,
 fm= frequency of the modal class,
 f1 = frequency of the class preceding the
   modal class,
 f2 = frequency of the class following modal
     class, and
 h = length of class interval of the modal
   class
Hence, the mode is given by

  ˆ                   18  6
  X    29.5                         10
               18  6   18  11
                 12
       29.5          10
               12  7
               120
       29.5 
                19
       29.5  6.3  35.8
ˆ
X  35.8



                                                        Mode:
                                      Y


                              20
           No. of Managers




                             15




                              10




                                  5



                                  0                                                             X
                                          19.5   29.5   39.5      49.5     59.5   69.5   79.5

                                                        Ages of Managers

                                          Mode = 35.8
                    PROPERTIES
                   OF THE MODE
•The mode is easily understood and easily ascertained in case
of a discrete frequency distribution.

•It is not affected by a few very high or low values.

The question arises, “When should we use the mode?”
The answer to this question is that the mode is a valuable
concept in certain situations such as the one described below:
                      EXAMPLE
Suppose the manager of a men’s clothing store is asked about
the average size of hats sold. He will probably think not of the
arithmetic or geometric mean size, or indeed the median size.
Instead, he will in all likelihood quote that particular size
which is sold most often. This average is of far more use to him
as a businessman than the arithmetic mean, geometric mean or
the median.
The modal size of all clothing is the size which the businessman
must stock in the greatest quantity and variety in comparison
with other sizes.
       On the other hand, sometimes a frequency distribution
contains two modes in which case it is called a bi-modal
distribution as shown below:
    THE BI-MODAL FREQUENCY
          DISTRIBUTION
f




0                            X

				
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