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Lec-2 Freq. Distt. ( Cont.&Disct.Data).ppt

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					      CLASSIFICATION AND
         TABULATION


          Tariq Mahmood Bajwa




University of Veterinary & Animal Sciences Lahore.

                                                     1
FREQUENCY DISTRIBUTION

 A frequency distribution is a tabular arrangement of data in
which various items are arranged into classes or groups and
the number of items falling in each class is stated. The number
of observations falling in a particular class is referred to as
class frequency or simply frequency and is denoted by "f". In
frequency distribution all the values falling in a class are
assumed to be equal to the midpoint of that class.
Data presented in the form of a frequency distribution is also
called grouped data. Data which have not been arranged in a
systematic order are called raw data or ungrouped data.



                                                                  2
CLASS LIMITS
The class limits are defined as the number or the values of the
variables which are used to separate two classes. The smaller
number is called lower class limit and larger number is called upper
class limit. For discrete variables, class boundaries are the same as
the class limits. Sometimes classes are taken as 20--25, 25--30 etc
In such a case, these class limits means " 20 but less than 25", "25
but less than 30" etc
Class Boundaries
The class boundaries are the precise numbers which separate one
class from another. The main object to defined class boundaries is
to removes the difficulty, if any, in knowing the class to which a
particular value should be assigned. The class boundary is located
midway between the upper limit of a class and the lower limit of
the next higher class.
                                                                   3
CLASS MARKS OR MIDPOINTS
 The class mark or the midpoint is that value which divides a
class into two equal parts. It is obtained by dividing the sum
of lower and upper class limits or class boundaries of a class
by 2.
CLASS INTERVAL
  Class interval is the length of a class. It is obtained by
I. The difference between the upper class boundary and the
     lower class boundary. (Not the difference between class
     limits).
II. The difference between either two successive lower class
     limits or two successive upper class limits.
III. The difference between two successive midpoints.

A uniform class interval is usually denoted by "h".
                                                                 4
 CONSTRUCTION OF A FREQUENCY DISTRIBUTION

 Decide the number of classes
No hard and fast rule for deciding on the no of classes.
Statistical experience tells us that no less than 5 and no more
than 20 classes are generally used.
The number of classes is determine by the formula i.e K=1+3.3
log(n). Where K denotes the number of classes and n denotes
the total number of observations.

Determine the range of variation of the data.
The difference between the largest and smallest values in the
data is called the range of the data. i.e
            R = largest observation - smallest observation
Where R denote the range of the data.

                                                                  5
Determine the approximate size of class interval
 The size of the class interval is determine by dividing the range
of the data by the number of classes i.e h= R/K
Where h denotes the size of the class interval. In case of
fractional results the next higher whole number is usually taken
as the size of the class interval.
Decide where to locate the class limits
The lower class limit of the first class is started just below the
smallest value in the data and then add class interval to get
lower class limit of the next class, repeat this process until the
lower class limit of the last class is achieved.
Distribute the data into appropriate classes
 Take an observation and marked a vertical bar "I"(Tally) against
the class it belongs.
                                                                     6
Example
The following data is the final plant height (cm) of thirty
plants of wheat. Construct a frequency distribution

87     91      89     88      89     91      87
92     90      98     95      97     96      100
101    96      98     99      98     100     102
99     101 105        103     107    105     106
107    112


                                                              7
Step- 1: Calculate the Range
        R = Largest observation - Smallest observation
          = 112 - 87 = 25
Step- 2: Number of classes
The number of classes is determine by the formula
K = 1+3.3 log (n) = 1+3.3 log(30)= 1+3.3(1.4771)= 5.87 = 6

Step-3: Size of class interval
       The size of the class interval h= R/K
                                     h = 25/6 = 4.17 = 5




                                                             8
Step- 4: Choose the lowest value
      Minimum Value = 87, so start the class interval from 86.
Step-5: Calculate the mid point
Average of lower and upper class limits

Step- 6: Convert the class limits to class boundaries
                                    h
                      midpiont 
                                    2
Step-7: Assigned the observations to the Classes
       Starting from first observation and assigned the
observation to the classes they belong. Tally mark is
made in the tally column against this class.

                                                                 9
The following data is the final plant height (cm) of thirty
plants of wheat.


87     91      89   88      89     91     87
92     90      98   95      97     96     100
101    96      98   99      98     100    102
99     101 105      103     107    105    106
107    112

                                                              10
Class        Class           Mid-     Entries                             Tally       f         c.f.
Limits       Boundaries      Points
86------90   85.5-----90.5     88     87,89,88,89,87,90                   IIII I           6        6


91------95   90.5-----95.5     93     91,91,92,95                         IIII             4        10

96----100    95.5----100.5     98     98,97,96,100,96,98,99,98,           IIII IIII       10        20
                                      100,99
101--105     100.5--105.5     103     101,102,101,105,103,105             IIII I          6         26

106--110     105.5--110.5     108     107,106,107                         III             3         29

111--115     110.5--115.5     113     112                                 I               1         30

                                                                  Total                   30



         Frequency distribution of the height of plants.
                                                                                               11
    FREQUENCY            DISTRIBUTION    TABLE

Class       Class          Mid Tally       Freq C.F
Limits      Boundaries     Points          uency
                                            (f)
 86---90     85.5---90.5    88    /////      6    6
 91---95     90.5---95.5    93    ////       4   10
96---100    95.5---100.5    98    //////// 10    20
101---105   100.5--105.5    103   /////      6   26
106---110   105.5– 110.5    108   ///        3   29
111---115   110.5--115.5    113   /          1   30
                                            30
                                                  12
                       Example
     Suppose we walk in the nursery class of a
  school and we count the no. of Books and copies
  that students have in their bags.

Suppose the no. of books and copies are
3,5,4,4,5,5,5,6,5,5,8,85,5,5,6,6,6,6,6,6,6,6,6,6,6,7,,7,7,7,
   7,7,8,,8,8, 7, 9,9,97,7,7,9,9,9, and so on.
Representation of Data in a Discrete
      Frequency Distribution
     X          Tally        Frequency
     3             |              1
     4            |||             3
     5         |||| ||||          9
     6       |||| |||| |||       13
     7         |||| ||||         10
     8            |||             3
     9           |||| |           6
   Total                         45
Relative Frequency Distribution
  X     Frequency    Relative/ %age
                      Frequency
  3         1        1/45 x 100 = 2.22%


  4         3        3/45 x 100 = 6.67%

  5         9        9/45 x 100 = 20%

  6        13       13/45 x 100 = 28.89%

  7        10       10/45 x 100 = 22.22%

  8         3        3/45 x 100 = 6.67%

  9         6       6/45 x 100 = 13.33%

Total      45
Cumulative Frequency Distribution
   X      Frequency   Cumulative
                      Frequency
   3          1            1

   4          3         1+3 = 4

   5          9         4+9 = 13

   6         13        13+13 = 26

   7         10        26+10 = 36

   8          3        36+3 = 39

   9          6        39+6 = 45

  Total      45
   Frequency
    The number of values falling in a particular category

   Cumulative frequency
    Sum of the observed frequency plus all above class frequencies

   Notations
    X,Y,Z, n, N,∑ (Summation)




                                                                     17

				
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