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CLASSIFICATION AND CLASSIFICATION AND TABULATIONTABULATION
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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.
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CLASS LIMITSThe 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 BoundariesThe 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.
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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 byI. 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".
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CONSTRUCTION OF A FREQUENCY DISTRIBUTION
Decide the number of classes 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.
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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/KWhere 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 limitsThe 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.
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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
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Step- 1: Calculate the Range
R = Largest observation - Smallest observation
= 112 - 87 = 25Step- 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
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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: 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.
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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
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Class Class LimitsLimits
ClassClass
BoundariBoundarieses
Mid-Mid-PointPointss
EntriesEntries TallyTally ff c.f.c.f.
86------86------9090
85.5-----85.5-----90.590.5
8888 87,89,88,89,87,90 87,89,88,89,87,90
IIII IIIII I 66 66
91------91------9595
90.5-----90.5-----95.595.5
9393 91,91,92,9591,91,92,95 IIIIIIII 44 1010
96----96----100100
95.5----95.5----100.5100.5
9898 98,97,96,100,96,98,99,98,97,96,100,96,98,99,98, 100,9998, 100,99
IIII IIII IIIIIIII
1010 2020
101--101--105105
100.5--100.5--105.5105.5
103103 101,102,101,105,103,1101,102,101,105,103,10505
IIII IIIII I 66 2626
106--106--110110
105.5--105.5--110.5110.5
108108 107,106,107107,106,107 IIIIII 33 2929
111--111--115115
110.5--110.5--115.5115.5
113113 112112 II 11 3030
TotalTotal
3030Frequency distribution of the height of plants.
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Class Class LimitsLimits
Class Class BoundariBoundarieses
Mid Mid PoinPointsts
TallyTally FreFrequequency ncy
(f) (f)
C.FC.F
86---9086---90
91---9591---95
96---96---100100
101---101---105105
106---106---110110
111---111---115115
85.5---85.5---90.590.5
90.5---90.5---95.595.5
95.5---95.5---100.5100.5
100.5--100.5--105.5105.5
105.5– 105.5– 110.5110.5
110.5--110.5--115.5115.5
8888
9393
9898
103103
108108
113113
//////////
////////
////////////////
//////////
//////
//
66
44
1010
66
33
11
66
1010
2020
2626
2929
3030
FREQUENCY DISTRIBUTION TABLE
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Example Example Suppose we walk in the nursery Suppose we walk in the nursery
class of a school and we count the no. class of a school and we count the no. of Books and copies that 45 students of Books and copies that 45 students have in their bags. have in their bags.
Suppose the no. of books and copies are Suppose the no. of books and copies are
9,9,6,3,5,4,7,6,7,5,6,5,5,8,7,5,5,6,6,6,9,69,9,6,3,5,4,7,6,7,5,6,5,5,8,7,5,5,6,6,6,9,6,7,6,6,4,5,5,6, 6,6,6, 7, 5,8, 7, 9, 9,4,7, ,7,6,6,4,5,5,6, 6,6,6, 7, 5,8, 7, 9, 9,4,7, 8,7,7,9,. 8,7,7,9,.
Representation of Data in a Representation of Data in a Discrete Frequency Discrete Frequency
DistributionDistributionXX TallyTally FrequencyFrequency
33 || 11
44 |||||| 33
55 |||| |||||||| |||| 99
66 |||| |||| ||||||| |||| ||| 1313
77 |||| |||||||| |||| 1010
88 |||||| 33
99 |||| ||||| | 66
TotalTotal 4545
Relative Frequency Relative Frequency
DistributionDistribution XX FrequencyFrequency Relative/ Relative/ %age%age
FrequencyFrequency
33 11 1/45 x 100 = 2.22%1/45 x 100 = 2.22%
44 33 3/45 x 100 = 6.67%3/45 x 100 = 6.67%
55 99 9/45 x 100 = 20%9/45 x 100 = 20%
66 1313 13/45 x 100 = 28.89%13/45 x 100 = 28.89%
77 1010 10/45 x 100 = 22.22%10/45 x 100 = 22.22%
88 33 3/45 x 100 = 6.67%3/45 x 100 = 6.67%
99 66 6/45 x 100 = 13.33%6/45 x 100 = 13.33%
TotalTotal 4545
Cumulative Frequency Cumulative Frequency DistributionDistribution
XX FrequencyFrequency Cumulative Cumulative FrequencyFrequency
33 11 11
44 33 1+3 = 41+3 = 4
55 99 4+9 = 134+9 = 13
66 1313 13+13 = 2613+13 = 26
77 1010 26+10 = 3626+10 = 36
88 33 36+3 = 3936+3 = 39
99 66 39+6 = 4539+6 = 45
TotalTotal 4545
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FrequencyFrequencyThe number of values falling in a particular categoryThe number of values falling in a particular category
Cumulative frequencyCumulative frequencySum of the observed frequency plus all above class frequenciesSum of the observed frequency plus all above class frequencies
NotationsNotations X,Y,Z, n, N,X,Y,Z, n, N,∑ (Summation)∑ (Summation)
