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Frequency Distribution

Frequency distribution is the tabular representation of the statistical data, usually in an ascending order, relating to measurable characteristics.

Frequency distribution is classified into two types:

  1. Simple or ungrouped frequency distribution
  2. Grouped frequency distribution

Grouped frequency distribution is further classified as continuous grouped frequency distribution and discontinuous grouped frequency distribution.

  • If the distribution of frequency is with respect to discrete variable (less data), it is known as ungrouped or simple frequency distribution.
     

     

    Example: Frequency distribution of couples based on the number of children they have:

     
    Number of Children (variable) Number of Couples (frequency)
    0 5
    1 10
    2 12
    3 20
    4 6
    5 2
    6 1
    Total 56
     

     

  • If the distribution of frequency is with respect to continuous variable, it is known as continuous grouped frequency distribution.

     

    A continuous frequency distribution is shown below:

     
    Age in Years (continuous variables) Number of Persons
    0 – 10 20
    10 – 20 39
    20 – 30 42
    30 – 40 34
    40 – 50 12
    50 – 60 3
    Total 150
  • If the distribution of frequency is with respect to discrete variable (more data), it is known as discontinuous grouped frequency distribution.
Frequency distribution of couples based on the number of children they have:
 
Number of Children (variable)
Number of Couples (frequency)
0–1
250
2–3
700
4–5
100
6–7
20
Total
1070

Some Important Terms in a Frequency Distribution

Class Interval
It is a range of values of a variable taken as an interval for dividing the total values of the variable in different classes while tabulating the frequency distribution of a sample.

 

In the above example, 0 – 10, 10 – 20, 20 – 30, etc. are all class intervals.

Class Limits

The two end values of class intervals are called class limits. In the above example, 0, 10, 20, 30, etc., are the class limits.

 

The smaller of the two end values is called as the lower class limit (LCL).

 

And the larger one is called as the upper class limit (UCL).

 

In the above example 0, 10, 20 are the lower class limits for the first, second and third class intervals and 10, 20, 30 are the upper class limits of the same class intervals.

Class Mark or Mid-value

It is the mid-value of the class.

 

It is given by, Description: 61292.png

 

Consider the above example, mid-value of the first class is Description: 61316.png, mid-value of the second class is Description: 61328.png etc.

Class Boundaries

It may be defined as the actual class limit of a class interval. For overlapping or mutually exclusive classification, the class boundaries coincide with the class limits.

 

For non-overlapping or mutually inclusive classification, which is usually applicable for a discrete variable, we have

 

Description: 61376.png 

 

Where ‘D’ is the difference between the UCL of the next class interval and LCL of the given class interval.

 

Example: Consider the below frequency distribution
 

Weight (kg) Number of Students
40 – 44 3
45 – 49 9
50 – 54 9
55 – 59 11
60 – 64 8
Total 40

In the given example, lower class boundary for the first class is Description: 61386.png
Similarly, the upper class boundary for the first class is Description: 61394.png

Class Width

Class size/ Class length

 

It is defined as the difference between upper class boundary and lower class boundary

 

Class width = UCB – LCB

 

In the above example, the class width is 44.5 – 39.5 = 5

Inclusive and Exclusive Class Intervals

If a class interval is such that the lower as well as the upper class interval are included in the same class interval, it is known as inclusive class interval. If a class interval is such that the lower class limit is included in the same class interval, whereas, the upper class limit is included in the succeeding class interval, it is known as exclusive class interval.
 

In the above example, the class intervals are 40 – 44, 45 – 49, 50 – 54, etc. are inclusive class intervals. If the class intervals are 40 – 44, 44 – 48, 48 – 52, etc. then, they are exclusive.

 

Note: Sometimes in a frequency distribution, the class intervals at the extremities may not have one of the limits. Such class intervals are called as open-ended class intervals.

Uni-variate and Multi-variate frequency distribution

Frequency distribution of a single variable is called uni-variate frequency distribution. Frequency distribution of more than one variable is called multi-variate frequency distribution.

 

Example: Frequency distribution of height of a set of students is uni-variate frequency distribution. Frequency distribution of their height as well as weight is a bi-variate frequency distribution.

Cumulative Frequency

In classification of statistical data, it is some time necessary to find the number of observations less than or more than the given value which is done by accumulating the frequencies up to or above the given value. This accumulated frequency is called cumulative frequency for the given data.

  • The number of observations less than the given value is called “Less than cumulative frequency.”
  • The number of observations exceeding the given value is called “More than cumulative frequency.”

     

    Example: 

Relative Frequency of a class

It may be defined as the ratio of the class frequency to the total frequency.

 

It is given by, Description: 61565.png

 

When the relative frequency is expressed as a percentage, it is known as percentage frequency.

Frequency density of class interval

It may be defined as the ratio of the frequency of that class interval to the corresponding class length. It is given by Description: 61597.png

Graphical Representation of a Frequency Distribution

  1. Histogram or area diagrams
    This is a very convenient graphical method to represent a frequency distribution. Histogram helps us to get an idea about frequency curve of the variable under study. A comparison among the frequencies for different class intervals is possible in case of histogram.
     
    In order to draw a histogram, the class limits are first converted to the corresponding class boundaries.
     
    Then a series of adjacent rectangles are drawn one against each class interval, with the class interval as base and the frequency or frequency density, as heights is used.
    Mode of the frequency distribution can be obtained using histograms.

     

    Example: For the following frequency of height of students, draw the histogram.


     

    Description: 62780.png

  2. Frequency polygon
    Frequency polygon is meant for a ungrouped frequency distribution.
     
    We also apply it for a grouped frequency distribution provided that the width of the class intervals remains the same.
     
     
    Frequency curve can be regarded as a limiting form of frequency polygon.
     
    In order to draw a frequency polygon, we plot (xi fi ) for i = 1, 2, 3,..... n with xi denoting the midpoint of the class interval and fi , the corresponding frequency, n being the number of class intervals.
     
     
    The plotted points are joined successively by line segments and the figure, so drawn, is given the shape of a polygon, a closed figure, by joining the two extreme ends of the drawn figure to two additional points (x0, 0) and (xn + 1, 0).
     
    Example
    For the following frequency distribution, draw a frequency polygon.
     
    Solution
    Income is taken along the x-axis. Since the class intervals are of equal width, the class frequencies are plotted against respective class mid-values. These points are joined by straight lines.
     
    Here, the extremities of the graph are joined with the mid-values of the class, preceding the first class and the class following the last class.
     
    Description: 62793.png
     
  3. Ogives or cumulative frequency graph
     
    By plotting cumulative frequency against the respective class boundary, we get ogives.
     
    There are two types of ogives:
  • Less than cumulative frequency curve or less than ogive obtained by taking less than cumulative frequency on the vertical axis
  • More than cumulative frequency curve or more than ogive obtained by taking more than cumulative frequency on the vertical axis
     
    Median of the frequency distribution can be obtained by drawing a perpendicular from the point of intersection of the less than and more than frequency curves.
     
    Example
    The following is the distribution of I.Q. of 75 children. Draw the ogives to the data and find the median.
    Solution
    The less-than ogive is obtained by plotting less-than cumulative frequencies against the upper class limits. The more-than ogive is obtained by plotting more-than cumulative frequencies against the lower class limits.
     
     
    Description: 62803.png 
  1. Frequency curve
     
    It is a smooth curve in which the total area is taken to be unity. The frequency curve for a distribution can be obtained by drawing a smooth and a free hand curve through the mid-point of the upper side of the rectangles forming the histogram. It is drawn free hand. The following are the types of frequency curve.
  • Bell-shaped curve: On a bell-shaped curve, the frequency, starting from a rather low value, gradually reaches the maximum value, somewhere near the central part and then gradually decreases to reach its lowest value at the other extremity.
  • U-shaped curve: For a U-shaped curve, the frequency is minimum near the central part and the frequency slowly, but steadily reaches its maximum at two extremities.
  • J-shaped curve: J-shaped curve starts with a minimum frequency and then gradually reaches maximum frequency at the other extremity.
  • Mixed curve: We may have a combination of these frequency curves, known as mixed curve. Multi modal curve is mixed curve.

Example
A sample of 100 apples from an orchard, on weighing, gave the following result.
 

 
Draw the frequency curve for the distribution.
Solution
Description: 62813.png 





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