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:
 Simple or ungrouped frequency distribution
 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.
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 Midvalue
It is the midvalue of the class.
It is given by,
Consider the above example, midvalue of the first class is , midvalue of the second class is 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 nonoverlapping or mutually inclusive classification, which is usually applicable for a discrete variable, we have
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
Similarly, the upper class boundary for the first class is
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 openended class intervals.
Univariate and Multivariate frequency distribution
Frequency distribution of a single variable is called univariate frequency distribution. Frequency distribution of more than one variable is called multivariate frequency distribution.
Example: Frequency distribution of height of a set of students is univariate frequency distribution. Frequency distribution of their height as well as weight is a bivariate 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,
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 byGraphical Representation of a Frequency Distribution
 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.
Mode of the frequency distribution can be obtained using histograms.Example: For the following frequency of height of students, draw the histogram.
 Frequency polygon
Frequency polygon is meant for a ungrouped frequency distribution.ExampleFor the following frequency distribution, draw a frequency polygon.SolutionIncome is taken along the xaxis. Since the class intervals are of equal width, the class frequencies are plotted against respective class midvalues. These points are joined by straight lines.  Ogives or cumulative frequency graph
 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
ExampleThe following is the distribution of I.Q. of 75 children. Draw the ogives to the data and find the median.
SolutionThe lessthan ogive is obtained by plotting lessthan cumulative frequencies against the upper class limits. The morethan ogive is obtained by plotting morethan cumulative frequencies against the lower class limits.
 Frequency curve
 Bellshaped curve: On a bellshaped 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.
 Ushaped curve: For a Ushaped curve, the frequency is minimum near the central part and the frequency slowly, but steadily reaches its maximum at two extremities.
 Jshaped curve: Jshaped 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.