The central tendency of a variable means a typical value around which other values tend to concentrate; hence this value representing the central tendency of the series is called measures of central tendency or average.
According to Clark, “Average is an attempt to find one single figure to describe whole of figures.”
Arithmetic Mean (X) : The most popular and widely used measure of representing the entire data by one value is known as arithmetic mean. Its value is obtained by adding together all the items and by dividing this total by the number of items.
Arithmetic mean may be of two types :
- Simple Arithmetic mean
- Weighted arithmetic mean
Calculation of Arithmetic Mean :
Median (M) : Median is that value of the variable which divides the group into two equal parts, one part comprising all values greater than, and the other all values less than the median Calculation of Median :
a) Individual Series :
Arrange the variables in ascending or descending order.
b) Discrete Series :
- Arrange the variables in ascending or descending order.
- Calculate cumulative frequencies.
c) Continuous Series :
- Arrange the variables in ascending or descending order.
- Calculate cumulative frequencies.
- Determine median class by using (N/2).
- Apply formula –
M = l1 + i (m – c)/1
l1 → lower limit of median group
i → class interval
m→ N/2 cumulative frequency preceding the median group.
f → frequency of median group
Mode (Z) : Mode is the value that appears most frequently in a series i.e. it is the value of the item around which frequencies are most densely concentrated.
Calculation of Mode :
a) Individual Series :
(i) By inspection – value repeated most.
(ii) By converting individual series into discrete series.
(iii) By empirical relationship between the averages -
Z = 3M – 2X
b) Discrete Series :
(i) By inspection – value having highest frequency.
(ii) By grouping.
(iii) By empirical relationship.
c) Continuous Series :
(i) First calculate model class by inspection or by grouping.
(ii) Then apply the following formula -
