Measure of Central Tendency Lecture 04

 Measure of Central Tendency 

Lecture 04

A measure of central tendency is defined as “the statistical measure that identifies a single value as representative of an entire distribution (population or sample).” It aims to provide an accurate description of the entire data. It is the single value that is most representative of the collected data.

For example, suppose the weekly earnings of a daily wage worker are given below:

Day	Monday	Tuesday	Wednesday	Thursday	Friday
Earning (Rs)	800	650	700	500	450

To summarise the weekly earnings in an understandable manner, one way is to find the average of weekly earnings.

We obtain the average earnings of a daily wage worker, which is Rs. 500.

A measure of central tendency measures the general position of a set of data in the range of observation. It is also known as a measure of location or position.

The measurement of central tendency is very useful in statistics. Some of their importance is given below:

i. The measure of central tendency gives us a single representative value for the entire data.

ii. The collected and classified data are so meaningful that measures of central tendency convert the whole data into just one figure and thus help in condensation.

iii. To compare two or more sets of data, we have to find the representative values of these data sets. These representative values can be obtained with the help of measures of central tendency.

iv. Many statistical analyses are based on measures of central tendency.

Characteristics of a Good Measure of Central Tendency

i. A good measure of central tendency should be rigidly defined.

 

A measure of central tendency should be rigidly defined. If a measure of central tendency is not rigidly defined, then the estimation of a measure of central tendency is based on the observer. The bias of the observer in such a case would affect the value of the measure of central tendency.

ii.                  A good measure of central tendency should be based on all the observation of data

 

A measure of central tendency should be based on all the observations of the given series. If some of the values of the series are not taken into account in its calculation, the average cannot be said to be representative one.

 

iii. A good measure of central tendency should be capable of further mathematical treatment.

 

A good measure of central tendency should be capable of further mathematical treatment. If it is not capable of further mathematical treatment, its application is very limited.

 

iv. A good measure of central tendency should be easy to calculate and understand.

 

A good measure of central tendency should be easy to calculate and easy to understand by persons of ordinary intelligence. If the calculation of a measure of central tendency involves a tedious procedure or is too abstract, it will be difficult to understand, and its use will be confined only to a limited person.

 

v.                  A good measure of central tendency should be less affected by fluctuation of sampling

A good measure of central tendency should not be affected by the fluctuation of sampling. The data from which the measure of central tendency is calculated should be a homogenous group.

   Main Measures of Central Tendency

The main measures of central tendency are given below:

1. Arithmetic Mean

2. Geometric Mean

3. Harmonic Mean

4. Median

5. Mode

MEAN (ARITHMETIC MEAN)

An arithmetic mean is a number that represents an “average” of a set of data. It is obtained by adding all the observations of the data and dividing by the total number of observations. The arithmetic mean for population is denoted by "μ“ and for sample data, it is denoted by X¯

The arithmetic mean for grouped data is obtained as:

Example 3.1: The enrolment in a college in the last five years was 1200, 1710, 1650, 1830, and 1900. Find the average enrolment per year.

Solution: As the data is ungrouped, we use the following formula:

Example 3.2: The following table represents the marks obtained by 10 students.

Marks	45	50	65	75	90
Number of students	2	3	2	2	1

Find the mean marks of the students.

Solution: As the data is given in a frequency table, a grouped data formula is convenient to solve our problem.

Note: It means that an average mark is 61, but it does not mean the mark of each student is 61. In general, it gives a message that the average marks of students are spread around 61.

Example 3.3: Consider the following grouped frequency distribution.

Wage (Rs)	300 -490	500 - 690	700 - 890	900 - 1090	1100 - 1290
Number of workers	30	50	89	65	15

Find the mean of wages.

Solution: As the data is given in classes, we use a grouped data formula.

• Read: Properties of AM