TOPIC TWO: MEASURES OF
CENTRAL TENDENCY
2.1 Objectives
By the end of the topic, the learner should be able to:
- Define measure of central tendency and state the objectives of
averaging.
- Calculate arithmetic mean using different methods.
- Compute combined mean for two or more data sets.
- Calculate Weighted Average for a given data set.
2.2 Introduction
Even after the data have been classified and tabulated one often
finds too much details for many uses that may be made of the information
available. We, therefore, frequently need further analysis of the
tabulated data. One of the powerful tools of analysis is to calculate a
single average value that represents the entire mass of data. An
“average” is a single value which is considered as the most
representative or typical value for a given set of data. Such a value is
neither the smallest nor the largest value, but is a number whose value
is somewhere in the middle of the group. For this reason an average is
frequently referred to as a measure of central tendency
or central value.
Definition: A measure of central tendency refers to
measurement of values around which data is scattered.
2.3 Objectives of
Averaging
There are two main objectives of study of averages:
- To get one single value that describes the characteristics
of the entire data.
- Measures of central value, by condensing the mass of data in one
single value, enables us to get an idea of the entire data.
- To facilitate comparison.
- Measures of central value, by reducing the mass of data in one
single value, enables comparisons to be made. Comparison can be made
either at a point of time or over a period of time.
2.4 Characteristics
of a Good Average
Since an average is a single value representing a group of values, it
is desirable that such a value satisfies the following properties:
- It should be easy to understand.
- Since statistical methods are designed to simplify complexity, it is
desirable that an average be such that it can be readily understood;
otherwise its use is bound to be very limited.
- It should be simple to compute.
- It should be simple to compute so that it can be used widely;
however, simplicity should not be sought at the expense of other
advantages.
- It should be based on all observations.
- The average should depend upon each and every observation so that if
any of the observations is dropped, the average itself is altered.
- It should be rigidly defined.
- An average should be properly defined so that it has one and only
one interpretation.
- It should be capable of further algebraic or statistical
treatment/analysis.
- We should prefer to have an average that could be used for further
statistical computations.
- It should have sampling stability.
- We should prefer to get a value which has what statisticians call
“sampling stability” — it should be least affected by the fluctuations
of sampling.
- It should not be affected by the presence of extreme
values.
- Although each and every observation should influence the value of
the average, none of the observations should influence it unduly.
In this course we will look at the following important measures of
central tendency which are generally used in various fields
e.g. business, education, etc:
- Arithmetic mean
- Median
- Mode
- Geometric mean
- Harmonic mean
2.5 Arithmetic
Mean
The most popular and widely used measure for representing the entire
data by one value is what most laymen call an “average” and what
statisticians call the arithmetic mean. Its value is
obtained by adding together all the observations and by dividing this
total by the number of observations.