MEASURES OF CENTRAL TENDANCY

The main measures of location are the:
1. mean
2.median
3. mode.
Let us first consider, in general terms, why we need these measures.
(a) Descriptive Use
The main purpose of a statistical analysis is to review unwieldy sets of data so that they may be understood and used in planning economic and business policies. A measure of location describes one feature of a set of data by a single number. You have to discriminate between the various "centres" as each has its advantages and disadvantages. You must inspect any set of data carefully and choose the "centre" which is best for the problem you have to solve.

(b)Comparison of Distributions
Suppose you wish to compare the distribution of the weights of men and women in a given population. The data has been summarised to give two frequency distributions and these distributions give the frequency curves . The
curves are the same shape and are symmetrical, so by symmetry, without any
calculations, you can read off a value of x from each distribution which you could call the "centre". Since every other visible feature of the curves is the same, these two values of x describe their difference.

MEANS
Arithmetic MeanThe arithmetic mean of a set of observations is the total sum of the observations divided by the number of observations. This is the most commonly used measure of location and it is often simply referred to as "the mean".
Advantages and Disadvantages of the Arithmetic Mean Advantages

1. It is a well known statistic and it is easily manipulated to calculate other useful         statistical measures.

2. It is easy to calculate as the only information you need is the sum of all the observations and the number of observations.

3.It uses the values of all the observations.

Disadvantages

(i) A few extreme values can cause distortion which makes it unrepresentative of the   data set.

(ii) When the data is discrete it may produce a value which appears to be unrealistic,

(iii) It cannot be read from a graph.

Geometric Mean

The geometric mean is seldom used outside of specialist applications. It is appropriate when dealing with aset of data such as that which shows exponential growth (that is where the rate of growth depends on the value of the variable itself), for example population levels arising from indigenous birth rates, or that which follows a geometric progression, such as changes in an index number over time, for example the Retail Price Index. It is sometimes quite difficult to decide where the use of the geometric mean over the arithmetic mean is the best choice. We will return to the use of geometric means in the next unit. The geometric mean is evaluated by taking the nth root of the product of all 'n' observations.

 

Harmonic Mean

Another measure of central tendency which is only occasionally used is the harmonic mean. It is most frequently employed for averaging speeds where the distances for each section of the journey are equal.

MEDIAN
Definition

If a set of n observations is arranged in order of size then, if n is odd, the median is the value of the middle observation; if n is even, the median is the value of the arithmetic mean of the two middle observations. Note that the same value is obtained whether the set is arranged in ascending or descending order of size, though the ascending order is most commonly used. This arrangement in order of size is often called ranking.

Advantages and Disadvantages of the Median(a)Advantages

1.  All the observations are used to order the data even though only the middle one        or two observations are used in the calculation.

2.Its value is not distorted by extreme values, open-ended classes or classes of irregular width.

3. It can be illustrated graphically in a very simple way.

Disadvantages

1.In a grouped frequency distribution the value of the median within the median class can only be an estimate, whether it is calculated or read from a graph. Although the median is easy to calculate it is difficult to manipulate arithmetically.

2. It is of little use in calculating other statistical measures.



MODE
Definition

If the variable is discrete, the mode is that value of the variable which occurs most frequently.This value can be found by ordering the observations or inspecting the simple frequency distribution or its histogram. If the variable is continuous, the mode is located in the class interval with the largest frequency, and its value must be estimated. As it is possible for several values of the variable or several class intervals to have the same frequency, a set of data may have several modes.

1. A set of observations with one mode is called unimodal.
2. A set of observations with two modes is called bimodal.
3. A set of observations with more than two modes is called multimodal.
Advantages and Disadvantages of the Mode

Advantages
1. It is not distorted by extreme values of the observations.
2.It is easy to calculate.
Disadvantages1. It cannot be used to calculate any further statistic.

2.It may have more than one value (although this feature helps to show the shape of the distribution).