What is meant by Central Tendency? A measure of central tendency gives information about the central position of data. This information is usually very useful in understanding a set of data and is used widely in a number of problems. It has many drawbacks as well, although it is an effective tool of Statistics. For instance, there may be a number of sets of data having the same value of central tendency although the series is quite different from one another. Let us have a look at the following data sets and their arithmetic means:

(i) 50,50,50,50,50 X = 50

(ii) 40,45,50,55,60 X = 50

(iii) 30,40,50,60,70 X = 50

In all the above data sets, the value of arithmetic mean is 50 whereas each data set has different nature of individual values. It means that central tendency is only describing the location of each set of data and no other information is available other than the mean. Therefore it becomes necessary in such situations to see the manner in which the individual values in a set of data are spread away from the central location. The manner in which individual values are dispersed from the average is called dispersion and the measures which enable us to calculate the amount of dispersion, are called measures of dispersion.

**MEASURES OF CENTRAL TENDENCY**

A single value that describes a set of data by identifying the central position within that set of data is called the measure of central tendency or measure of central location. There are various measures which are used to find the central position of data. These measures are given as follows:

(1)Arithmetic Mean (2) Geometric Mean

(3) Harmonic Mean (4) Median (5) Mode

We will explain these measures one by one.

**ARITHMETIC MEAN**

Arithmetic mean is a value obtained by dividing the sum of observations by the number of observations. Suppose we have __n__ observations such as, so the Arithmetic Mean denoted by the symbol is X:

**Weighted Arithmetic Mean:**

The simple arithmetic mean is calculated where individual observations of the data same importance, but in real life, we observe that different commodities have different as these commodities are not purchased in the same quantity for e.g. we purchase more quantity of flour from the market as compared to salt. It means that we attach more importance to flour than Salt.

**weighted arithmetic mean**

Example7: A student obtained 40, 50, 80 and 70 marks in the subjects of Economics, Statistics Mathematics, and Urdu respectively. The weights assigned to these subjects were 2, 2, 2 and l’ respectively. Calculate the weighted arithmetic mean for marks in these subjects. Solution:

Subject x w

Economics 40 2 80

Statistics 80 2 160

Mathematics 60 2 120

Urdu 70 1 70

7 430

X= 61.43

**PROPERTIES OF ARITHMETIC MEAN**

The arithmetic mean has the following properties:

1) Sum of the deviations taken from arithmetic mean is always zero i.e.

2) Sum of the squared deviations taken from the arithmetic mean is minimum than a sum of the squared deviations taken from any other observation i.e. XI, R.

3) If we have k subgroups having means Xk with observations in each subgroup respectively, the combined mean for all n observations denoted by the symbol Xc is given by the formula:

**MERITS OF ARITHMETIC MEAN:**

1) It is easy to calculate.

2) It is easy to understand.

3) It is based on all observations of data.

4) It is capable of further algebraic treatment.

5) It is a stable average as it is not much affected by changes in the sample.

**DEMERITS OFARITHMETIC MEAN:**

l) It is affected by extreme observations.

2) It is not a suitable average when open end classes are present in the data.

3) The answer an f arithmetic mean is sometimes unrealistic .e.g. 7.3 men or 9.5 eggs etc.

**GEOMETRIC MEAN:**

The geometric mean of n positive observations is the nth root of their product. Suppose we have n observations such as X1′ X1……. Xn, the geometric mean of these n observations is given by.

G M. = Antilog 1/n logx

**MERITS OF GEOMETRIC MEAN**

1) It is based on all observations of data.

2) It is not affected by extreme observations.

3) It is capable of further algebraic treatment.

4) It is a stable average as it is not much affected by changes in the sample.

5) Geometric mean is a suitable average when data consist of percentages.

**DEMERITS OF GEOMETRIC MEAN**

1) It is not easy to calculate.

2) It is not easy to understand.

3) It is not a suitable average when open end classes are present in the data.

4) It cannot be determined if any observation is zero in the data.

**HARMONIC MEAN:**

Harmonic mean is the reciprocal of the arithmetic mean of reciprocal of the observations. Suppose we have n observations such X1′ X1……. Xn, the harmonic mean of these n observations is given by:

H M. = n/ Sum 1/x.

**MERITS OF HARMONIC MEAN**

1) It is based on all observations of data.

2) It is not affected by extreme large observations.

3) It is capable of further algebraic treatment.

4) It is a stable average as it is not much affected by changes in the sample.

**DEMERITS OF HARMONIC MEAN: **1) It is not easy to calculate.

2) It is not easy to understand.

3) It is not a suitable average when open end classes are present in the data.

4) It cannot be determined if any observation is zero in the data.

Relation Between A.M., G.M., and H.M.

**MEDIAN**

Median is the middle value of arranging a set of data in ascending or descending order. When data contains odd number of values, a median is exactly the middle value. But when data contains even number of value, the median is the mean of two middle values.

X=n/2 th

**MERITS OF MEDIAN:**

1) It is easy to understand.

2) It is not affected by extreme observations.

3) It is a suitable average when open-end classes are present in the data.

4) It can also be determined from a diagram.

5) It is possible to find median when data are of qualitative nature.

**DEMERITS OF MEDIAN:**

1) It is not based on all observations of data.

2) It is not easy to calculate. Arranging the observations in ascending or descending order is a very difficult task.

3) It is not a stable average as it is affected by changes in the sample.

4) It is not capable of further algebraic treatment.

**QUARTILES, QUINTILES, DECILES AND PERCENTILES**

** **Quartiles: Quartiles consist of a set of three values QI.Q2 and Qa that divide the arranged set of data into four equal parts. The data are arranged in ascending order only.

Quintiles: Quintiles consist of a set of four values QI, Q2, and Qa that divide the arranged set of data into five equal parts. The data are arranged in ascending order only.

Deciles: Deciles consist of a set of nine values DI, D2, . … .D’ that divide the arranged set of data into ten equal parts. The data are arranged in ascending order only.

Percentiles: Percentiles consist of a set of ninety-nine values PI, P-, and P99 that divide the arranged set of data into hundred equal parts. The data are arranged in ascending order only.

**MODE:**

The most repeated observation in a set of data is called mode. The symbol used for mode is X.