## Statistics for Economics Class 11 Notes Chapter 5 Measures of Central Tendency

Central Tendency

A central tendency refers to a central value or a representative value of a statistical series.

According to Clark, “An average is a figure that represents the whole group”.

Types of Statistical Averages

Averages are broadly classified into two categories

- Mathematical Averages
- Positional Averages

**Arithmetic Mean**

Arithmetic Mean is the number which is obtained by adding the values of all the items of a series and dividing the total by the number of items.

Arithmetic Mean is generally written as X. It may be expressed in the form of following formula

\(\overline{X}=\frac{x_{1}+x_{2}+x_{3}+\ldots \ldots x_{N}}{N} \text { or } \frac{\Sigma \overline{X}}{N}\)

**Types of Arithmetic Mean**

- Simple Arithmetic Mean
- Weighted Arithmetic Mean

**Methods of Calculating Simple Arithmetic Mean**

(i) Individual Series In the case of individual series, Arithmetic Mean may be calculated by two methods

- Direct Method According to this method, we find the Arithmetic mean from the following formula

\(\overline{X}=\frac{\Sigma X}{N} \text { or } \overline{X}=\frac{\text { Total value of the item }}{\text { Number of items }}\) - Short-cut Method By short cut method, we find the Arithmetic Mean from the following formula

\(\overline{X}=A+\frac{\Sigma d}{N}\)

Here, \(\overline{X}\) = Arithmetic Mean, A = Assumed average of Ed = Net sum of the deviations of the different values from the assumed average; and N = Number of items in the series,

(ii) Discrete Series There are three methods of calculating mean of the discrete series

- Direct Method Direct method of estimating mean of the discrete frequency series uses the formula

\(\overline{X}=\frac{\Sigma f X}{\Sigma f}\) - Short-cut Method Short cut method of estimating mean of the discrete frequency series uses the following formula

\(\overline{X}=A+\frac{\Sigma f d}{\Sigma f}\) - Step-deviation Method This method is a variant of short-cut method. It is adopted when deviations from the assumed mean have some common factor

\(\overline{X}=A+\frac{\Sigma f d}{\Sigma f} \times c\)

(iii) Frequency Distribution

There are three methods of calculating mean in frequency distribution

(a) Direct Method Direct method of estimating mean of the discrete frequency series uses the formula

\(\overline{X}=\frac{\Sigma f m}{\Sigma f}\)

m = mid-value, mid-value = \(\frac{L_{1}+L_{2}}{2}\)

L_{1} = lower limit of the class

L_{2} = upper limit of the class

(b) Short-cut Method Short cut method of estimating mean of the frequency distribution uses the formula

\(\overline{X}=A+\frac{\Sigma f d}{\Sigma f}\)

(c) Step Deviation Method According to this method, we find the Arithmetic Mean by the following formula

\(\overline{X}=A+\frac{\Sigma f d^{\prime}}{\Sigma f} \times c\)

(d) Weighted Arithmetic Mean It is the mean of weighted items of the series. Different items are accorded different weights depending on their relative importance. The weighted sum of the items is divided by the sum of the weights.

**Calculation of Weighted Mean**

According to this way, we find weighted mean from the following information

\(\overline{X}_{W}=\frac{\Sigma W X}{\Sigma W}\)

(i) Merits

- Simplicity
- Certainty
- Based on all items
- Algebraic treatment
- Stability
- Basis of comparison
- Accuracy test

(ii) Demerits

- Effect of extreme value
- Mean value may not figure in the series at all
- Laughable conclusions
- Unsuitability
- Misleading conclusions

**Median**

“The Median is that value of the variable which divides the group into two equal parts, one part comprising all values greater than the Median value and the other part comprising all the values smaller than the Median value”.

(i) Calculation of Median

(a) Individual Series Calculation of Median in individual series involves the following formula

M = Size of \(\left(\frac{N+1}{2}\right)\)th item

When N of the series is an even number, Median is estimated using the following formula

(b) Discrete Series Calculation of Median in case of discrete series or frequency array involves the following formula

M = Size of \(\left(\frac{N+1}{2}\right)\)th item

(c) Frequency Distribution Series

The following formula is applied to determine the Median Value

**Quartiles**

If a statistical series is divided in to four equal parts, the end value of each part is called a Quartile.

(i) Calculation of Quartiles Quartile values (Q_{1} and Q_{3}) are estimated differently for different sets of series,

(a) Individual and Discrete Series

(b) Frequency Distribution Series In frequency distribution series, the class interval of Q_{1} and Q_{3} are first identified as under

**Percentiles**

Percentiles divide the series into 100 equal parts, and is generally expressed as P.

Percentiles are estimated for different types of series as under

(i) Individual and Discrete Series

(ii) Frequency Distribution Series

**Mode**

The value of the variable which occurs most frequently in a distribution is called the mode.

According to Croxton and Cowden, “ The mode may be regarded as the most typical of a series of value”.

(i) Calculation of Mode

- Individual Series There are two ways of calculating Mode in individual series
- By inspection
- By converting individual series into discrete series

- Discrete Series There are two methods for calculation of mode indiscrete frequency series
- Inspection Method
- Grouping Method

- Frequency Distribution Series The exact value of Mode can be calculated with the following formula

\(Z=L_{1}+\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}} x_{i}\)

Relative Position of Arithmetic Mean, Median and Mode Suppose we express,

Arithmetic Mean = M_{e}

Median = M_{i}

Mode = M_{o}

The relative magnitude of the three are M_{e} > M_{i} > M_{o} or M_{e} < M_{i} < M_{o} The Median is always between the Arithmetic Mean and the Mode.