Quick Revision Notes & Formula Sheet

Class: 11th Economics
Chapter: Measures of Central Tendency
Board: MP Board

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1. Introduction

A Measure of Central Tendency (or statistical average) is a single value that represents the entire data set and tends to lie somewhere in the center of the distribution.

Objectives of Central Tendency:

• To get a single value that describes the characteristic of the whole data.
• To facilitate comparison between different sets of data.
• To help in further statistical analysis.

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2. Essentials of an Ideal Average

1. Clear and stable definition: It should be rigidly defined.
2. Easy to understand and calculate: Simplicity in computation.
3. Based on all observations: It should use every item in the series.
4. Sampling stability: It should not be unduly affected by fluctuations of sampling.
5. Capable of further algebraic treatment.

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3. Arithmetic Mean ($\bar{X}$)

Arithmetic mean is the most popular and widely used average. It is the sum of all observations divided by the total number of observations.

Formulas:

A. Individual Series (व्यक्तिगत श्रेणी)

• Direct Method: $$\bar{X} = \frac{\sum X}{N}$$
• Assumed Mean Method: $$\bar{X} = A + \frac{\sum d}{N}$$ (where $d = X - A$, $A$ = Assumed Mean, $N$ = Total number of observations)

B. Discrete Series (खंडित श्रेणी)

• Direct Method: $$\bar{X} = \frac{\sum fX}{\sum f}$$
• Assumed Mean Method: $$\bar{X} = A + \frac{\sum fd}{\sum f}$$ (where $d = X - A$, $f$ = Frequency)
• Step-Deviation Method: $$\bar{X} = A + \left( \frac{\sum fd'}{\sum f} \right) \times c$$ (where $d' = \frac{X - A}{c}$, $c$ = Common factor / Class interval width)

C. Continuous Series (सतत श्रेणी)

• Direct Method: $$\bar{X} = \frac{\sum fm}{\sum f}$$ (where $m$ = Mid-value of the class interval, $m = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$)
• Step-Deviation Method: $$\bar{X} = A + \left( \frac{\sum fd'}{\sum f} \right) \times c$$ (where $d' = \frac{m - A}{c}$)

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4. Median ($M$)

Median is the positional average that divides the arranged (ascending/descending) series into two equal parts. It is the value of the middle item.

Formulas:

• Location of Median (Size of item): $$\text{Size of } \left( \frac{N+1}{2} \right)^{\text{th}} \text{ item}$$

A. Individual & Discrete Series:

1. Arrange data in ascending/descending order.
2. Find cumulative frequencies ($cf$) in case of discrete series.
3. Apply the formula: $M = \text{Size of } \left( \frac{N+1}{2} \right)^{\text{th}} \text{ item}$.

B. Continuous Series:

1. Find $N/2$ (Note: Here we use $N/2$, not $(N+1)/2$).
2. Identify the Median Class where $N/2$ lies in the cumulative frequency ($cf$) column.
3. Apply the Interpolation Formula: $$M = L_1 + \frac{\frac{N}{2} - cf}{f} \times h$$
   • Where:
     • $L_1$ = Lower limit of the median class
     • $cf$ = Cumulative frequency of the class preceding the median class
     • $f$ = Frequency of the median class
     • $h$ or $i$ = Width/size of the median class interval

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5. Mode ($Z$)

Mode is the value that occurs most frequently in the data series (the value with the highest frequency).

Formulas:

• Inspection Method: By simply looking at the series to find the value with the maximum frequency.

A. Continuous Series:

1. Identify the Modal Class (class with the maximum frequency).
2. Apply the Formula: $$Z = L_1 + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$
   • Where:
     • $L_1$ = Lower limit of the modal class
     • $f_1$ = Frequency of the modal class
     • $f_0$ = Frequency of the class preceding the modal class
     • $f_2$ = Frequency of the class succeeding the modal class
     • $h$ = Width of the modal class interval

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6. Relationship between Mean, Median, and Mode

For a moderately asymmetrical (skewed) distribution, the empirical relationship is given by Karl Pearson's formula:

$$\text{Mode} = 3(\text{Median}) - 2(\text{Mean})$$ $$\text{or } Z = 3M - 2\bar{X}$$

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7. Quick Comparison of Averages

Feature	Arithmetic Mean ($\bar{X}$)	Median ($M$)	Mode ($Z$)
Definition	Mathematical average	Positional middle value	Most frequent value
Affected by Extreme Values?	Yes, heavily affected	No, unaffected	No
Graphical Location	Cannot be located graphically	Can be located using Ogives	Can be located using Histogram
Best suited for	Algebraic calculations, symmetrical data	Qualitative data, open-end classes	Finding standard/popular sizes (e.g., shoe/garment sizes)