NIOS Economics • Module 6 Lesson 18

Analysis of Data: Arithmetic Mean

Interactive Exam Preparation Suite • Central Tendency & Statistical Averages

4 Notes Sections
10 MCQs
10 Flashcards
Lesson 18 Overview

Central Tendency & Arithmetic Mean Calculations

Statistical methods organize raw economic data into meaningful summary values. The central tendency represents the point around which observations cluster.

1

Meaning & Functions of Central Tendency

Definition & Core Concept

Central Tendency refers to the tendency of raw statistical data to cluster towards a central location or value.

📌 Textbook Core Definition:

"Tendency of data to cluster towards the central location or value is called central tendency."

4 Purposes and Functions of Averages

  • • Brief Summary: To convert collected raw data into a brief, easily understandable value.
  • • Facilitate Comparison: To enable direct comparison between two or more sets of data or groups.
  • • Representative Figure: To present a single typical value that represents the whole dataset.
  • • Policy Formulation: To provide a reliable base for future policy and programme decisions.
2

Arithmetic Mean Definition & Mathematical Formula

Arithmetic Mean (commonly called average) is obtained by dividing the total sum of all items by the total number of items.

$$\bar{X} = \frac{\sum x}{N}$$

Where \(\bar{X}\) = Arithmetic Mean, \(\sum x\) = Sum of all item values, \(N\) = Number of items.

📌 Textbook Illustration (Sachin's Innings): Runs scored in 5 innings = 59, 78, 100, 50, 63. Total runs \(\sum x = 350\), \(N = 5\). Average \(\bar{X} = \frac{350}{5} = 70\). Note that Sachin never scored exactly 70 in any single inning, yet 70 is the true statistical representative!
3

Calculation Methods in 3 Data Series

Series Type 1

Individual Series

Raw items without frequencies.

Direct: \(\bar{X} = \frac{\sum x}{N}\)

Shortcut: \(\bar{X} = A + \frac{\sum dx}{N}\)

\(A\) = Assumed Mean, \(dx = x - A\)

Series Type 2

Discrete Series

Exact values with corresponding frequencies (\(f\)).

Direct: \(\bar{X} = \frac{\sum fx}{N}\)

Shortcut: \(\bar{X} = A + \frac{\sum fdx}{N}\)

\(N = \sum f\), \(dx = x - A\)

Series Type 3

Continuous Series

Class intervals with frequencies (\(f\)).

Mid Value: \(x = \frac{L_1 + L_2}{2}\)

Direct: \(\bar{X} = \frac{\sum fx}{N}\)

Step Deviation: \(\bar{X} = A + \frac{\sum fdx'}{N} \times c\)

\(dx' = \frac{x-A}{c}\), \(c\) = class width

4

Precautions & Limitations of Arithmetic Mean

⚠️

1. Theoretical & Unrealistic Values

Arithmetic mean is a theoretical calculated figure which may not reflect actual possibilities.

📌 Textbook Example: If 10 families have 27 children in total, the average is \(27 \div 10 = 2.7\) children per family. In real life, a family can have 2 or 3 children, but never 2.7 children!
⚠️

2. Inapplicable to Qualitative Data

Arithmetic mean cannot be directly computed for qualitative attributes that cannot be measured numerically.

📌 Attributes Limited: Qualities like honesty, bravery, beauty, loyalty, and intelligence cannot be directly averaged using arithmetic mean.