Chapter 5: Measures of Central Tendency

A measure of central tendency is a single representative value around which the bulk of the data values cluster. The main measures are the Arithmetic Mean, Median, and Mode.

Arithmetic Mean (\( \bar{X} \))

Arithmetic mean is the sum of all observations divided by the number of observations. It is the most common measure of central tendency.

For discrete series (Ungrouped data): $$ \bar{X} = \frac{\sum x}{N} $$

For frequency distribution (Grouped data) via Direct Method: $$ \bar{X} = \frac{\sum f x}{\sum f} $$

For Grouped data via Step-Deviation Method: $$ \bar{X} = A + \frac{\sum f d’}{\sum f} \times c $$ Where \(A\) is the assumed mean, \(d’ = \frac{x - A}{c}\), and \(c\) is the common class magnitude.

Median (\(M\))

The median is the exactly middle value of a series when the data is arranged in ascending or descending order. It divides the distribution into two equal parts.

For continuous series: $$ M = L + \frac{\frac{N}{2} - c.f.}{f} \times i $$ Where:

• \(L\) = Lower limit of the median class
• \(N\) = Sum of frequencies
• \(c.f.\) = Cumulative frequency of the class preceding the median class
• \(f\) = Frequency of the median class
• \(i\) = Class interval of the median class

Mode (\(Z\))

The mode is the value that occurs most frequently in a statistical distribution. It is the point of maximum concentration.

For continuous series: $$ Z = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times i $$ Where:

• \(L\) = 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
• \(i\) = Size of the class interval

Competency-Based Questions

Numerical Application

Q1. The weekly wages of 5 workers are ₹1500, ₹2000, ₹1800, ₹2200, and ₹2500. Calculate the Arithmetic Mean of their wages.

Answer: Given wages \(x\): 1500, 2000, 1800, 2200, 2500 Number of workers \(N\) = 5 $$ \bar{X} = \frac{\sum x}{N} $$ $$ \bar{X} = \frac{1500 + 2000 + 1800 + 2200 + 2500}{5} = \frac{10000}{5} = 2000 $$ The Arithmetic Mean of the weekly wages is ₹2000.

Analytical Questions

Q2. Under what circumstances is the Median considered a better measure of central tendency than the Arithmetic Mean? Provide a logical rationale.

Answer: The Median is considered better when a dataset contains extreme values or outliers. Reasoning: The Arithmetic mean relies on all observations, meaning a single extremely high or low value excessively shifts the mean, causing it to misrepresent the bulk of the data. For instance, in income data, a handful of billionaires heavily skew the mean upward. The median, being a positional average, remains unaffected by extreme values and precisely reflects the middle point of the distribution, offering a superior representation of the typical value.