Measures of Central Tendency

1. INTRODUCTION

This chapter introduces the concept of measures of central tendency, which are statistical tools used to summarise a large set of data by a single representative value. In practical life, we often encounter large data sets such as marks obtained by students, rainfall in an area, production in a factory, or income of persons in a locality. To understand and compare such data effectively, it is essential to summarise it into a single typical value that represents the entire data set. This summarisation is achieved through measures of central tendency or averages. Consider Baiju, a farmer in Balapur village, Bihar, who owns 1 acre of land. To assess Baiju's economic condition relative to other farmers in the village, one needs to compare his landholding size with others. This comparison can be done using measures of central tendency: 1. Arithmetic Mean: To see if Baiju's landholding is above the average size. 2. Median: To check if Baiju owns more land than half of the farmers. 3. Mode: To find if Baiju's landholding size is the most common among farmers. Thus, measures of central tendency help in summarising and representing data by a single value, facilitating comparison and interpretation. The three most commonly used averages are Arithmetic Mean, Median, and Mode. Although Geometric Mean and Harmonic Mean also exist, this chapter focuses on the first three.

• Measures of central tendency summarise data by a single representative value.
• They help in comparing and interpreting large data sets effectively.
• Examples include average marks, rainfall, production, and income.
• Three main averages: Arithmetic Mean, Median, and Mode.
• Geometric and Harmonic means exist but are not covered here.
• Central tendency values represent typical or representative data points.
• 📌 Measures of Central Tendency: Statistical tools to summarise data by a single value.
• 📌 Arithmetic Mean: Sum of all observations divided by number of observations.
• 📌 Median: Middle value dividing data into two equal halves.

2. ARITHMETIC MEAN

Arithmetic Mean (AM) is the most widely used measure of central tendency. It is calculated by adding all the values in a data set and dividing the sum by the total number of observations. It provides an average value that represents the data set. For example, consider the monthly incomes (in Rs) of six families: 1600, 1500, 1400, 1525, 1625, and 1630. The arithmetic mean income is calculated as: = (1600 + 1500 + 1400 + 1525 + 1625 + 1630) / 6 = Rs 1,547 This means, on average, a family earns Rs 1,547. More formally, if there are N observations X1, X2, ..., XN, the arithmetic mean is given by: Arithmetic Mean = (X1 + X2 + ... + XN) / N = ΣX / N Where ΣX denotes the sum of all observations and N is the total number of observations. The arithmetic mean is denoted by X̄ (X-bar). The calculation of arithmetic mean can be done for ungrouped data (individual observations) and grouped data (data presented in classes or categories with frequencies). For ungrouped data, the direct method involves straightforward summation and division. For grouped data, the mean is calculated using frequencies and mid-values or class marks.

• Arithmetic Mean is the sum of all observations divided by the number of observations.
• It is denoted by X̄ (X-bar).
• Represents the average or central value of the data.
• Applicable to both ungrouped and grouped data.
• Affected by extreme values (outliers).
• Widely used due to simplicity and ease of calculation.
• 📌 Arithmetic Mean: Average value obtained by dividing sum of observations by total observations.
• 📌 Ungrouped Data: Data presented as individual observations.
• 📌 Grouped Data: Data presented in classes or categories with frequencies.