ICSE Class 10 Maths Chapter 24 Measures of Central Tendency

Measures Of Central Tendency

(Mean, Median, Quartiles and Mode)

24.1 Introduction

The numerical expressions which represent the characteristics of a group (a large collection of numerical data) are called Measures of Central Tendency (or, Averages).

An average which is used to represent a whole series should neither have the lowest value nor the highest value in the group, but a value somewhere between two limits, possibly in the centre, where most of the items of the group cluster.

There are many types of statistical averages, out of them the following averages will be studied in this chapter.

1. Arithmetic Average or Mean 2. Median 3. Mode

24.2 Arithmetic Mean

The arithmetic mean (or, simply, mean) of a set of numbers is obtained by dividing the sum of numbers of the set by the number of numbers.

For example:

The mean of n numbers x₁, x₂, x₃, -, xₙ is

\[\frac{x_1 + x_2 + x_3 + \cdots + x_n}{n} = \frac{\sum x}{n}\]

The Greek letter Σ (called sigma) represents the sum of numbers.

Example 1

Problem: The weights (in kilogram) of 5 persons are 67, 65, 71, 57 and 45. Find the arithmetic mean of their weights.

Solution:

According to the definition:

\[\text{Arithmetic mean} = \frac{\sum x}{n} = \frac{67 + 65 + 71 + 57 + 45}{5} \text{ kg}\]

\[= \frac{305}{5} \text{ kg} = 61 \text{ kg}\]

24.3 Arithmetic Mean of Tabulated Data

For a given discrete frequency distribution, the arithmetic mean can be obtained by using any one of the following three methods:

1. Direct method. 2. Short-cut method. 3. Step-deviation method.

1. Direct Method

Steps:

1. Prepare a frequency table with three columns: (a) In the first column from the left, write the values of the variate (x). (b) In the second column from the left, write the corresponding frequency (f) of each variate in column (a). (c) In the third column, write the product of each x with its frequency (f) i.e. write each value of fx.

2. Add all the entries in the second column to get Σf (sum of all the frequencies).

3. Add all the entries in the third column to get Σfx.

4. Then required mean = \(\frac{\sum fx}{\sum f}\), by using direct method.

Example 2

Problem: Find the mean of:

Solution:

Σf = 20 and, Σfx = 137

\[\text{Mean} = \frac{\sum fx}{\sum f} = \frac{137}{20} = 6.85\]

Example 3

Problem: Using direct method, find the mean of following frequency distribution:

Solution:

Σf = n = 100 and Σfx = 2750

\[\text{Mean}(\bar{x}) = \frac{\sum fx}{n} = \frac{2750}{100} = 27.50\]

In general, mean is denoted by \(\bar{x}\).

Teacher's Note

Understanding mean helps students calculate average test scores, average heights in a class, or average daily pocket money - all practical ways to summarize data in daily life.

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