ICSE Class 10 Maths Chapter 28 Measures of Central Tendency

Chapter 28

Measures of Central Tendency

Points to Remember

1. Average of a Data

For a given data, a single value of the variable representing the entire data, which describes the characteristics of the data, is called an average of the data.

An average tends to lie centrally with the values of the variable arranged in ascending order of magnitude. So, we call an average a measure of central tendency of the data.

Mainly, we are interested in three types of averages:

(i) Mean (ii) Median (iii) Mode.

2. Arithmetic Mean

The average of numbers in arithmetic is known as the Arithmetic Mean of these numbers in statistics.

Mean of An Ungrouped Data

The Arithmetic Mean or simply the Mean of n observations x₁, x₂, x₃, ......., xₙ is given by the formula:

Mean = \[\frac{(x_1 + x_2 + x_3 + ..... + x_n)}{n} = \frac{\Sigma x_i}{n}\]

where the symbol Σ, called sigma stands for the summation of the terms.

3. Some Useful Results

Let the mean of x₁, x₂, x₃, ......, xₙ be A. Then

(i) Mean of (x₁ + k), (x₂ + k), (x₃ + k) ........., (xₙ + k) is (A + k);

(ii) Mean of (x₁ - k), (x₂ - k), (x₃ - k) ........, (xₙ - k) is (A - k);

(iii) Mean of kx₁, kx₂, kx₃ ........., kxₙ is kA, where k ≠ 0.

4. Mean of grouped data:

(A) Direct method.

When the variates x₁, x₂, x₃ ........., xₙ have frequencies f₁, f₂, f₃ ....... fₙ respectively, then the mean is given by the formula:

Mean = \[\frac{f_1x_1 + f_2x_2 + f_3x_3....... + f_nx_n}{f_1 + f_2 + f_3 + ....... + f_x} = \frac{\Sigma f_ix_i}{\Sigma f_i}\]

(B) Shortcut Method.

Using this method larger quantities get converted into smaller ones, making the process of multiplication and division easier.

Method. From the given data, we suitable choose a term, usually the middle term and call it the assumed mean, to be denoted by A. We find the deviations, d₁ = (xᵢ - A) for each term: then

Mean = \[A + \frac{\Sigma f_id_i}{\Sigma f_i}\]

5. Mean of grouped data in the form of classes:

(A) Direct Method:

Step 1. For each class, find the class mark xᵢ by using the relation, \[x = \frac{1}{2}\] (lower limit + upper limit).

Step 2. Use the formula, Mean = \[\frac{\Sigma f_ix_i}{\Sigma f_i}\]

(B) Short Cut Method or Deviation Method:

Step 1. For each class, find the class mark xᵢ.

Step 2. Let A be the assumed mean.

Step 3. Find dᵢ = (xᵢ - A).

Step 4. Use the formula, Mean = \[\left( A + \frac{\Sigma f_id_i}{\Sigma f_i} \right)\]

(C) Step-Deviation Method:

Step 1. For each class, find the class mark x₁.

Step 2. Let A be the assumed mean.

Step 3. Calculate, \[u_i = \frac{(x_i - A)}{c}\], where c is the class size.

Step 4. Use the formula, Mean = \[\left( A + c \cdot \frac{\Sigma f_iu_i}{\Sigma f_i} \right)\]

Teacher's Note

Understanding averages helps us make sense of real-world data, like calculating your average test score or finding the mean temperature for the month.

Exercise 28

Q.1. Find the mean of each of the following sets of numbers:

(i) 10, 4, 6, 9, 12 (ii) 14, 11, 23, 7, 18, 14, 5, 8 (iii) 5-8, 6-3, 7-1, 9-4, 4-9 (iv) 0-2, 0-02, 2, 2-02

Sol.

(i) Sum of variates = 10 + 4 + 6 + 9 + 12 = 41 and number of variates = 5

Mean = \[\frac{\Sigma x_i}{n} = \frac{41}{5} = 8.2\]

(ii) Sum of variates = 14 + 11 + 23 + 7 + 18 + 14 + 5 + 8 = 100

Number of variates = 8

Mean = \[\frac{\Sigma x_i}{n} = \frac{100}{8} = 12.5\]

(iii) Sum of variates = 5-8 + 6-3 + 7-1 + 9-4 + 4-9 = 33-5

Number of variates = 5

Mean = \[\frac{\Sigma x_i}{n} = \frac{33.5}{5} = 6.7\]

(iv) Sum of variates = 0-2 + 0-02 + 2 + 2-02 = 4-24

Number of variates (n) = 4

Mean = \[\frac{\Sigma x_i}{n} = \frac{4.24}{4} = 1.06\] Ans.

Q.2. Find the arithmetic mean of:

(i) first eight natural numbers;

(ii) first five prime numbers;

(iii) first six positive even integers;

(iv) first five positive integral multiples of 3;

(v) all factors of 20.

Sol.

(i) First eight natural numbers are 1, 2, 3, 4, 5, 6, 7, 8

Sum = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

Mean = \[\frac{\Sigma x_i}{n} = \frac{36}{8} = 4.5\]

(ii) First 5 prime numbers are 2, 3, 5, 7, 11

Sum = 2 + 3 + 5 + 7 + 11 = 28

Mean = \[\frac{\Sigma x_i}{n} = \frac{28}{5} = 5.6\]

(iii) First 6 positive even integers are 2, 4, 6, 8, 10, 12

Sum = 2 + 4 + 6 + 8 + 10 + 12 = 42

Mean = \[\frac{\Sigma x_i}{n} = \frac{42}{6} = 7\]

(iv) First 5 positive integral multiples of 3 are 3, 6, 9, 12, 15

Sum = 3 + 6 + 9 + 12 + 15 = 45

Mean = \[\frac{\Sigma x_i}{n} = \frac{45}{5} = 9\]

(v) All factors of 20 are 1, 2, 4, 5, 10, 20

Sum = 1 + 2 + 4 + 5 + 10 + 20 = 42

Mean = \[\frac{\Sigma x_i}{n} = \frac{42}{6} = 7\] Ans.

Q.3. The daily minimum temperature recorded (in degrees F) at a place during a week was as under:

Find the mean temperature of the week.

Sol.

Total temperature during 7 days = 35-5 + 30-8 + 28-3 + 31-1 + 23-8 + 29-9 + 32-7 = 212-1 F°

Mean temperature = \[\frac{\Sigma x_i}{n} = \frac{212.1}{7} = 30.3\] F°

Teacher's Note

Weather forecasters use mean temperatures to describe typical conditions for a season or month, helping us decide what clothes to wear or plan outdoor activities.

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