ICSE Class 8 Maths Chapter 36 Arithmetic Mean Mode and Median

Chapter 36: Arithmetic-Mean, Mode and Median

36.1 Review

Arithmetic Mean (or mean)

The Arithmetic mean in statistics has the same meaning as average in Arithmetic.

Arithmetic Mean of Raw Data

Add the given data and divide the sum by the number of data.

Example 1

Find the arithmetic mean of: 7, 10, 15, 11 and 9.

Solution

Sum of given data = 7 + 10 + 15 + 11 + 9 = 52

No. of given data = 5

Arithmetic mean of given data = \(\frac{52}{5} = 10.4\)

Thus, the mean of n numbers \(x_1, x_2, x_3, x_4, \ldots, x_n\) is:

\[\frac{x_1 + x_2 + x_3 + x_4 + \ldots + x_n}{n} = \frac{\Sigma x}{n}\]

The Greek letter \(\Sigma\) (called sigma) represents the sum of numbers (the given statistical data). In example 1, given above:

\(\Sigma x = 7 + 10 + 15 + 11 + 9 = 52\) and n = 5

Arithmetic mean = \(\frac{\Sigma x}{n} = \frac{52}{5} = 10.4\)

Teacher's Note

Understanding the arithmetic mean helps students calculate average scores, grades, or expenses in everyday situations like determining class performance or monthly spending patterns.

Arithmetic Mean of Tabulated Data

Let the frequencies of the numbers \(x_1, x_2, x_3, \ldots, x_n\) be \(f_1, f_2, f_3, \ldots, f_n\) respectively.

Then, the Arithmetic mean = \(\frac{f_1 \cdot x_1 + f_2 \cdot x_2 + \ldots + f_n \cdot x_n}{f_1 + f_2 + f_3 + \ldots + f_n}\)

= \(\frac{\Sigma f \cdot x}{\Sigma f}\) where \(\Sigma f =\) total number of observations = n

Example 2

Find the mean (arithmetic mean) of:

Solution

\(\Sigma f \cdot x = f_1 \cdot x_1 + f_2 \cdot x_2 + f_3 \cdot x_3 + \ldots + f_n \cdot x_n\)

= 3 × 10 + 4 × 15 + 2 × 20 + 5 × 25 + 6 × 30 = 435

\(\Sigma f = 3 + 4 + 2 + 5 + 6 = 20\)

Mean = \(\frac{\Sigma f \cdot x}{\Sigma f} = \frac{435}{20} = 21.75\)

Teacher's Note

Frequency distribution tables are used in real life such as analyzing customer ages in a store or student test scores, making calculations more efficient when data repeats.

Arithmetic Mean of Grouped Data

Example 3

Find the mean of:

Solution

Steps

1. Find the mean-value (class-mark) of each class-interval.

2. Represent the mean value by x.

3. Find the mean using the same method as in example 2 (given above).

Required Mean = \(\frac{\Sigma fx}{\Sigma f}\)

= \(\frac{2180}{50} = 43.6\)

Let the mean of some given numbers is m. If each of these numbers is:

(i) increased by x, mean of resulting numbers = m + x

(ii) decreased by x, mean of resulting numbers = m - x

(iii) multiplied by x, mean of resulting numbers = m × x, or

(iv) divided by x, mean of resulting numbers = m ÷ x.

Example 4

The mean of some numbers is 20. Find the mean of resulting numbers when each number is:

(i) increased by 5

(ii) decreased by 8

(iii) multiplied by 2

(iv) divided by 4

Solution

(i) The resulting mean = 20 + 5 = 25

(ii) The resulting mean = 20 - 8 = 12

(iii) The resulting mean = 20 × 2 = 40

(iv) The resulting mean = \(\frac{20}{4} = 5\)

Teacher's Note

These properties show how mean values change proportionally when we apply uniform operations, useful in scenarios like calculating new salary ranges after raises or cost adjustments.

36.2 Mode

The number which appears maximum times in the given statistical data is called mode. In other words, mode is the number whose frequency is maximum.

Example 4

Find the mode of following data: 15, 20, 15, 30, 20, 20, 30, 15, 20, 20.

Solution

Since, in the given data, the number 20 appears maximum times. Mode = 20

Example 5

Find the mode from the following frequency distribution:

Solution

Since, the frequency of number 14 is maximum. Mode = 14

Teacher's Note

Mode is used in real life to identify the most popular item, like the most frequently purchased shoe size in a store or the most common age group visiting a cinema.

36.3 Median

If the given statistical data be arranged in ascending or descending order of their magnitudes (values); then the value of the middle term is called median.

If the number of given data is odd, there will be only one middle term, which is the median of given data. But if, the number of given data is even, there will be two middle terms and the median will be the average of these two terms.

Whether the given data be arranged in ascending or descending order, the value of their median is unique.

Example 6

Find the median of the following data:

(i) 15, 12, 10, 9, 8, 13, 17

(ii) 3, 5, 9, 10, 11, 4, 5, 8, 12, 15.

Solution

(i) On arranging the given data in ascending order of their magnitudes, we get:

8, 9, 10, 12, 13, 15, 17

Since, the number of terms is odd (seven):

Median = middle term = 12

(ii) On arranging the given data in descending order of their magnitudes, we get:

15, 12, 11, 10, 9, 8, 5, 5, 4, 3

As the number of terms is even, the two middle terms are 9 and 8.

Median = \(\frac{9 + 8}{2} = 8.5\)

Teacher's Note

The median is useful in real life when finding the middle value of data such as house prices in a neighborhood or income distribution, as it is less affected by extreme values than the mean.

Test Yourself

1. Mean of x + 2, 3 - 2x and x + 1 = ........................... = ....................

2. The mean of a certain set of data = 16. If each number in this data is first increased by 4 and then divided by 5; the new mean of the resulting numbers = ............................... = ..................

3. For 10, 15, 10, 20, 15, 10, 10, 15 and 20 mode = ...................

If each given number is:

(a) increased by 3; the mode of resulting numbers = ...................

(b) divided by 5, the mode of resulting numbers = ..................

4. The median of 8, 3, 4, 7 and 6 = ...................

If each given number is:

(a) decreased by 2, the median of the resulting numbers = ...................

(b) multiplied by 2, the median of the resulting numbers = ...................

5. For 20, 25, 30, 25 and 25,

(a) mean = ................................... = ...................

(b) mode = ...................

(c) median = ...................

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