Measures of Central Tendency — Assertion-Reason Type Questions

Assertion (A): Mean of the prime numbers lying between 5 and 20 is 13.4

Reason (R): Mean = $\dfrac{\text{sum of all observations}}{\text{number of observations}}$

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

The prime numbers between 5 and 20 are: 7, 11, 13, 17, 19 .

There are 5 such primes.

By formula,

Mean = $\dfrac{\text{sum of all observations}}{\text{number of observations}}$

∴ Reason (R) is true.

Mean = $\dfrac{7 + 11 + 13 + 17 + 19}{5} = \dfrac{67}{5}$ = 13.4

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): A survey was conducted by a group of students as a part of their environment awareness program, in which they collected the following data regarding the number of plants in 10 houses in a locality:

The mean of the data is 6.9

Reason (R): If the observation x₁, x₂, x₃,........., x_k has frequencies f₁, f₂, f₃, ........, f_k, then mean = $\dfrac{f_1x_1 + f_2x_2 + f_3x_3 + ........ + f_kx_k}{f_1 + f_2 + f_3 + ........ + f_k}$

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Mean = $\dfrac{∑f_ix_i}{∑f_i} = \dfrac{f_1x_1 + f_2x_2 + f_3x_3 + ........ + f_kx_k}{f_1 + f_2 + f_3 + ........ + f_k}$

∴ Reason (R) is true.

Mean = $\dfrac{66}{10}$​ = 6.6

∴ Assertion (A) is false.

∴ Assertion (A) is false, but Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): The number of goals scored by a football team in a series of matches are 3, 1, 0, 7, 5, 3, 3, 4, 1, 2, 0, 2

The median of the data is 2.5.

Reason (R): Median of an ungrouped data is the variate which has maximum frequency.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given goals: 3, 1, 0, 7, 5, 3, 3, 4, 1, 2, 0, 2

Arrange the data in ascending order: 0, 0, 1, 1, 2, 2, 3, 3, 3, 4, 5, 7

There are 12 values (even).

By formula,

$\text{Median }= \dfrac{\dfrac{n}{2}\text{th observation} + \Big(\dfrac{n}{2} + 1\Big)\text{th observation}}{2}\\[1em] = \dfrac{\dfrac{12}{2}\text{th observation} + \Big(\dfrac{12}{2} + 1\Big)\text{th observation}}{2}\\[1em] = \dfrac{6\text{th observation} + (6 + 1)\text{th observation}}{2}\\[1em] = \dfrac{6\text{th observation} + 7\text{th observation}}{2}\\[1em] = \dfrac{(2 + 3)}{2} = \dfrac{5}{2} \\[1em] = 2.5.$

∴ Assertion (A) is true.

The median is a measure of central tendency that splits your ordered data into two equal parts—half the observations lie below it, and half lie above it.

∴ Reason (R) is false.

∴ Assertion (A) is true, but Reason (R) is false.

Hence, option 1 is the correct option.

For the given 25 variables : x₁, x₂, x₃,........., x₂₅.

Assertion (A): To find median of the given data, the variate need to be arranged in ascending or descending order.

Reason (R): The median is the central most term of the arranged data.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

For the given 25 variables : x₁, x₂, x₃,........., x₂₅.

To compute the median, you must first arrange the data in either ascending or descending order. Only with ordered data, the middle value can be identified.

∴ Assertion (A) is true.

The median is a measure of central tendency that splits your ordered data into two equal parts—half the observations lie below it, and half lie above it.

Thus, the median is the central most term of the arranged data.

∴ Reason (R) is true.

∴ Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.