Statistics — Assertion-Reason Type Questions

Assertion (A): Runs scored by batsman A are 55, 60, 30, 80 while runs scored by batsman B are 81, 87, 76, 92. Then B has higher range than A.

Reason (R): Range is the difference between maximum and minimum values of a variable.

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

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

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

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Range = Maximum Value - Minimum Value.

∴ Reason (R) is true.

For Batsman A:

Runs: 55, 60, 30, 80

Minimum value (Min) = 30

Maximum value (Max) = 80

Range of A = Max − Min = 80 − 30 = 50.

For Batsman B:

Runs: 81, 87, 76, 92

Minimum value (Min) = 76

Maximum value (Max) = 92

Range of B = Max − Min = 92 − 76 = 16.

Thus, B does not have higher range than A.

∴ Assertion (A) is false.

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

Hence, option 2 is the correct option.

Assertion (A): Mean of first 5 odd natural numbers is 5.

Reason (R): Mean is the middle value of a set of observations.

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

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

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

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

The first 5 odd natural numbers are: 1, 3, 5, 7, 9.

The mean (arithmetic average) is calculated as the sum of all observations divided by the number of observations.

Sum = 1 + 3 + 5 + 7 + 9 = 25

Number of observations = 5

Mean = $\dfrac{\text{Sum of observations}}{\text{Number of observations}} = \dfrac{25}{5}$ = 5

∴ Assertion (A) is true.

The mean is the arithmetic average (sum of values divided by the count).

The median is the middle value of a set of observations when they are arranged in ascending or descending order.

∴ Reason (R) is false.

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

Hence, option 1 is the correct option.

Assertion (A): If 6 students score 78, 62, 91, 37, 80 and 66 marks in a subject, then median score is 72.

Reason (R): If number of observations is even, then

Median = $\dfrac{\dfrac{n}{2}\text{th observation} + \Big(\dfrac{n}{2} + 1\Big)\text{th observation}}{2}$, after putting all observations in ascending or descending order.

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

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

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

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

According to assertion: if 6 students score 78, 62, 91, 37, 80 and 66 marks in a subject, then median score is 72.

Arrange the scores in ascending order: 37, 62, 66, 78, 80, 91

There are 6 scores, so n = 6. This is an even number.

If number of observations is even, then

Median = $\dfrac{\dfrac{n}{2}\text{th observation} + \Big(\dfrac{n}{2} + 1\Big)\text{th observation}}{2}$, after putting all observations in ascending or descending order.

∴ Reason (R) is true.

Substituting the value,

$\text{Median }= \dfrac{\dfrac{6}{2}\text{th observation} + \Big(\dfrac{6}{2} + 1\Big)\text{th observation}}{2}\\[1em] = \dfrac{3\text{rd observation} + (3 + 1)\text{th observation}}{2}\\[1em] = \dfrac{3\text{th observation} + 4\text{th observation}}{2}\\[1em] = \dfrac{66 + 78}{2}\\[1em] = \dfrac{144}{2}\\[1em] = 72.$

∴ Assertion (A) is true.

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

Hence, option 3 is the correct option.