Mean & Median of Ungrouped Data & Frequency Polygon — Assertion-Reason Type Questions

Assertion (A) : The mean of 19 numbers is 38. If the mean of the first 10 numbers is 36 and that of the last 10 is 40, then the 10th number is 38.

Reason (R) : Mean = $\dfrac{\text{Sum of observations}}{\text{Number of observations}}$

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Given,

Mean of 19 number = 38

⇒ Total sum = 19 × 38 = 722

Mean of first 10 numbers = 36

⇒ Total sum = 10 × 36 = 360

Mean of last 10 numbers = 40

⇒ Total sum = 10 × 40 = 400

The 10th number is included in both groups:

⇒ First 10 numbers → includes 10th

⇒ Last 10 numbers → also includes 10th

So,

⇒ 10th number = (Sum of first 10 numbers + Sum of last 10 numbers)- Sum of all the 19 numbers

= (360 + 400) - 722

= 760 - 722 = 38.

∴ Assertion (A) is true.

We know that,

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

∴ Reason (R) is true.

Both Assertion (A) and Reason (R) are true.

Hence, option 3 is the correct option.

Assertion (A) : The mean of 15 observations was found to be 21. Later it was detected that one value 15 was wrongly copied as 18, while calculating the mean. The correct mean is 20.

Reason (R) : The mean of n observations x₁, x₂, x₃, ....., x_n is $\bar{x}$. If each observation is increased by p, then the new mean is increased by p, i.e., the new mean is $\bar{x}$ + p.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Given,

Wrong mean = 21

Total observations = 15

So,

Total wrong sum = 21 × 15 = 315

One value 15 was wrongly copied as 18

∴ 18 - 15 = 3

New total sum = 315 - 3 = 312

New mean = $\dfrac{312}{15}$

= 20.8.

But given mean = 20.

∴ Assertion (A) is false.

If given observation is increased by p, then

$\Rightarrow \text{New Mean} = \dfrac{(x_1 + p) + (x_2 + p) + \dots + (x_n + p)}{n} \\[1em] \Rightarrow \text{New Mean} = \dfrac{(x_1 + x_2 + \dots + x_n) + np}{n} \\[1em] \Rightarrow \text{New Mean} = \bar{x} + p \\[1em]$

∴ Reason (R) is true.

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

Hence option 2 is the correct option.