Assertion (A): Mean of n observations is x and mean of another set of n observations is y, the combined mean of all the observations is $\dfrac{x + y}{2}$

Reason (R): Total number of observations = nx + ny

∴ Mean of all the observations = $\dfrac{nx + ny}{2n}$

1. A is true, but R is false.

2. A is false, but R is true.

3. Both A and R are true, and R is the correct reason for A.

4. Both A and R are true, and R is the incorrect reason for A.

Given,

For 1st set :

Mean = x

Number of observations = n

By formula,

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

Substituting values we get :

$\Rightarrow x = \dfrac{\text{Sum of observations}}{n} \\[1em] \Rightarrow \text{Sum of observations} = x \times n \\[1em] \Rightarrow \text{Sum of observations} = xn.$

For 2nd set :

Mean = y

Number of observations = n

Substituting values we get :

$\Rightarrow y = \dfrac{\text{Sum of observations}}{n} \\[1em] \Rightarrow \text{Sum of observations} = y \times n \\[1em] \Rightarrow \text{Sum of observations} = yn. \\[1em] \text{Mean of all observations } = \dfrac{\text{Sum of observations of 1st set + Sum of observations of 2nd set}}{\text{Total number of observations}}\\[1em] = \dfrac{nx + ny}{n + n}\\[1em] = \dfrac{n(x + y)}{2n}\\[1em] = \dfrac{x + y}{2}.$

∴ Both A and R are true, and R is the correct reason for A.

Hence, option 3 is the correct option.