The mean of the numbers 1, 7, 5, 3, 4, 4 is m. The numbers 3, 2, 4, 2, 3, 3, p have mean m - 1 and median q. Find (i) p (ii) q (iii) the mean of p and q.

(i) Given, the mean of the numbers 1, 7, 5, 3, 4, 4 is m.

Sum of terms = 1 + 7 + 5 + 3 + 4 + 4 = 24.

By definition,

$\text{Mean } = \dfrac{\text{Sum of terms}}{\text{No. of terms}} \\[1em] \therefore \text{m} = \dfrac{24}{6} = 4.$

Given, the mean of the numbers 3, 2, 4, 2, 3, 3, p is (m - 1) i.e. 3.

Sum of terms = 3 + 2 + 4 + 2 + 3 + 3 + p = 17 + p.

By definition,

$\text{Mean } = \dfrac{\text{Sum of terms}}{\text{No. of terms}} \\[1em] \therefore 3 = \dfrac{17 + p}{7} \\[1em] \Rightarrow 21 = p + 17 \\[1em] \Rightarrow p = 21 - 17 = 4.$

Hence, the value of p is 4.

(ii) Given, the median of the numbers 3, 2, 4, 2, 3, 3, 4 is q.

Arranging the numbers in ascending order we get,

2, 2, 3, 3, 3, 4, 4.

Here, n (no. of observations ) = 7, which is odd.

$\therefore \text{Median} = \dfrac{n + 1}{2} \text{th observation} \\[1em] = \dfrac{7 + 1}{2} \\[1em] = \dfrac{8}{2} \\[1em] = 4 \text{th observation}$

Given, median = q = 4th observation = 3.

⇒ q = 3.

Hence, the value of q is 3.

(iii) Mean of p and q i.e. 4 and 3 is,

$\text{Mean } = \dfrac{\text{Sum of terms}}{\text{No. of terms}} \\[1em] = \dfrac{4 + 3}{2} \\[1em] = \dfrac{7}{2} \\[1em] = 3.5$

Hence, mean of p and q is 3.5