The mean of 1, 7, 5, 3, 4 and 4 is m. The numbers 3, 2, 4, 2, 3, 3 and p have mean m - 1 and median q. Find p and q.

Given,

Mean of 1, 7, 5, 3, 4 and 4 is m.

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

Mean (m) = $\dfrac{\text{Sum of observations}}{\text{No. of observations}} = \dfrac{24}{6}$ = 4.

Given,

Numbers 3, 2, 4, 2, 3, 3 and p have mean m - 1 or mean = 3.

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

Mean (m) = $\dfrac{\text{Sum of observations}}{\text{No. of observations}}$

⇒ 3 = $\dfrac{17 + p}{7}$

⇒ 21 = 17 + p

⇒ p = 4.

Observations in ascending order are = 2, 2, 3, 3, 3, 4, 4.

Here, n = 7, which is odd.

By formula,

Median = $\dfrac{n + 1}{2}$ th term

= $\dfrac{7 + 1}{2} = \dfrac{8}{2}$ = 4th term = 3.

∴ q = 3.

Hence, p = 4 and q = 3.