Compute the arithmetic mean, median and mode of the following distribution and explain their relationships:

(a) If z₁ = x₁ + y₁, z₂ = x₂ + y₂,…….z_n = x_n + y_n, then prove that z = x + y , where the symbols have their usual meaning.

(b) Prove that the logarithm of geometric mean of observations is the arithmetic mean of logarithms of the observations

Monthly income (Rs) 0-75 75-150 Frequency 15 200 150-225 250 225-300 300-375 225 10 375-450 5