Q.Calculate the Arithmetic Mean of the following weekly income distribution (₹'00): Income (X): 10, 20, 30, 40 with corresponding number of workers (f): 4, 6, 7, 3.

The standard inequality among the three means is $A.M \ge G.M \ge H.M$.

For any set of positive real numbers, the arithmetic mean (A.M.), geometric mean (G.M.) and harmonic mean (H.M.) always satisfy

$A.M \ge G.M \ge H.M,$

with equality if and only if all the observations are equal.

For two positive numbers $a,b$ this is easy to see: $A.M=\dfrac{a+b}{2}$, $G.M=\sqrt{ab}$, $H.M=\dfrac{2ab}{a+b}$, and one can verify $\dfrac{a+b}{2}\ge\sqrt{ab}\ge\dfrac{2ab}{a+b}$. A useful related identity is $G.M=\sqrt{A.M\times H.M}$.

This relationship among the averages is a key result in the measures-of-central-tendency section of the TN HSC Class-11 Business Statistics syllabus.

The median is the value below which half the data lie, which is exactly the second quartile $Q_2$, so the answer is option (d).

Quartiles divide an ordered data set into four equal parts:

• $Q_1$ (first quartile) cuts off the lowest 25%,
• $Q_2$ (second quartile) cuts off the lowest 50%,
• $Q_3$ (third quartile) cuts off the lowest 75%.

The median is the middle value that splits the data into two equal halves — that is precisely the point with 50% below it, so

$\text{Median} = Q_2.$

(Equivalently, Median $= D_5 = P_{50}$.)