Question

The arithmetic mean of marks in Mathematics for four divisions A, B, C and D were 80, 75, 70 and 72 respectively. Their standard deviations were 12, 6, 8 and 10 respectively. Then, which division has more uniformity.

• A. A
• B. B
• C. C
• D. D

Hint:

To compare fractions like $\frac{12}{80}$ and $\frac{6}{75}$ instantly without completing long division, write them with equal numerators: change $\frac{6}{75}$ to $\frac{12}{150}$. Since $\frac{12}{150}$ has a much larger denominator than $\frac{12}{80}$, its value is significantly smaller, confirming the lowest C.V. immediately!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem provides the arithmetic mean ($\mu$) and standard deviation ($\sigma$) of math marks for four separate classroom divisions (A, B, C, and D). We need to determine which division shows the greatest "uniformity" (consistency) in its marks.

Step 2: Key Formula or Approach:
Uniformity or consistency of statistical data is inversely proportional to its Coefficient of Variation (C.V.). The data set with the absolute

lowest Coefficient of Variation is classified as the most uniform and consistent. The formula for the Coefficient of Variation is: $$\text{C.V.} = \frac{\sigma}{\mu}$$

Step 3: Detailed Explanation:
Let's systematically compute the C.V. value for each of the four school divisions:

• For Division A: $\mu_A = 80$, $\sigma_A = 12$ $$\text{C.V.}_A = \frac{12}{80} = 0.150$$

• For Division B: $\mu_B = 75$, $\sigma_B = 6$ $$\text{C.V.}_B = \frac{6}{75} = 0.080$$

• For Division C: $\mu_C = 70$, $\sigma_C = 8$ $$\text{C.V.}_C = \frac{8}{70} \approx 0.114$$

• For Division D: $\mu_D = 72$, $\sigma_D = 10$ $$\text{C.V.}_D = \frac{10}{72} \approx 0.139$$
Comparing the computed results: $$0.080 < 0.114 < 0.139 < 0.150 \implies \text{C.V.}_B < \text{C.V.}_C < \text{C.V.}_D < \text{C.V.}_A$$ Since Division B has the smallest Coefficient of Variation value ($0.080$), it possesses the highest uniformity in its distribution of marks.

Step 4: Final Answer:
Division B has more uniformity, which corresponds to option (B).