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.
Show 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!
Step 1: Pick the right measure. Uniformity (consistency) is judged by the coefficient of variation $\text{C.V.}=\dfrac{\sigma}{\mu}\times100\%$; the smaller the C.V., the more uniform the data. Step 2: Compute C.V. for A and B. $\text{C.V.}_A=\dfrac{12}{80}\times100=15\%$; $\text{C.V.}_B=\dfrac{6}{75}\times100=8\%$. Step 3: Compute C.V. for C and D. $\text{C.V.}_C=\dfrac{8}{70}\times100\approx11.4\%$; $\text{C.V.}_D=\dfrac{10}{72}\times100\approx13.9\%$. Step 4: Order the values. Arranged increasingly: $8\%<11.4\%<13.9\%<15\%$, that is $B<C<D<A$. Step 5: Identify the smallest. Division B has the lowest coefficient of variation at $8\%$. Step 6: Interpret. The smallest C.V. means the marks are most tightly clustered about the mean, so Division B is the most uniform. \[ \boxed{\text{Division B}} \]