Following data shows the information about marks obtained in Physics, Chemistry, Mathematics and Biology by 100 students in a class. Then subject shows the highest variability in marks. Choose the correct subject from the options below:
Show Hint
To easily compare fractions like $\frac{4}{23}$ and $\frac{5}{27}$ without long division, cross-multiply their terms: $4 \times 27 = 108$ and $5 \times 23 = 115$. Since $115 \gt 108$, $\frac{5}{27}$ is larger, confirming Biology is more volatile than Math!
Step 1: Understanding the Question:
The problem presents statistical data (Mean and Standard Deviation) of 100 students across four distinct subjects. We are asked to determine which subject demonstrates the highest relative variability in marks.
Step 2: Key Formula or Approach:
To compare the relative variability between datasets with different means, calculating the Standard Deviation alone is insufficient. We must compute the Coefficient of Variation (C.V.), which normalizes the standard deviation against the group mean:
$$\text{C.V.} = \frac{\text{Standard Deviation } (\sigma)}{\text{Mean } (\mu)}$$
The dataset featuring the highest numerical value for C.V. has the highest relative dispersion and variability.
Step 3: Detailed Explanation:
Let's systematically calculate the Coefficient of Variation (C.V.) for all four fields:
1. For Physics:
$$\text{C.V.}_{\text{Physics}} = \frac{3}{20} = 0.150$$
2. For Chemistry:
$$\text{C.V.}_{\text{Chemistry}} = \frac{2}{25} = 0.080$$
3. For Mathematics:
$$\text{C.V.}_{\text{Mathematics}} = \frac{4}{23} \approx 0.174$$
4. For Biology:
$$\text{C.V.}_{\text{Biology}} = \frac{5}{27} \approx 0.185$$
Comparing the computed coefficients:
$$0.080 \lt 0.150 \lt 0.174 \lt 0.185$$
The value $0.185$ is the largest coefficient in the group, indicating that Biology marks display the greatest variability.
Step 4: Final Answer:
The subject showing the highest variability is Biology, corresponding to option (C).
