Discussion
Two different brands, namely Brand A and Brand B, of mobile batteries are tested and the following data on their lifetimes (in months) are obtained:
| Brand A | 27 | 28 | 30 | 32 | 33 |
| Brand B | 17 | 19 | 20 | 21 | 23 |
Let \(CV_{A}\) and \(CV_{B}\) denote the sample coefficients of variation for Brand A and Brand B, respectively. Let \(s_{A}\) and \(s_{B}\) denote the sample standard deviations for Brand A and Brand B, respectively. Which one of the following statements is true?
Answer: B
The true statement is \(CV_{A} \lt CV_{B}\) and \(s_{A} \gt s_{B}\).
Brand A has mean 30 and deviations \(-3, -2, 0, 2, 3\); Brand B has mean 20 and deviations \(-3, -1, 0, 1, 3\). With divisor \(n - 1 = 4\):
\[ \begin{aligned} s_{A}^{2} &= \frac{9 + 4 + 0 + 4 + 9}{4} = 6.5, \ s_{A} \approx 2.55 \\ s_{B}^{2} &= \frac{9 + 1 + 0 + 1 + 9}{4} = 5, \ s_{B} \approx 2.24 \end{aligned} \]
So \(s_{A} \gt s_{B}\). The coefficient of variation divides by the mean:
\[ \begin{aligned} CV_{A} &\approx \frac{2.55}{30} = 0.085 \\ CV_{B} &\approx \frac{2.24}{20} = 0.112 \end{aligned} \]
Brand A varies more in absolute terms but less relative to its much larger mean, so \(CV_{A} \lt CV_{B}\). The conclusion is the same with divisor \(n\), since it scales both standard deviations equally.
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