Question

For the frequency distribution :
Variate (x) : \(x_1\) \(x_2\) \(x_3\) .... \(x_{15}\)
Frequency (f) : \(f_1\) \(f_2\) \(f_3\) ...... \(f_{15}\)
where \(0 \lt x_1 \lt x_2 \lt x_3 \lt ... \lt x_{15} = 10\) and \(\sum_{i=1}^{15} f_i \gt 0\), the standard deviation cannot be :

• A. 6
• B. 1
• C. 4
• D. 2

Hint:

All observations lie in \((0, 10]\), so each is within less than 5 of the number 5. Use this to bound the variance.

Answer 1

The Correct Option is A

All values satisfy \(0 \lt x_i \le 10\), hence \(|x_i - 5| \lt 5\) for every i.
The variance about the mean is at most the mean square deviation about any other point, in particular about 5:
\(\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} \le \frac{\sum f_i (x_i - 5)^2}{\sum f_i} \lt 25\)
So \(\sigma \lt 5\). Therefore the standard deviation cannot be 6.
(1, 2 and 4 are all below 5 and are attainable by suitable frequencies.)
Answer: (1) 6