Question

The numerical value of the standard deviation can never be:

(i)Zero
(ii)Negative
(iii)Larger than the variance
Which of the following statements is true:
a. (i) & (ii)
b. (ii) & (iii)
c. (i), (ii) & (iii)
d. None of the above

Answer 1

The numerical value of the standard deviation can be zero when all data values are identical, but it can never be negative or larger than the variance. Hence, the true statements are (i) & (ii), making option a the correct choice.

Step-by-step explanation:

The numerical value of the standard deviation can never be negative since it is derived from the squares of the differences between each data point and the mean. If the standard deviation were zero, this would indicate that all data points are identical. Therefore, statements (i) and (ii) are true. Regarding statement (iii), the standard deviation is the square root of the variance, and hence it can never be larger than the variance.

Thus, the correct answer to the student's question is:

1. Zero: This is possible if all data values are the same.
2. Negative: This is impossible because standard deviation is based on squared values.
3. Larger than the variance: This is impossible as the standard deviation is the square root of the variance.

As a result, the correct statement is option a: (i) & (ii).