The standard deviation of 17 numbers is...

MathematicsVariance and Standard DeviationJEE Advanced 1980Easy

View in:English Hindi

The standard deviation of 17 numbers is zero. Then

(A)

the numbers are in geometric progression with common ratio not equal to one.

(B)

eight numbers are positive, eight are negative and one is zero.

(C)

either (a) or (b)

(D)

none of these

Answer:D

Visualized Solution (Hindi)

Understanding Standard Deviation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } σ

Standard Deviation ( $σ$ ) is a measure of the dispersion or spread of a set of values.
A low standard deviation indicates that the values tend to be close to the mean.
A standard deviation of zero indicates that there is no spread at all.

The Mathematical Condition for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } σ = 0

The formula for standard deviation is $σ = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}$ .
For $σ = 0$ , we must have $\sum (x_{i} - \overset{x}{ˉ})^{2} = 0$ .
Since $(x_{i} - \overset{x}{ˉ})^{2} \geq 0$ for all $i$ , the sum can only be zero if each individual term is zero.
This implies $x_{i} - \overset{x}{ˉ} = 0$ for all $i$ , meaning $x_{1} = x_{2} = ... = x_{17}$ .

Evaluating Option (a)

Option (a) states numbers are in G.P. with $r \neq = 1$ .
If $r \neq = 1$ , then $x_{2} = x_{1} \cdot r \neq = x_{1}$ .
Since the numbers are not all equal, $σ$ cannot be zero.
Therefore, Option (a) is incorrect.

Evaluating Option (b)

Option (b) states 8 positive, 8 negative, and 1 zero.
In this set, the numbers are clearly distinct (e.g., $5 \neq = - 5 \neq = 0$ ).
Since the numbers are not all equal, $σ$ cannot be zero.
Therefore, Option (b) is incorrect.

Final Conclusion

Standard deviation is zero if and only if all observations are equal.
Options (a) and (b) describe sets where numbers are not necessarily equal.
Thus, neither (a) nor (b) is a valid consequence.
The correct answer is Option (d): none of these.

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