Question:
Which of the following statements are correct about standard deviation ?
A. It is the root-mean-square deviation from mean.
B. It is always positive.
C. It is dependent of the change of origin.
D. It is independent of the scale of measurement.
Choose the correct answer from the options given below :
Which of the following statements are correct about standard deviation ?
A. It is the root-mean-square deviation from mean.
B. It is always positive.
C. It is dependent of the change of origin.
D. It is independent of the scale of measurement.
Choose the correct answer from the options given below :
A. It is the root-mean-square deviation from mean.
B. It is always positive.
C. It is dependent of the change of origin.
D. It is independent of the scale of measurement.
Choose the correct answer from the options given below :
Show Hint
Standard Deviation is Shift-Invariant (adding a constant does nothing) but Scale-Variant (multiplying changes it).
Updated On: May 20, 2026
- A, B, C only
- B, C, D only
- A, B only
- A, D only
Show Solution
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The Correct Option is C
Solution and Explanation
Concept:
Standard deviation ($\sigma$) is a fundamental measure of dispersion that quantifies the amount of variation or spread in a set of data values. It has specific mathematical properties regarding how it responds to data transformations.
Step 1: Evaluate Statement A and B.
Statement A is correct. By definition, standard deviation is the square root of the variance, which is the average of the squared deviations from the mean (root-mean-square deviation). Statement B is also correct because it is derived from squared values and a square root; it can be zero (if all data points are identical) or positive, but never negative.
Step 2: Evaluate Statement C (Change of Origin).
This is incorrect. Standard deviation is independent of the change of origin. This means if you add or subtract a constant value from every data point in a set, the standard deviation remains exactly the same because the relative "spread" hasn't changed.
Step 3: Evaluate Statement D (Change of Scale).
This is incorrect. Standard deviation is dependent on the scale of measurement. If you multiply every data point by a constant $k$, the standard deviation is also multiplied by $|k|$. Therefore, it is not independent of scale.
Step 1: Evaluate Statement A and B.
Statement A is correct. By definition, standard deviation is the square root of the variance, which is the average of the squared deviations from the mean (root-mean-square deviation). Statement B is also correct because it is derived from squared values and a square root; it can be zero (if all data points are identical) or positive, but never negative.
Step 2: Evaluate Statement C (Change of Origin).
This is incorrect. Standard deviation is independent of the change of origin. This means if you add or subtract a constant value from every data point in a set, the standard deviation remains exactly the same because the relative "spread" hasn't changed.
Step 3: Evaluate Statement D (Change of Scale).
This is incorrect. Standard deviation is dependent on the scale of measurement. If you multiply every data point by a constant $k$, the standard deviation is also multiplied by $|k|$. Therefore, it is not independent of scale.
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