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Consider the following statements: 1) Mean is independent of change in scale and change in origin. 2) Variance is independent of change in scale but not in origin. Which of the above statements is/are correct?

Consider the following statements: 1) Mean is independent of change in scale and change in origin. 2) Variance is independent of change in scale but not in origin. Which of the above statements is/are correct? Correct Answer Neither 1 nor 2

Concept:

Properties of mean:

  •  If C is added to each observation, then the new mean is also increased by C.

  • If C is multiplied to each observation, then the new mean is also C times.

  • If C is divided to each observation, then the new mean is also old mean divided by C.


Properties of variance:

  • Var(X + C) = Var(X), where C is a constant.

  • Var(CX) = C2 × Var(X), where C is a constant.


Calculation:

Mean:

Change in scale:

We know that, if C is multiplied to each observation, then new mean is also C times old mean.

So, mean is dependent on change in scale.

Change in origin:

We know that, if C is added/subtracted to each observation, then new mean is also increased/decreased by C.

So, mean is dependent on change in origin.

Variance:

Change in scale:

We know that, Var(CX) = C2 × Var(X), where C is a constant.

So, variance is dependent on change in scale.

Change in origin:

We know that, Var(X + C) = Var(X), where C is a constant.

So, variance is independent of change in origin.

Hence, both statement (1) & (2) are incorrect.

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