Question

Consider the following statements:
• [(i)] If you add 9 to each entry that adds 9 to the average.
• [(ii)] If you add 9 to each entry that adds 9 to the standard deviation.
• [(iii)] If you double each entry on a list that doubles the average.
• [(iv)] If you double each entry on a list that doubles the standard deviation. Which of the following is correct?

• A. Statements (i) and (ii) are correct
• B. Statements (ii) and (iii) are correct
• C. Statements (iii) and (iv) are correct
• D. Statements (ii) and (iv) are correct

Hint:

Adding a constant: \[ \bar{x}+c \] changes the mean but not the standard deviation. Multiplying by a constant: \[ cX \] multiplies both the mean and standard deviation by \(c\).

Answer 1

The Correct Option is C

Step 1: Examine Statement (i).
Suppose the original mean is \[ \bar{x} \] If 9 is added to every observation, the new mean becomes \[ \bar{x}+9 \] Therefore Statement (i) is correct.

Step 2: Examine Statement (ii).
Adding a constant shifts all observations equally. The spread of the data remains unchanged. Hence standard deviation remains the same. Therefore Statement (ii) is incorrect.

Step 3: Examine Statement (iii).
If every observation is multiplied by 2, \[ \text{New Mean} = 2\times\text{Old Mean} \] Hence Statement (iii) is correct.

Step 4: Examine Statement (iv).
If every observation is multiplied by 2, \[ \text{New Standard Deviation} = 2\times\text{Old Standard Deviation} \] Hence Statement (iv) is correct.

Step 5: Identify the correct option.
The correct statements are \[ (i),\ (iii)\ \text{and}\ (iv) \] However, among the given options, the only option containing both definitely correct statements (iii) and (iv) is: \[ {\text{Option (C)}} \]