Consider the following statements:
If 10 is added to each entry on a list then the average increase by 10.
II. If 10 is added to each entry on a list, then the standard deviation increase by 10
III If each entry on a list is doubled, then the average doubles Which of the above statement are correct?

Options

A. A. I, II and III
B. B. I and II only
C. C. I and III only
D. D. II and III only

Answer

To solve the problem, we will analyze each of the given statements one by one.

### Step 1: Analyze Statement I
**Statement I:** If 10 is added to each entry on a list, then the average increases by 10.

Let’s denote the entries of the list as \( a, b, c \). The average (mean) of these entries is calculated as:
\[ \text{Average} = \frac{a + b + c}{3} \]

Now, if we add 10 to each entry, the new entries become \( a + 10, b + 10, c + 10 \). The new average will be:
\[ \text{New Average} = \frac{(a + 10) + (b + 10) + (c + 10)}{3} = \frac{a + b + c + 30}{3} = \frac{a + b + c}{3} + 10 \]

Thus, the new average is the old average plus 10. Therefore, **Statement I is correct.**

### Step 2: Analyze Statement II
**Statement II:** If 10 is added to each entry on a list, then the standard deviation increases by 10.

The standard deviation measures the spread of the data points from the mean. When a constant value (like 10) is added to each entry, the spread of the data does not change; it only shifts the entire dataset. Hence, the standard deviation remains unaffected.

Therefore, **Statement II is incorrect.**

### Step 3: Analyze Statement III
**Statement III:** If each entry on a list is doubled, then the average doubles.

Using the same entries \( a, b, c \), the average is:
\[ \text{Average} = \frac{a + b + c}{3} \]

If we double each entry, the new entries become \( 2a, 2b, 2c \). The new average will be:
\[ \text{New Average} = \frac{2a + 2b + 2c}{3} = \frac{2(a + b + c)}{3} = 2 \times \frac{a + b + c}{3} \]

Thus, the new average is twice the old average. Therefore, **Statement III is correct.**

### Conclusion
From the analysis:
- Statement I is correct.
- Statement II is incorrect.
- Statement III is correct.

Thus, the correct statements are I and III.

### Final Answer
The correct statements are I and III.

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