The standard deviation of some consecutive integers is found to be 2. Which of the following statements best describes the nature of the consecutive integers?

Options

A. The integers are any set of eight consecutive integers
B. The integers are any set of eight consecutive positive integers
C. The integers are any set of seven consecutive integers
D. None of the above

Explanation

To solve the problem, we need to determine which statement best describes the nature of the consecutive integers given that their standard deviation is 2.

### Step-by-Step Solution:

1. **Understanding Consecutive Integers**:
- Let the first integer be \( x \). Therefore, the consecutive integers can be represented as:
\[ x, x+1, x+2, x+3, x+4, x+5, x+6, \ldots \]

2. **Choosing a Set of Consecutive Integers**:
- We will consider a set of 7 consecutive integers. Thus, the integers are:
\[ x, x+1, x+2, x+3, x+4, x+5, x+6 \]

3. **Calculating the Mean**:
- The mean \( \mu \) of these integers is calculated as follows:
\[ \text{Mean} = \frac{\text{Sum of integers}}{\text{Number of integers}} = \frac{x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6)}{7} \]
- The sum simplifies to:
\[ 7x + (0 + 1 + 2 + 3 + 4 + 5 + 6) = 7x + 21 \]
- Therefore, the mean is:
\[ \mu = \frac{7x + 21}{7} = x + 3 \]

4. **Calculating the Variance**:
- The variance \( \sigma^2 \) is calculated using the formula:
\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \]
- Here, \( n = 7 \) and \( x_i \) are the consecutive integers. We calculate \( (x_i - \mu)^2 \) for each integer:
- For \( x \): \( (x - (x + 3))^2 = (-3)^2 = 9 \)
- For \( x+1 \): \( (x + 1 - (x + 3))^2 = (-2)^2 = 4 \)
- For \( x+2 \): \( (x + 2 - (x + 3))^2 = (-1)^2 = 1 \)
- For \( x+3 \): \( (x + 3 - (x + 3))^2 = 0^2 = 0 \)
- For \( x+4 \): \( (x + 4 - (x + 3))^2 = 1^2 = 1 \)
- For \( x+5 \): \( (x + 5 - (x + 3))^2 = 2^2 = 4 \)
- For \( x+6 \): \( (x + 6 - (x + 3))^2 = 3^2 = 9 \)

5. **Summing the Squared Differences**:
- Now, we sum these squared differences:
\[ 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28 \]

6. **Calculating the Variance**:
- The variance is:
\[ \sigma^2 = \frac{28}{7} = 4 \]

7. **Calculating the Standard Deviation**:
- The standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{4} = 2 \]

8. **Conclusion**:
- Since we have shown that the standard deviation of 7 consecutive integers is indeed 2, the correct statement is:
- The integers are any set of 7 consecutive integers.

### Final Answer:
The correct option is: **The integers are any set of 7 consecutive integers.**