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# Find the variance of the numbers k, k + 1, k + 2.

### Question

Find the variance of the numbers k, k + 1, k + 2.

### Options

A)
1/3
B)
3
C)
2/3 D)
1

### Explanation:

To understand this question, it's important to know what variance is. Variance is a statistical measurement of the spread between numbers in a data set. It measures how far each number in the set is from the mean (average) and, thus, from every other number in the set.

The formula for variance for a sample is:

$\frac{\sum (x_i - \mu)^2}{n}$

Where:

• $$x_i$$ is each value from the data set
• $$\mu$$ is the mean of the data set
• $$n$$ is the number of data points

In this case, our data set consists of the numbers $$k$$, $$k + 1$$, and $$k + 2$$. First, let's find the mean, $$\mu$$:

$\mu = \frac{k + (k + 1) + (k + 2)}{3} = \frac{3k + 3}{3} = k + 1$

Next, we substitute these values into the formula for variance:

$\frac{(k - (k + 1))^2 + ((k + 1) - (k + 1))^2 + ((k + 2) - (k + 1))^2}{3} = \frac{(-1)^2 + 0^2 + 1^2}{3} = \frac{1 + 0 + 1}{3} = \frac{2}{3}$

Therefore, the variance of the numbers $$k$$, $$k + 1$$, and $$k + 2$$ is $$\frac{2}{3}$$, which corresponds to Option C.