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

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.