The variance of a set of data points measures the spread or dispersion of the data points around the mean. The formula for the variance of a set of data points \(x_1, x_2, \ldots, x_n\) with mean \(\mu\) is given by:

\(\text{Variance} = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}\)

In this case, the data points are \(x, 2x, 3x, 4x,\) and \(5x\), and the mean of these data points is:

\(\mu = \frac{x + 2x + 3x + 4x + 5x}{5} = \frac{15x}{5} = 3x\)

Substituting the values into the variance formula:

\(\text{Variance} = \frac{(x - 3x)^2 + (2x - 3x)^2 + (3x - 3x)^2 + (4x - 3x)^2 + (5x - 3x)^2}{5}\)

Simplify the expressions inside the parentheses:

\(\text{Variance} = \frac{(-2x)^2 + (-x)^2 + (0)^2 + (x)^2 + (2x)^2}{5}\)

\(\text{Variance} = \frac{4x^2 + x^2 + 0 + x^2 + 4x^2}{5}\)

\(\text{Variance} = \frac{10x^2}{5} = 2x^2\)

Therefore, the variance of the data points \(x, 2x, 3x, 4x,\) and \(5x\) is \(2x^2\).