To find the variance of the numbers \( x, 2x, 3x, 4x, \) and \( 5x \), we will follow a systematic approach. Let's break it down step-by-step. ### Step 1: Understanding Variance Variance is a measure of how much the values in a dataset differ from the mean (average) of that dataset. The formula for variance \( \sigma^2 \) for a set of values \( x_1, x_2, \ldots, x_n \) is given by: \[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \] where \( \mu \) is the mean of the dataset, and \( n \) is the number of values. ### Step 2: Calculate the Mean First, we need to calculate the mean \( \mu \) of the numbers \( x, 2x, 3x, 4x, \) and \( 5x \). 1. **Sum of the values**: \[ x + 2x + 3x + 4x + 5x = 15x \] 2. **Number of values**: There are 5 values. 3. **Mean**: \[ \mu = \frac{15x}{5} = 3x \] ### Step 3: Calculate the Variance Now, we will calculate the variance using the mean we just found. 1. **Calculate each squared deviation from the mean**: - For \( x \): \[ (x - 3x)^2 = (-2x)^2 = 4x^2 \] - For \( 2x \): \[ (2x - 3x)^2 = (-x)^2 = x^2 \] - For \( 3x \): \[ (3x - 3x)^2 = 0^2 = 0 \] - For \( 4x \): \[ (4x - 3x)^2 = (x)^2 = x^2 \] - For \( 5x \): \[ (5x - 3x)^2 = (2x)^2 = 4x^2 \] 2. **Sum of squared deviations**: \[ 4x^2 + x^2 + 0 + x^2 + 4x^2 = 10x^2 \] 3. **Calculate the variance**: \[ \sigma^2 = \frac{1}{5} \times 10x^2 = 2x^2 \] ### Conclusion The variance of the numbers \( x, 2x, 3x, 4x, \) and \( 5x \) is \( 2x^2 \). ### Explanation of Options - **Option A: \( x\sqrt{2} \)** - This option is incorrect because it does not represent the calculated variance. The variance is a function of \( x^2 \), not \( x\sqrt{2} \). - **Option B: \( 2x^2 \)** - This is the correct answer as derived from our calculations. - **Option C: \( x^2 \)** - This option is incorrect because it underestimates the variance. The variance accounts for the spread of the data, which is greater than just \( x^2 \). - **Option D: \( 3x \)** - This option is incorrect as it does not represent a measure of variance. Variance is expressed in squared units, not linear units like \( 3x \). ### Revision Summary - Variance measures how much data points differ from the mean. - The mean of \( x, 2x, 3x, 4x, 5x \) is \( 3x \). - The variance is calculated as \( \sigma^2 = \frac{1}{n} \sum (x_i - \mu)^2 \). - The variance of the given numbers is \( 2x^2 \).