Find the mean and variance of first 10 multiples of 3.

Answer

To find the mean and variance of the first 10 multiples of 3, we will follow these steps:

### Step 1: Identify the first 10 multiples of 3.
The first 10 multiples of 3 are:
- 3 × 1 = 3
- 3 × 2 = 6
- 3 × 3 = 9
- 3 × 4 = 12
- 3 × 5 = 15
- 3 × 6 = 18
- 3 × 7 = 21
- 3 × 8 = 24
- 3 × 9 = 27
- 3 × 10 = 30

So, the observations are: **3, 6, 9, 12, 15, 18, 21, 24, 27, 30**.

### Step 2: Calculate the mean.
The formula for the mean (average) is given by:

\[ \text{Mean} = \frac{\sum x_i}{n} \]

Where:
- \( \sum x_i \) is the sum of all observations.
- \( n \) is the total number of observations.

Calculating the sum of the observations:

\[ \sum x_i = 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30 = 165 \]

Now, substitute the values into the mean formula:

\[ \text{Mean} = \frac{165}{10} = 16.5 \]

### Step 3: Calculate the variance.
The formula for variance is:

\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n} \]

Where:
- \( x_i \) are the observations,
- \( \bar{x} \) is the mean,
- \( n \) is the total number of observations.

First, we need to calculate \( x_i - \bar{x} \) for each observation:

\[ \begin{align*} 3 - 16.5 & = -13.5 \\ 6 - 16.5 & = -10.5 \\ 9 - 16.5 & = -7.5 \\ 12 - 16.5 & = -4.5 \\ 15 - 16.5 & = -1.5 \\ 18 - 16.5 & = 1.5 \\ 21 - 16.5 & = 4.5 \\ 24 - 16.5 & = 7.5 \\ 27 - 16.5 & = 10.5 \\ 30 - 16.5 & = 13.5 \\ \end{align*} \]

Next, we calculate \( (x_i - \bar{x})^2 \):

\[ \begin{align*} (-13.5)^2 & = 182.25 \\ (-10.5)^2 & = 110.25 \\ (-7.5)^2 & = 56.25 \\ (-4.5)^2 & = 20.25 \\ (-1.5)^2 & = 2.25 \\ (1.5)^2 & = 2.25 \\ (4.5)^2 & = 20.25 \\ (7.5)^2 & = 56.25 \\ (10.5)^2 & = 110.25 \\ (13.5)^2 & = 182.25 \\ \end{align*} \]

Now, sum these squared differences:

\[ \sum (x_i - \bar{x})^2 = 182.25 + 110.25 + 56.25 + 20.25 + 2.25 + 2.25 + 20.25 + 56.25 + 110.25 + 182.25 = 742.5 \]

Finally, substitute this into the variance formula:

\[ \text{Variance} = \frac{742.5}{10} = 74.25 \]

### Final Answers:
- **Mean**: 16.5
- **Variance**: 74.25