Find the variance of first 10 multiples of 3

Answer

To find the 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:
- \(3 \times 1 = 3\)
- \(3 \times 2 = 6\)
- \(3 \times 3 = 9\)
- \(3 \times 4 = 12\)
- \(3 \times 5 = 15\)
- \(3 \times 6 = 18\)
- \(3 \times 7 = 21\)
- \(3 \times 8 = 24\)
- \(3 \times 9 = 27\)
- \(3 \times 10 = 30\)

Thus, the multiples are: \(3, 6, 9, 12, 15, 18, 21, 24, 27, 30\).

### Step 2: Calculate the sum of these multiples
To find the sum of these multiples, we can use the formula for the sum of an arithmetic series:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Where:
- \(n = 10\) (number of terms)
- \(a = 3\) (first term)
- \(d = 3\) (common difference)

Calculating:
\[ S_{10} = \frac{10}{2} \times (2 \times 3 + (10-1) \times 3) \]
\[ = 5 \times (6 + 27) = 5 \times 33 = 165 \]

### Step 3: Calculate the mean
The mean (\(\mu\)) is given by:
\[ \mu = \frac{\text{Sum of the terms}}{n} \]
\[ \mu = \frac{165}{10} = 16.5 \]

### Step 4: Calculate the squared deviations from the mean
Now, we need to calculate the squared deviations from the mean for each term:
- \( (3 - 16.5)^2 = (-13.5)^2 = 182.25 \)
- \( (6 - 16.5)^2 = (-10.5)^2 = 110.25 \)
- \( (9 - 16.5)^2 = (-7.5)^2 = 56.25 \)
- \( (12 - 16.5)^2 = (-4.5)^2 = 20.25 \)
- \( (15 - 16.5)^2 = (-1.5)^2 = 2.25 \)
- \( (18 - 16.5)^2 = (1.5)^2 = 2.25 \)
- \( (21 - 16.5)^2 = (4.5)^2 = 20.25 \)
- \( (24 - 16.5)^2 = (7.5)^2 = 56.25 \)
- \( (27 - 16.5)^2 = (10.5)^2 = 110.25 \)
- \( (30 - 16.5)^2 = (13.5)^2 = 182.25 \)

### Step 5: Sum of squared deviations
Now, we sum all the squared deviations:
\[ 182.25 + 110.25 + 56.25 + 20.25 + 2.25 + 2.25 + 20.25 + 56.25 + 110.25 + 182.25 = 742.5 \]

### Step 6: Calculate the variance
The variance (\(\sigma^2\)) is given by:
\[ \sigma^2 = \frac{\text{Sum of squared deviations}}{n} \]
\[ \sigma^2 = \frac{742.5}{10} = 74.25 \]

### Final Answer
The variance of the first 10 multiples of 3 is \(74.25\).

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