Find the variance of the numbers 3,7,10,18,22.

Answer

To find the variance of the numbers 3, 7, 10, 18, and 22, we will follow these steps:

### Step 1: Calculate the Mean (μ)
The mean (μ) is calculated using the formula:
\[ \mu = \frac{\sum x_i}{n} \]
where \(x_i\) are the numbers and \(n\) is the total number of values.

- Here, the numbers are: 3, 7, 10, 18, 22
- Calculate the sum:
\[ 3 + 7 + 10 + 18 + 22 = 70 \]
- The total number of values \(n\) is 5.

Now, calculate the mean:
\[ \mu = \frac{70}{5} = 14 \]

### Step 2: Calculate the Variance (σ²)
The variance is calculated using the formula:
\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \]
We will calculate each term \((x_i - \mu)^2\):

1. For \(x_1 = 3\):
\[ (3 - 14)^2 = (-11)^2 = 121 \]
2. For \(x_2 = 7\):
\[ (7 - 14)^2 = (-7)^2 = 49 \]
3. For \(x_3 = 10\):
\[ (10 - 14)^2 = (-4)^2 = 16 \]
4. For \(x_4 = 18\):
\[ (18 - 14)^2 = (4)^2 = 16 \]
5. For \(x_5 = 22\):
\[ (22 - 14)^2 = (8)^2 = 64 \]

Now, sum these squared differences:
\[ 121 + 49 + 16 + 16 + 64 = 266 \]

Now, calculate the variance:
\[ \sigma^2 = \frac{266}{5} = 53.2 \]

### Final Answer
The variance of the numbers 3, 7, 10, 18, and 22 is **53.2**.

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