Variance of the data 2,4,6,8,10 is

Options

A. 6
B. 7
C. 8
D. None of these

Explanation

To find the variance of the data set \(2, 4, 6, 8, 10\), we will follow these steps:

### Step 1: Calculate the Mean (\( \bar{x} \))
The mean is calculated using the formula:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]
where \(n\) is the number of data points and \(x_i\) are the data points.

For our data:
\[ \bar{x} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6 \]

### Step 2: Calculate the Squared Differences from the Mean
Next, we calculate the squared differences from the mean for each data point:
\[ (x_i - \bar{x})^2 \]
- For \(2\): \((2 - 6)^2 = (-4)^2 = 16\)
- For \(4\): \((4 - 6)^2 = (-2)^2 = 4\)
- For \(6\): \((6 - 6)^2 = (0)^2 = 0\)
- For \(8\): \((8 - 6)^2 = (2)^2 = 4\)
- For \(10\): \((10 - 6)^2 = (4)^2 = 16\)

### Step 3: Sum the Squared Differences
Now, we sum all the squared differences:
\[ \sum (x_i - \bar{x})^2 = 16 + 4 + 0 + 4 + 16 = 40 \]

### Step 4: Calculate the Variance
Finally, we calculate the variance using the formula:
\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]
Substituting the values we have:
\[ \sigma^2 = \frac{40}{5} = 8 \]

### Conclusion
The variance of the data set \(2, 4, 6, 8, 10\) is \(8\).

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