Calculate variance and standard deviation of the following data : 10, 12, 8, 14, 16

Answer

To calculate the variance and standard deviation of the given data set \(10, 12, 8, 14, 16\), we will follow these steps:

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

For our data:
\[ \bar{x} = \frac{10 + 12 + 8 + 14 + 16}{5} = \frac{60}{5} = 12 \]

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

1. For \(10\): \((10 - 12)^2 = (-2)^2 = 4\)
2. For \(12\): \((12 - 12)^2 = (0)^2 = 0\)
3. For \(8\): \((8 - 12)^2 = (-4)^2 = 16\)
4. For \(14\): \((14 - 12)^2 = (2)^2 = 4\)
5. For \(16\): \((16 - 12)^2 = (4)^2 = 16\)

Now, sum these squared differences:
\[ \sum (x_i - \bar{x})^2 = 4 + 0 + 16 + 4 + 16 = 40 \]

Now, divide by the number of observations \(n = 5\):
\[ \sigma^2 = \frac{40}{5} = 8 \]

### Step 3: Calculate the Standard Deviation
Standard deviation is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{8} \approx 2.83 \]

### Final Answers
- Variance: \(8\)
- Standard Deviation: \(\sqrt{8} \approx 2.83\)