Find the mean and variance of the following data : 2,4,5,6,8,17

Answer

To find the mean and variance of the data set \(2, 4, 5, 6, 8, 17\), we will follow these steps:

### Step 1: Calculate the Mean

The formula for the mean (\(\bar{x}\)) is given by:

\[ \bar{x} = \frac{\Sigma x_i}{n} \]

where \(\Sigma x_i\) is the sum of all observations and \(n\) is the number of observations.

1. **Find the sum of the observations**:
\[ \Sigma x_i = 2 + 4 + 5 + 6 + 8 + 17 \]
\[ \Sigma x_i = 42 \]

2. **Count the number of observations**:
\[ n = 6 \]

3. **Calculate the mean**:
\[ \bar{x} = \frac{42}{6} = 7 \]

### Step 2: Calculate the Variance

The formula for variance (\(s^2\)) is given by:

\[ s^2 = \frac{\Sigma x_i^2}{n} - \left(\frac{\Sigma x_i}{n}\right)^2 \]

1. **Calculate \(\Sigma x_i^2\)**:
We need to square each observation and then sum them up:
\[ \Sigma x_i^2 = 2^2 + 4^2 + 5^2 + 6^2 + 8^2 + 17^2 \]
\[ = 4 + 16 + 25 + 36 + 64 + 289 \]
\[ = 434 \]

2. **Calculate the variance**:
- First, calculate \(\frac{\Sigma x_i}{n}\):
\[ \frac{\Sigma x_i}{n} = \frac{42}{6} = 7 \]
- Now, square this value:
\[ \left(\frac{\Sigma x_i}{n}\right)^2 = 7^2 = 49 \]
- Finally, substitute into the variance formula:
\[ s^2 = \frac{434}{6} - 49 \]
\[ = 72.3333 - 49 \]
\[ = 23.3333 \]

### Final Results

- Mean: \(7\)
- Variance: \(23.3333\)