Find the variance and S.D. of 3,5,7,9,11, 13

Answer

To find the variance and standard deviation of the given data set \(3, 5, 7, 9, 11, 13\), we will follow these steps:

### Step 1: Calculate the Mean
The mean (\(\mu\)) is calculated using the formula:
\[ \mu = \frac{\text{Sum of all observations}}{\text{Total number of observations}} \]

For our data:
\[ \mu = \frac{3 + 5 + 7 + 9 + 11 + 13}{6} \]
Calculating the sum:
\[ 3 + 5 + 7 + 9 + 11 + 13 = 48 \]
Now, divide by the number of observations (which is 6):
\[ \mu = \frac{48}{6} = 8 \]

### Step 2: Calculate the Variance
Variance (\(\sigma^2\)) is calculated using the formula:
\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \]
Where \(x_i\) are the observations, \(\mu\) is the mean, and \(n\) is the number of observations.

Now, we will calculate each term \((x_i - \mu)^2\):
- For \(x_1 = 3\): \((3 - 8)^2 = (-5)^2 = 25\)
- For \(x_2 = 5\): \((5 - 8)^2 = (-3)^2 = 9\)
- For \(x_3 = 7\): \((7 - 8)^2 = (-1)^2 = 1\)
- For \(x_4 = 9\): \((9 - 8)^2 = (1)^2 = 1\)
- For \(x_5 = 11\): \((11 - 8)^2 = (3)^2 = 9\)
- For \(x_6 = 13\): \((13 - 8)^2 = (5)^2 = 25\)

Now, sum these squared differences:
\[ 25 + 9 + 1 + 1 + 9 + 25 = 70 \]

Now, divide by the number of observations (6):
\[ \sigma^2 = \frac{70}{6} \approx 11.67 \]

### Step 3: Calculate the Standard Deviation
The standard deviation (\(\sigma\)) is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{11.67} \approx 3.414 \]

### Final Results
- Variance (\(\sigma^2\)) = 11.67
- Standard Deviation (\(\sigma\)) = 3.414

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