If the variance of the following data `:` 6,8,10,12,14,16,18,20,22,24 is K, then the value of `(K)/( 11)` is

Answer

To find the value of \( \frac{K}{11} \) where \( K \) is the variance of the given data set \( 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 \), we will follow these steps:

### Step 1: Calculate the Mean
The mean \( \bar{x} \) is calculated using the formula:

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

Where \( n \) is the number of data points. Here, the data points are:

\[ 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 \]

Calculating the sum:

\[ \sum x_i = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 = 150 \]

Since there are 10 data points (\( n = 10 \)):

\[ \bar{x} = \frac{150}{10} = 15 \]

### Step 2: Calculate the Variance
The variance \( K \) is given by the formula:

\[ K = \frac{\sum (x_i - \bar{x})^2}{n} \]

We need to compute \( (x_i - \bar{x})^2 \) for each data point:

- For \( x_1 = 6 \): \( (6 - 15)^2 = (-9)^2 = 81 \)
- For \( x_2 = 8 \): \( (8 - 15)^2 = (-7)^2 = 49 \)
- For \( x_3 = 10 \): \( (10 - 15)^2 = (-5)^2 = 25 \)
- For \( x_4 = 12 \): \( (12 - 15)^2 = (-3)^2 = 9 \)
- For \( x_5 = 14 \): \( (14 - 15)^2 = (-1)^2 = 1 \)
- For \( x_6 = 16 \): \( (16 - 15)^2 = (1)^2 = 1 \)
- For \( x_7 = 18 \): \( (18 - 15)^2 = (3)^2 = 9 \)
- For \( x_8 = 20 \): \( (20 - 15)^2 = (5)^2 = 25 \)
- For \( x_9 = 22 \): \( (22 - 15)^2 = (7)^2 = 49 \)
- For \( x_{10} = 24 \): \( (24 - 15)^2 = (9)^2 = 81 \)

Now, summing these squared differences:

\[ \sum (x_i - \bar{x})^2 = 81 + 49 + 25 + 9 + 1 + 1 + 9 + 25 + 49 + 81 = 330 \]

Now we can calculate the variance:

\[ K = \frac{330}{10} = 33 \]

### Step 3: Calculate \( \frac{K}{11} \)
Now, we need to find \( \frac{K}{11} \):

\[ \frac{K}{11} = \frac{33}{11} = 3 \]

### Final Answer
Thus, the value of \( \frac{K}{11} \) is \( \boxed{3} \).