The variance of 6,8,10,12,14,16,18,20,22,24 is

Options

A. 33
B. 37
C. 45
D. 54

Answer

To calculate the variance of the data set \(6, 8, 10, 12, 14, 16, 18, 20, 22, 24\), we will follow these steps:

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

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

where \( \sum x_i \) is the sum of all data points and \( n \) is the number of data points.

- Sum of the data points:
\[ 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 = 150 \]

- Number of data points (\( n \)):
\[ n = 10 \]

Now, calculate the mean:
\[ \bar{x} = \frac{150}{10} = 15 \]

### Step 2: Calculate the Variance
The variance is calculated using the formula:

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

Now, we will calculate \( (x_i - \bar{x})^2 \) 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 \( x_{10} = 24 \):
\[ (24 - 15)^2 = (9)^2 = 81 \]

### Step 3: Sum of Squared Deviations
Now, sum all the squared deviations:
\[ \sum (x_i - \bar{x})^2 = 81 + 49 + 25 + 9 + 1 + 1 + 9 + 25 + 49 + 81 = 330 \]

### Step 4: Calculate the Variance
Now, substitute back into the variance formula:
\[ \sigma^2 = \frac{330}{10} = 33 \]

### Final Answer
The variance of the data set \(6, 8, 10, 12, 14, 16, 18, 20, 22, 24\) is \(33\).

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