The standard deviation of data 6,5,9,13,12,8 and 10 is

Options

A. `sqrt((52)/(7))`
B. `(52)/(7)`
C. `sqrt(6)`
D. 6

Explanation

To find the standard deviation of the data set \(6, 5, 9, 13, 12, 8, 10\), we will follow these steps:

### Step 1: Calculate the Mean
First, we need to calculate the mean (\(\bar{x}\)) of the data set.

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

Where:
- \(\sum x_i\) is the sum of all data points.
- \(n\) is the number of data points.

Calculating the sum of the data points:

\[ \sum x_i = 6 + 5 + 9 + 13 + 12 + 8 + 10 = 63 \]

The number of data points \(n\) is 7.

Now, calculate the mean:

\[ \bar{x} = \frac{63}{7} = 9 \]

### Step 2: Calculate Each Deviation from the Mean
Next, we calculate the deviation of each data point from the mean and then square each deviation.

\[ \begin{align*} (6 - 9)^2 & = (-3)^2 = 9 \\ (5 - 9)^2 & = (-4)^2 = 16 \\ (9 - 9)^2 & = (0)^2 = 0 \\ (13 - 9)^2 & = (4)^2 = 16 \\ (12 - 9)^2 & = (3)^2 = 9 \\ (8 - 9)^2 & = (-1)^2 = 1 \\ (10 - 9)^2 & = (1)^2 = 1 \\ \end{align*} \]

### Step 3: Sum of Squared Deviations
Now, we sum these squared deviations:

\[ \sum (x_i - \bar{x})^2 = 9 + 16 + 0 + 16 + 9 + 1 + 1 = 52 \]

### Step 4: Calculate Variance
The variance (\(\sigma^2\)) is given by:

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

Substituting the values:

\[ \sigma^2 = \frac{52}{7} \]

### Step 5: Calculate Standard Deviation
The standard deviation (\(\sigma\)) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{52}{7}} = \frac{\sqrt{52}}{\sqrt{7}} = \frac{2\sqrt{13}}{\sqrt{7}} \]

### Final Answer
Thus, the standard deviation of the data set \(6, 5, 9, 13, 12, 8, 10\) is:

\[ \sqrt{\frac{52}{7}} \]

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