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, and 10, we will follow these steps:

### Step 1: Calculate the Mean
First, we need to find the mean (average) of the data set.

\[ \text{Mean} = \frac{\sum x_i}{n} \]

Where \( n \) is the number of data points.

\[ \sum x_i = 6 + 5 + 9 + 13 + 12 + 8 + 10 = 63 \]
\[ n = 7 \]
\[ \text{Mean} = \frac{63}{7} = 9 \]

### 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: Calculate the Sum of Squared Deviations
Now, we sum up all the squared deviations.

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

### Step 4: Calculate the Variance
The variance is the average of the squared deviations. Since we are working with a sample, we will divide by \( n - 1 \).

\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n - 1} = \frac{52}{7 - 1} = \frac{52}{6} \approx 8.67 \]

### Step 5: Calculate the Standard Deviation
The standard deviation is the square root of the variance.

\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{\frac{52}{6}} = \frac{\sqrt{52}}{\sqrt{6}} \approx \sqrt{8.67} \approx 2.94 \]

### Final Answer
Thus, the standard deviation of the data set is approximately \( \sqrt{\frac{52}{6}} \).

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