The mean and S.D of `1, 2, 3, 4, 5, 6` is

Options

A. `(7)/(2),sqrt((35)/(2))`
B. `3,3`
C. `(7)/(2),sqrt(3)`
D. `3,(35)/(12)`

Explanation

To find the mean and standard deviation of the data set \(1, 2, 3, 4, 5, 6\), 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 \(x_i\) represents each value in the data set and \(n\) is the number of values.

For our data set:
\[ \sum x_i = 1 + 2 + 3 + 4 + 5 + 6 = 21 \]
The number of values \(n = 6\).

Now, substituting the values into the mean formula:
\[ \bar{x} = \frac{21}{6} = \frac{7}{2} = 3.5 \]

### Step 2: Calculate the Standard Deviation

The standard deviation (\(SD\)) is calculated using the formula:
\[ SD = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} \]

First, we need to calculate \((x_i - \bar{x})^2\) for each \(x_i\):
- For \(x_1 = 1\): \((1 - 3.5)^2 = (-2.5)^2 = 6.25\)
- For \(x_2 = 2\): \((2 - 3.5)^2 = (-1.5)^2 = 2.25\)
- For \(x_3 = 3\): \((3 - 3.5)^2 = (-0.5)^2 = 0.25\)
- For \(x_4 = 4\): \((4 - 3.5)^2 = (0.5)^2 = 0.25\)
- For \(x_5 = 5\): \((5 - 3.5)^2 = (1.5)^2 = 2.25\)
- For \(x_6 = 6\): \((6 - 3.5)^2 = (2.5)^2 = 6.25\)

Now, summing these squared differences:
\[ \sum (x_i - \bar{x})^2 = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5 \]

Now, substituting into the standard deviation formula:
\[ SD = \sqrt{\frac{17.5}{6}} = \sqrt{2.9167} \approx 1.7078 \]

### Final Results

- Mean: \(3.5\)
- Standard Deviation: \(1.7078\)