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

Options

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

Answer

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:
\[ \bar{x} = \frac{\sum x}{n} \]
where \(\sum x\) is the sum of all data points and \(n\) is the number of data points.

1. **Sum of the data points:**
\[ \sum x = 1 + 2 + 3 + 4 + 5 + 6 = 21 \]

2. **Number of data points:**
\[ n = 6 \]

3. **Calculate the mean:**
\[ \bar{x} = \frac{21}{6} = \frac{7}{2} \]

### Step 2: Calculate the Standard Deviation
The standard deviation (\(\sigma\)) is calculated using the formula:
\[ \sigma = \sqrt{\frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2} \]

1. **Calculate \( \sum x^2 \):**
\[ x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91 \]

2. **Calculate the first part of the standard deviation formula:**
\[ \frac{\sum x^2}{n} = \frac{91}{6} \]

3. **Calculate the second part of the standard deviation formula:**
\[ \left(\frac{\sum x}{n}\right)^2 = \left(\frac{21}{6}\right)^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]

4. **Combine these to find the standard deviation:**
\[ \sigma = \sqrt{\frac{91}{6} - \frac{49}{4}} \]

5. **Find a common denominator to subtract:**
The least common multiple of 6 and 4 is 12.
\[ \frac{91}{6} = \frac{182}{12}, \quad \frac{49}{4} = \frac{147}{12} \]

6. **Subtract the two fractions:**
\[ \frac{182}{12} - \frac{147}{12} = \frac{35}{12} \]

7. **Take the square root:**
\[ \sigma = \sqrt{\frac{35}{12}} \]

### Final Results
- Mean: \(\frac{7}{2}\)
- Standard Deviation: \(\sqrt{\frac{35}{12}}\)

### Summary
The mean of the data set \(1, 2, 3, 4, 5, 6\) is \(\frac{7}{2}\) and the standard deviation is \(\sqrt{\frac{35}{12}}\).