Find the mean, variance and standard deviation for the numbers 4, 6, 10, 12, 7, 8, 13, 12.

Answer

To find the mean, variance, and standard deviation for the numbers 4, 6, 10, 12, 7, 8, 13, and 12, we will follow these steps:

### Step 1: Calculate the Mean
The mean (average) is calculated using the formula:
\[ \text{Mean} ( \bar{x} ) = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

First, we need to find the sum of the observations:
\[ 4 + 6 + 10 + 12 + 7 + 8 + 13 + 12 = 72 \]

Next, we count the number of observations:
\[ \text{Number of observations} = 8 \]

Now we can calculate the mean:
\[ \bar{x} = \frac{72}{8} = 9 \]

### Step 2: Calculate the Variance
Variance is calculated using the formula:
\[ \text{Variance} ( \sigma^2 ) = \frac{\sum (x_i - \bar{x})^2}{n} \]

Where \( x_i \) represents each observation, \( \bar{x} \) is the mean, and \( n \) is the number of observations.

We will calculate \( (x_i - \bar{x})^2 \) for each observation:

1. \( (4 - 9)^2 = (-5)^2 = 25 \)
2. \( (6 - 9)^2 = (-3)^2 = 9 \)
3. \( (10 - 9)^2 = (1)^2 = 1 \)
4. \( (12 - 9)^2 = (3)^2 = 9 \)
5. \( (7 - 9)^2 = (-2)^2 = 4 \)
6. \( (8 - 9)^2 = (-1)^2 = 1 \)
7. \( (13 - 9)^2 = (4)^2 = 16 \)
8. \( (12 - 9)^2 = (3)^2 = 9 \)

Now, we sum these squared differences:
\[ 25 + 9 + 1 + 9 + 4 + 1 + 16 + 9 = 74 \]

Now we can calculate the variance:
\[ \sigma^2 = \frac{74}{8} = 9.25 \]

### Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{9.25} \approx 3.04 \]

### Final Results
- Mean: \( 9 \)
- Variance: \( 9.25 \)
- Standard Deviation: \( 3.04 \)