Calculate the variance and standard deviation of the observations : 11, 12, 13, . . . .. 20.

Answer

To calculate the variance and standard deviation of the observations 11, 12, 13, ..., 20, we will follow these steps:

### Step 1: List the observations
The observations given are:
\[ 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 \]

### Step 2: Calculate the mean (\( \bar{x} \))
The mean is calculated using the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \( n \) is the number of observations.

Calculating the sum:
\[ \sum x_i = 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 155 \]
The number of observations \( n = 10 \).

Now, calculating the mean:
\[ \bar{x} = \frac{155}{10} = 15.5 \]

### Step 3: Calculate \( x_i - \bar{x} \) for each observation
Now we will calculate \( x_i - \bar{x} \) for each observation:
- For \( 11 \): \( 11 - 15.5 = -4.5 \)
- For \( 12 \): \( 12 - 15.5 = -3.5 \)
- For \( 13 \): \( 13 - 15.5 = -2.5 \)
- For \( 14 \): \( 14 - 15.5 = -1.5 \)
- For \( 15 \): \( 15 - 15.5 = -0.5 \)
- For \( 16 \): \( 16 - 15.5 = 0.5 \)
- For \( 17 \): \( 17 - 15.5 = 1.5 \)
- For \( 18 \): \( 18 - 15.5 = 2.5 \)
- For \( 19 \): \( 19 - 15.5 = 3.5 \)
- For \( 20 \): \( 20 - 15.5 = 4.5 \)

### Step 4: Calculate \( (x_i - \bar{x})^2 \)
Now we will square each of the results from Step 3:
- For \( 11 \): \( (-4.5)^2 = 20.25 \)
- For \( 12 \): \( (-3.5)^2 = 12.25 \)
- For \( 13 \): \( (-2.5)^2 = 6.25 \)
- For \( 14 \): \( (-1.5)^2 = 2.25 \)
- For \( 15 \): \( (-0.5)^2 = 0.25 \)
- For \( 16 \): \( (0.5)^2 = 0.25 \)
- For \( 17 \): \( (1.5)^2 = 2.25 \)
- For \( 18 \): \( (2.5)^2 = 6.25 \)
- For \( 19 \): \( (3.5)^2 = 12.25 \)
- For \( 20 \): \( (4.5)^2 = 20.25 \)

### Step 5: Calculate the sum of squares
Now we will sum these squared differences:
\[ \sum (x_i - \bar{x})^2 = 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 = 112.5 \]

### Step 6: Calculate the variance (\( \sigma^2 \))
The variance is given by:
\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]
Substituting the values:
\[ \sigma^2 = \frac{112.5}{10} = 11.25 \]

### Step 7: Calculate the standard deviation (\( \sigma \))
The standard deviation is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{11.25} \approx 3.3541 \]

### Final Results
- Variance: \( 11.25 \)
- Standard Deviation: \( 3.3541 \)