The standard deviation of the observations 22, 26, 28, 20, 24, 30 is

Options

A. 2
B. 2.4
C. 3
D. 3.42

Answer

To find the standard deviation of the observations 22, 26, 28, 20, 24, and 30, we can follow these steps:

### Step 1: Calculate the Mean
The mean (average) of a set of observations is calculated by summing all the observations and dividing by the total number of observations.

\[ \text{Mean} ( \bar{x} ) = \frac{\sum x_i}{n} \]

Where:
- \( \sum x_i \) is the sum of all observations.
- \( n \) is the number of observations.

Calculating the mean:

\[ \sum x_i = 22 + 26 + 28 + 20 + 24 + 30 = 150 \]
\[ n = 6 \]
\[ \bar{x} = \frac{150}{6} = 25 \]

### Step 2: Calculate the Squared Deviations
Next, we need to calculate the squared deviations from the mean for each observation. This is done using the formula:

\[ (x_i - \bar{x})^2 \]

Calculating the squared deviations:

1. For \( x_1 = 22 \):
\[ (22 - 25)^2 = (-3)^2 = 9 \]
2. For \( x_2 = 26 \):
\[ (26 - 25)^2 = (1)^2 = 1 \]
3. For \( x_3 = 28 \):
\[ (28 - 25)^2 = (3)^2 = 9 \]
4. For \( x_4 = 20 \):
\[ (20 - 25)^2 = (-5)^2 = 25 \]
5. For \( x_5 = 24 \):
\[ (24 - 25)^2 = (-1)^2 = 1 \]
6. For \( x_6 = 30 \):
\[ (30 - 25)^2 = (5)^2 = 25 \]

### Step 3: Sum of Squared Deviations
Now, we sum all the squared deviations:

\[ \sum (x_i - \bar{x})^2 = 9 + 1 + 9 + 25 + 1 + 25 = 70 \]

### Step 4: Calculate the Variance
The variance is calculated by dividing the sum of squared deviations by the number of observations:

\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n} \]
\[ \text{Variance} = \frac{70}{6} \approx 11.67 \]

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

\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{11.67} \approx 3.42 \]

### Final Answer
Thus, the standard deviation of the observations 22, 26, 28, 20, 24, and 30 is approximately **3.42**.

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