What is the standard deviation of numbers 7, 9, 11, 13, 15?

Options

A. 2.2
B. 2.4
C. 2.6
D. 2.8

Explanation

To find the standard deviation of the numbers 7, 9, 11, 13, and 15, we will follow these steps:

### Step 1: Calculate the Mean
First, we need to calculate the mean (average) of the given numbers.

\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Total number of observations}} \]

Calculating the sum:

\[ 7 + 9 + 11 + 13 + 15 = 55 \]

Now, divide by the total number of observations (which is 5):

\[ \text{Mean} = \frac{55}{5} = 11 \]

### Step 2: Calculate the Deviations from the Mean
Next, we will find the deviation of each number from the mean and then square those deviations.

- For 7: \(7 - 11 = -4\) and \((-4)^2 = 16\)
- For 9: \(9 - 11 = -2\) and \((-2)^2 = 4\)
- For 11: \(11 - 11 = 0\) and \(0^2 = 0\)
- For 13: \(13 - 11 = 2\) and \(2^2 = 4\)
- For 15: \(15 - 11 = 4\) and \(4^2 = 16\)

### Step 3: Sum of Squared Deviations
Now, we will sum these squared deviations:

\[ 16 + 4 + 0 + 4 + 16 = 40 \]

### Step 4: Calculate the Variance
To find the variance, we divide the sum of squared deviations by the number of observations:

\[ \text{Variance} = \frac{\text{Sum of squared deviations}}{\text{Total number of observations}} = \frac{40}{5} = 8 \]

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

\[ \text{Standard Deviation} = \sqrt{8} \approx 2.83 \]

### Final Answer
The standard deviation of the numbers 7, 9, 11, 13, and 15 is approximately \(2.83\).

---