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

Options

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

Answer

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

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

\[ \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total count of numbers}} \]

Calculating the sum:

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

Now, divide by the total count of numbers, which is 5:

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

### Step 2: Calculate the Deviations from the Mean
Next, we calculate the deviation of each number from the mean:

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

### Step 3: Square the Deviations
Now, we square each of the deviations:

- For -4: \( (-4)^2 = 16 \)
- For -2: \( (-2)^2 = 4 \)
- For 0: \( (0)^2 = 0 \)
- For 2: \( (2)^2 = 4 \)
- For 4: \( (4)^2 = 16 \)

### Step 4: Calculate the Variance
Next, we find the variance by calculating the average of these squared deviations.

\[ \text{Variance} = \frac{\text{Sum of squared deviations}}{\text{Total count of numbers}} \]

Calculating the sum of squared deviations:

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

Now, divide by the total count of numbers (5):

\[ \text{Variance} = \frac{40}{5} = 8 \]

### 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**.

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