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

Options

A. 2.4
B. 2.5
C. 2.7
D. 2.8

Explanation

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

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

\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Total number of observations}} = \frac{7 + 9 + 11 + 13 + 15}{5} \]

Calculating the sum:

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

Now, divide by the number of observations:

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

### Step 2: Calculate the Deviations from the Mean
Next, we find the deviation of each observation from the mean and square it.

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

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

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

### Step 4: Calculate the Variance
The variance is the average of the squared deviations. Since we have 5 observations, we divide by 5:

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

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

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

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

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