The standard deviation for the data 7, 11, 15 is

Options

A. 2.4
B. 2.5
C. 2.7
D. 3.26

Answer

To find the standard deviation for the data set 7, 11, and 15, we will follow these steps:

### Step 1: Calculate the Mean
The mean (average) is calculated using the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \( \sum x_i \) is the sum of all observations and \( n \) is the number of observations.

For our data:
\[ \sum x_i = 7 + 11 + 15 = 33 \]
The number of observations \( n = 3 \).

Thus, the mean is:
\[ \bar{x} = \frac{33}{3} = 11 \]

### Step 2: Calculate the Squared Differences from the Mean
Next, we calculate the squared differences from the mean for each observation:
\[ (x_i - \bar{x})^2 \]
- For \( x_1 = 7 \):
\[ (7 - 11)^2 = (-4)^2 = 16 \]
- For \( x_2 = 11 \):
\[ (11 - 11)^2 = 0^2 = 0 \]
- For \( x_3 = 15 \):
\[ (15 - 11)^2 = 4^2 = 16 \]

### Step 3: Sum the Squared Differences
Now, we sum the squared differences:
\[ \sum (x_i - \bar{x})^2 = 16 + 0 + 16 = 32 \]

### Step 4: Calculate the Variance
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]
Substituting the values:
\[ \sigma^2 = \frac{32}{3} \approx 10.67 \]

### Step 5: Calculate the Standard Deviation
The standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{32}{3}} \approx \sqrt{10.67} \approx 3.26 \]

### Final Answer
The standard deviation for the data set 7, 11, and 15 is approximately **3.26**.

---