The standard deviation for the data 7,9,11,13,15, is

Options

A. 2.4
B. 2.5
C. 2.7
D. 2.82

Answer

To find the standard deviation of the data set \(7, 9, 11, 13, 15\), we will follow these steps:

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

For our data:
\[ \sum x_i = 7 + 9 + 11 + 13 + 15 = 55 \]
The number of data points \(n = 5\).

Thus, the mean is:
\[ \bar{x} = \frac{55}{5} = 11 \]

### Step 2: Calculate the Sum of Squares of Deviations from the Mean
Next, we calculate the squared deviations from the mean for each data point:
\[ (x_i - \bar{x})^2 \]
Calculating for each data point:
- 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 \(15\): \((15 - 11)^2 = (4)^2 = 16\)

Now, sum these squared deviations:
\[ \sum (x_i - \bar{x})^2 = 16 + 4 + 0 + 4 + 16 = 40 \]

### Step 3: Calculate the Variance
The variance (\( \sigma^2 \)) is given by:
\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]
Substituting the values we have:
\[ \sigma^2 = \frac{40}{5} = 8 \]

### Step 4: Calculate the Standard Deviation
The standard deviation (\( \sigma \)) is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{8} \approx 2.83 \]

### Conclusion
Thus, the standard deviation of the data set \(7, 9, 11, 13, 15\) is approximately \(2.83\).

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