Questions: Find the standard deviation for the group of data items. 11,12,13,14,15 The standard deviation is (Round to two decimal places as needed.)

Solution Steps

The mean \( \mu \) of the dataset is calculated as follows:

\[ \mu = \frac{\sum x_i}{n} = \frac{11 + 12 + 13 + 14 + 15}{5} = \frac{65}{5} = 13.0 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \]

Calculating each term:

• For \( x_1 = 11 \): \( (11 - 13)^2 = 4 \)
• For \( x_2 = 12 \): \( (12 - 13)^2 = 1 \)
• For \( x_3 = 13 \): \( (13 - 13)^2 = 0 \)
• For \( x_4 = 14 \): \( (14 - 13)^2 = 1 \)
• For \( x_5 = 15 \): \( (15 - 13)^2 = 4 \)

Now summing these values:

\[ \sum (x_i - \mu)^2 = 4 + 1 + 0 + 1 + 4 = 10 \]

Thus, the variance is:

\[ \sigma^2 = \frac{10}{5} = 2.0 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{2.0} \approx 1.41 \]

Final Answer