Questions: Consider the following data: 7,10,12,2,9,14 Calculate the value of the sample standard deviation. Round your answer to one decimal place.

Solution Steps

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

\[ \mu = \frac{\sum x_i}{n} = \frac{7 + 10 + 12 + 2 + 9 + 14}{6} = \frac{54}{6} = 9.0 \]

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

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \]

First, we compute \( (x_i - \mu)^2 \) for each data point:

• For \( x_1 = 7 \): \( (7 - 9)^2 = 4 \)
• For \( x_2 = 10 \): \( (10 - 9)^2 = 1 \)
• For \( x_3 = 12 \): \( (12 - 9)^2 = 9 \)
• For \( x_4 = 2 \): \( (2 - 9)^2 = 49 \)
• For \( x_5 = 9 \): \( (9 - 9)^2 = 0 \)
• For \( x_6 = 14 \): \( (14 - 9)^2 = 25 \)

Now, summing these values:

\[ \sum (x_i - \mu)^2 = 4 + 1 + 9 + 49 + 0 + 25 = 88 \]

Now, substituting into the variance formula:

\[ \sigma^2 = \frac{88}{6-1} = \frac{88}{5} = 17.6 \]

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

\[ \sigma = \sqrt{17.6} \approx 4.2 \]

Final Answer