Questions: Consider the following data: -7,12,-7,14,-7,-11,-8 Step 1 of 3: Calculate the value of the sample variance. Round your answer to one decimal place.

Solution Steps

To find the mean \( \mu \) of the dataset, we use the formula:

\[ \mu = \frac{\sum x_i}{n} \]

where \( \sum x_i \) is the sum of all data points and \( n \) is the number of data points. For our dataset:

\[ \sum x_i = -7 + 12 - 7 + 14 - 7 - 11 - 8 = -14 \] \[ n = 7 \]

Thus, the mean is calculated as:

\[ \mu = \frac{-14}{7} = -2.0 \]

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

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

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

• For \( -7 \): \( (-7 - (-2))^2 = (-5)^2 = 25 \)
• For \( 12 \): \( (12 - (-2))^2 = (14)^2 = 196 \)
• For \( -7 \): \( (-7 - (-2))^2 = (-5)^2 = 25 \)
• For \( 14 \): \( (14 - (-2))^2 = (16)^2 = 256 \)
• For \( -7 \): \( (-7 - (-2))^2 = (-5)^2 = 25 \)
• For \( -11 \): \( (-11 - (-2))^2 = (-9)^2 = 81 \)
• For \( -8 \): \( (-8 - (-2))^2 = (-6)^2 = 36 \)

Now, summing these values:

\[ \sum (x_i - \mu)^2 = 25 + 196 + 25 + 256 + 25 + 81 + 36 = 644 \]

Now, we can calculate the variance:

\[ \sigma^2 = \frac{644}{7-1} = \frac{644}{6} \approx 107.3 \]

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

\[ \sigma = \sqrt{107.3} \approx 10.4 \]

Final Answer