Questions: Consider the following data: -9, 13, -14, -14, -9, -14, -9 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 the given data:

\[ \sum x_i = -9 + 13 - 14 - 14 - 9 - 14 - 9 = -56 \] \[ n = 7 \]

Thus, the mean is calculated as:

\[ \mu = \frac{-56}{7} = -8.0 \]

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

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

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

\[ \begin{align_} (-9 - (-8))^2 & = 1^2 = 1 \\ (13 - (-8))^2 & = 21^2 = 441 \\ (-14 - (-8))^2 & = (-6)^2 = 36 \\ (-14 - (-8))^2 & = (-6)^2 = 36 \\ (-9 - (-8))^2 & = 1^2 = 1 \\ (-14 - (-8))^2 & = (-6)^2 = 36 \\ (-9 - (-8))^2 & = 1^2 = 1 \\ \end{align_} \]

Now, summing these values:

\[ \sum (x_i - \mu)^2 = 1 + 441 + 36 + 36 + 1 + 36 + 1 = 556 \]

Now, substituting into the variance formula:

\[ s^2 = \frac{556}{7-1} = \frac{556}{6} = 92.0 \]

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

\[ s = \sqrt{s^2} = \sqrt{92.0} \approx 9.6 \]

Final Answer