Questions: For the data shown, answer the questions. Round to 4 decimal places where possible. 3.5, 14.4, 3.3, 6.9, 27.6, 22.5, 15.7, 19 Find the mean: Find the median: Find the standard deviation:

Solution Steps

The mean \( \mu \) of the dataset is calculated using the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} \]

For the given data:

\[ \mu = \frac{112.9}{8} = 14.1125 \]

To find the median, we first sort the data:

\[ \text{Sorted data} = [3.3, 3.5, 6.9, 14.4, 15.7, 19, 22.5, 27.6] \]

Since there are \( N = 8 \) data points, the rank for the median is calculated as:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (8 + 1) = 4.5 \]

This indicates that the median is the average of the 4th and 5th values:

\[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{14.4 + 15.7}{2} = 15.05 \]

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

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

Substituting the values, we find:

\[ \sigma^2 = 80.0441 \]

The standard deviation \( \sigma \) is then:

\[ \sigma = \sqrt{80.0441} = 8.9467 \]

Final Answer