Questions: Calculate the variance of the following data: 1,2,3,3,5,6,8,9 Express your answer to 1 decimal place, with rounding.

Solution Steps

To find the mean \( \mu \) of the dataset \( x = \{1, 2, 3, 3, 5, 6, 8, 9\} \), we use the formula:

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

Calculating the sum:

\[ \sum_{i=1}^8 x_i = 1 + 2 + 3 + 3 + 5 + 6 + 8 + 9 = 37 \]

The number of data points \( N = 8 \). Thus, the mean is:

\[ \mu = \frac{37}{8} = 4.6 \]

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

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

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

\[ \begin{align_} (1 - 4.6)^2 & = 12.96 \\ (2 - 4.6)^2 & = 6.76 \\ (3 - 4.6)^2 & = 2.56 \\ (3 - 4.6)^2 & = 2.56 \\ (5 - 4.6)^2 & = 0.16 \\ (6 - 4.6)^2 & = 1.96 \\ (8 - 4.6)^2 & = 11.56 \\ (9 - 4.6)^2 & = 19.36 \\ \end{align_} \]

Now, summing these squared differences:

\[ \sum (x_i - \mu)^2 = 12.96 + 6.76 + 2.56 + 2.56 + 0.16 + 1.96 + 11.56 + 19.36 = 58.4 \]

Finally, we calculate the variance:

\[ \sigma^2 = \frac{58.4}{8} = 7.3 \]

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

\[ \sigma = \sqrt{7.3} \approx 2.7 \]

Final Answer