Questions: Marathon Winners The data show the times of a random sample of local marathon runners in hours. Find the range, variance, and standard deviation for the data. Use a TI-83 Plus/TI-84 Plus calculator. 3.3, 2.3, 2.6, 2.4, 2.5, 2.6 Send data to Excel

Solution Steps

To find the range, variance, and standard deviation of the given data set, we will follow these steps:

1. Range: Calculate the difference between the maximum and minimum values in the data set.
2. Variance: Compute the average of the squared differences from the mean of the data set.
3. Standard Deviation: Take the square root of the variance to find the standard deviation.

The range of a data set is the difference between the maximum and minimum values. For the given data set \([3.3, 2.3, 2.6, 2.4, 2.5, 2.6]\), the maximum value is \(3.3\) and the minimum value is \(2.3\).

\[ \text{Range} = 3.3 - 2.3 = 1.0 \]

Variance is the average of the squared differences from the mean. First, calculate the mean \(\mu\) of the data set:

\[ \mu = \frac{3.3 + 2.3 + 2.6 + 2.4 + 2.5 + 2.6}{6} = \frac{15.7}{6} \approx 2.6167 \]

Next, calculate the squared differences from the mean and their average:

\[ \text{Variance} = \frac{(3.3 - 2.6167)^2 + (2.3 - 2.6167)^2 + (2.6 - 2.6167)^2 + (2.4 - 2.6167)^2 + (2.5 - 2.6167)^2 + (2.6 - 2.6167)^2}{6} \approx 0.1047 \]

The standard deviation is the square root of the variance:

\[ \text{Standard Deviation} = \sqrt{0.1047} \approx 0.3236 \]

Final Answer