Exam Time The following data represent the amount of time (in minutes) a random sample of eight students took to complete the online portion of an exam in Sullivan’s Statistics course. Compute the range, sample variance, and sample standard deviation time.


60.5, 128.0, 84.6, 122.3, 78.9, 94.7, 85.9, 89.9

Verified step by step guidance

1. Step 1: Identify the data points given: 60.5, 128.0, 84.6, 122.3, 78.9, 94.7, 85.9, 89.9. These represent the times in minutes for each student.
2. Step 2: Calculate the range by subtracting the smallest value from the largest value. Use the formula: \(\text{Range} = \max(x_i) - \min(x_i)\), where \(x_i\) are the data points.
3. Step 3: Find the sample mean \(\bar{x}\) using the formula: \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\), where \(n\) is the number of data points (8 in this case).
4. Step 4: Calculate the sample variance \(s^2\) using the formula: \(s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2\). This involves subtracting the mean from each data point, squaring the result, summing all squared differences, and dividing by \(n-1\).
5. Step 5: Compute the sample standard deviation \(s\) by taking the square root of the sample variance: \(s = \sqrt{s^2}\). This gives a measure of the spread of the data in the same units as the original data.

Key Concepts

Range

The range is a measure of dispersion that shows the difference between the maximum and minimum values in a data set. It provides a quick sense of the spread of the data but is sensitive to outliers. For example, in the given times, the range is calculated by subtracting the smallest time from the largest.

Sample Variance

Sample variance measures the average squared deviation of each data point from the sample mean, indicating how spread out the data are. It is calculated by summing the squared differences between each value and the mean, then dividing by n-1 (degrees of freedom). This adjustment accounts for using a sample rather than the entire population.

Sample Standard Deviation

The sample standard deviation is the square root of the sample variance, providing a measure of spread in the same units as the data. It quantifies the typical distance of data points from the mean, helping to understand variability. A smaller standard deviation indicates data points are closer to the mean.