Estimating the Sample Mean and Standard Deviation for Grouped Data In Exercises 41–44, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set.
Weekly Study Hours The distribution of the number of hours that a random sample of college students study per week is shown in the pie chart. Use 32 as the midpoint for “30+ hours.”

Verified step by step guidance

1. Step 1: Create a frequency distribution table. Use the pie chart to list the study hour intervals (e.g., 0–4 hours, 5–9 hours, etc.) and their corresponding frequencies (e.g., 5, 12, etc.). Assign midpoints to each interval. For example, the midpoint for 0–4 hours is 2, for 5–9 hours is 7, and so on. Use 32 as the midpoint for '30+ hours.'
2. Step 2: Calculate the estimated sample mean. Use the formula for the mean of grouped data: \( \text{Mean} = \frac{\sum (f \cdot x)}{\sum f} \), where \( f \) is the frequency and \( x \) is the midpoint of each interval. Multiply each frequency by its corresponding midpoint, sum these products, and divide by the total frequency.
3. Step 3: Calculate the estimated sample variance. Use the formula for variance of grouped data: \( \text{Variance} = \frac{\sum f \cdot (x - \text{Mean})^2}{\sum f} \). Subtract the mean from each midpoint, square the result, multiply by the frequency, and sum these values. Divide by the total frequency.
4. Step 4: Calculate the estimated sample standard deviation. Take the square root of the variance obtained in Step 3. The formula is \( \text{Standard Deviation} = \sqrt{\text{Variance}} \).
5. Step 5: Interpret the results. The sample mean represents the average weekly study hours for the group, and the standard deviation indicates the variability in study hours among the students.

Key Concepts

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. It organizes data into categories or intervals, showing the number of observations within each category. In this case, the pie chart represents the frequency of college students studying within specified hour ranges, which helps visualize the distribution of study hours.

Sample Mean

The sample mean is the average of a set of values, calculated by summing all the observations and dividing by the number of observations. For grouped data, the mean can be estimated using the midpoints of each interval multiplied by their respective frequencies, then divided by the total number of observations. This provides a central value that represents the data set.

Sample Standard Deviation

The sample standard deviation measures the amount of variation or dispersion in a set of values. It quantifies how much the values deviate from the sample mean. For grouped data, it is calculated using the squared differences between each midpoint and the mean, weighted by the frequencies, and then taking the square root of the average of these squared differences.