Construct a frequency distribution for the data set using the indicated number of classes. In the table, include the midpoints, relative frequencies, and cumulative frequencies. Which class has the greatest class frequency and which has the least class frequency.
Textbook Spending
Number of classes: 6
Data set: Amounts (in dollars) spent on textbooks for a semester 91 472 279 249 530 376 188 341 266 199 142 273 189 130 489 266 248 101 375 486 190 398 188 269 43 30 127 354 84 319

Verified step by step guidance

1. Step 1: Determine the range of the data set. The range is calculated as the difference between the maximum and minimum values in the data set. Identify the maximum value (530) and the minimum value (30), then compute the range as Range = Maximum - Minimum.
2. Step 2: Calculate the class width. Divide the range by the number of classes (6) and round up to the nearest whole number. Use the formula: Class Width = ⌈Range / Number of Classes⌉.
3. Step 3: Create the class intervals. Start with the minimum value (30) as the lower limit of the first class. Add the class width to determine the upper limit of the first class. Repeat this process to create six non-overlapping class intervals.
4. Step 4: Tally the data into the class intervals to determine the frequency for each class. Count how many data points fall into each class interval and record these frequencies.
5. Step 5: Calculate the midpoints, relative frequencies, and cumulative frequencies for each class. Midpoint = (Lower Limit + Upper Limit) / 2. Relative Frequency = (Class Frequency / Total Frequency). Cumulative Frequency = Sum of frequencies up to and including the current class.

Key Concepts

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. It organizes data into classes or intervals, showing the number of observations (frequency) that fall within each class. This helps in visualizing the distribution of data and identifying patterns, such as the most and least common values.

Midpoints

Midpoints are the central values of each class interval in a frequency distribution. They are calculated by averaging the upper and lower boundaries of each class. Midpoints are useful for further statistical analysis, such as calculating the mean or estimating the total of the dataset, as they provide a representative value for each class.

Cumulative Frequency

Cumulative frequency is the running total of frequencies up to a certain class in a frequency distribution. It shows the number of observations that fall below the upper boundary of each class. This concept is important for understanding the distribution of data and for determining percentiles, as it allows for quick identification of how many data points lie below a specific value.