A frequency distribution organizes many data values into a compact table so patterns become easier to see. Instead of reading a long list of numbers, you can compare how often values fall in each class interval. This matters in statistics because grouped data is the starting point for histograms, summaries, and estimates.
It helps turn raw measurements into a clear visual and numerical story.
For grouped data, each class interval has a frequency, a class midpoint, and often a product of midpoint times frequency. The midpoint represents all values in that interval when estimating statistics such as the mean. A histogram uses the same intervals on the horizontal axis and the frequencies as bar heights.
Because grouping loses some detail, grouped statistics are estimates rather than exact values from the original data.
Key Facts
- A frequency distribution lists class intervals and the number of data values in each interval.
- Class width = upper class limit of a class minus upper class limit of the previous class, when widths are equal.
- Class midpoint = (lower class limit + upper class limit) / 2.
- Estimated mean for grouped data: x̄ = Σ(fm) / Σf, where f is frequency and m is midpoint.
- Relative frequency = f / n, where n is the total number of data values.
- In a histogram, adjacent bars touch because class intervals represent continuous ranges.
Vocabulary
- Frequency
- Frequency is the number of data values that fall in a category or class interval.
- Class interval
- A class interval is a range of values used to group data in a frequency distribution.
- Class midpoint
- A class midpoint is the value halfway between the lower and upper limits of a class interval.
- Histogram
- A histogram is a graph that displays class intervals on the horizontal axis and frequencies as touching bars.
- Grouped mean
- A grouped mean is an estimated average found by using class midpoints and frequencies instead of every original data value.
Common Mistakes to Avoid
- Using overlapping class intervals, such as 10 to 20 and 20 to 30, is wrong because a value of 20 could belong to two classes. Use intervals that assign every value to exactly one class.
- Adding the midpoints instead of multiplying each midpoint by its frequency is wrong because classes with more data values should have more influence on the mean. Use Σ(fm), not just Σm.
- Dividing by the number of classes instead of the total frequency is wrong because the mean depends on the total number of data values. Divide by Σf, not by the number of rows in the table.
- Leaving gaps between histogram bars is wrong when the data are continuous or grouped into intervals. Touching bars show that the intervals cover a continuous scale.
Practice Questions
- 1 A data set has the grouped distribution 0 to 9: f = 3, 10 to 19: f = 5, 20 to 29: f = 2. Find each class midpoint and estimate the mean using x̄ = Σ(fm) / Σf.
- 2 Test scores are grouped as 50 to 59: 4 students, 60 to 69: 6 students, 70 to 79: 10 students, 80 to 89: 5 students, 90 to 99: 3 students. Find the total frequency and the relative frequency for the 70 to 79 class.
- 3 A student creates intervals 0 to 10, 10 to 20, 20 to 30 for a histogram of measurement data. Explain the problem with these intervals and describe a better way to define the classes.