What a frequency distribution is
A frequency distribution takes a raw list of numbers and organizes it into a compact table that shows how the data is spread out. Instead of staring at dozens or hundreds of individual values, you sort them into a handful of equal-width intervals called classes (or bins) and count how many values fall into each one. The result is a table with one row per class showing its frequency (the count), its relative frequency (the proportion of the total), and its cumulative frequency (the running total from the first class down). This is the standard first step in descriptive statistics and the data behind every histogram.
The method this calculator uses
This tool builds a grouped frequency distribution with equal-width classes, the method taught in introductory statistics courses:
- Range = maximum value − minimum value.
- Class width = ⌈Range ÷ k⌉, where k is the number of classes you choose. The result is rounded up to the next whole number so that every data point, including the maximum, lands inside a class.
- Class boundaries start at the minimum value; each successive class begins where the previous one ends (lower ≤ value < upper), and the final class includes its upper endpoint so the maximum is counted.
- Frequency = the number of values in each class. The frequencies always add up to n, the total number of data points.
- Relative frequency = frequency ÷ n. Expressed as a proportion (0–1) or a percentage; relative frequencies always sum to 1 (100%).
- Cumulative frequency = the running total of frequencies through and including the current class. The last cumulative value equals n.
How many classes should you use?
There is no single correct number, but two conventions are widely used. The simplest is to pick a round number between 5 and 20 depending on how much data you have. A more formal choice is Sturges' rule: k = 1 + 3.322 · log₁₀(n). For 20 data points that gives k ≈ 5.3, so you would round to 5 or 6 classes. For 100 points it gives k ≈ 7.6, so 7 or 8. Too few classes flatten out the shape of the data; too many leave sparse, jagged bins that overstate random noise.
Why frequency distributions matter
Grouping data reveals its shape — where it clusters, whether it is symmetric or skewed, and whether there are gaps or outliers — none of which is visible in an unsorted list. A frequency table is also the direct input to a histogram, a frequency polygon, or an ogive (a cumulative-frequency curve). Relative frequencies let you compare two data sets of different sizes on the same footing, and cumulative frequencies answer "how many values are at or below this level?" — the basis for percentiles and medians estimated from grouped data.
A worked example
Take the 20 test scores 23, 45, 12, 67, 34, 89, 21, 55, 43, 78, 32, 61, 48, 27, 90, 14, 52, 39, 71, 26. The minimum is 12 and the maximum is 90, so the range is 78. With k = 5 classes the width is ⌈78 ÷ 5⌉ = ⌈15.6⌉ = 16. The classes run 12–28, 28–44, 44–60, 60–76, 76–92, and the counts are 6, 4, 4, 3, 3, which sum back to 20. The relative frequencies are 0.30, 0.20, 0.20, 0.15, 0.15 (30%, 20%, 20%, 15%, 15%), and the cumulative frequencies build to 6, 10, 14, 17, 20.