Frequency Distribution Calculator
Paste or type your data set and this calculator builds a complete frequency distribution table with frequency, relative frequency, cumulative frequency, cumulative relative frequency, and midpoint columns. Switch between ungrouped (each unique value) and grouped (equal-width class intervals) modes. Key descriptive statistics - mean, median, mode, range, and standard deviation - update live with every keystroke.
What is a frequency distribution?
A frequency distribution is an organised summary of how often each value (or range of values) appears in a data set. Instead of looking at a raw list of numbers, you group observations into categories and record a count for each. This makes patterns such as clusters, gaps, and outliers immediately visible. Frequency distributions are the foundation of exploratory data analysis and underpin histograms, bar charts, frequency polygons, and many inferential statistics tests.
Ungrouped vs. grouped frequency distributions
An ungrouped frequency distribution lists every distinct value in the data set alongside the number of times it appears. It is ideal for discrete data with a small number of unique values - for example, survey ratings from 1 to 5, dice rolls, or quiz scores. A grouped (or classed) frequency distribution organises continuous or widely spread data into equal-width class intervals. Each interval has a lower bound, an upper bound, a midpoint (the average of the two bounds), and a frequency count. Grouped tables are better for large data sets or those measured on a continuous scale such as height, weight, or time. This calculator supports both modes: select "Ungrouped" for discrete data and "Grouped" for continuous or large data sets.
How to read the frequency distribution table
Each row in the table represents one value or class interval. The frequency column shows the raw count of observations in that row. The relative frequency column expresses that count as a proportion of the total (count divided by n), and the values in this column always sum to 1.0 (or 100%). The cumulative frequency is a running total that keeps adding each row's count to the previous total; it equals n in the final row. The cumulative relative frequency is the same running total expressed as a proportion, ending at 1.0 in the last row. For grouped distributions, the midpoint column gives the centre of each class interval and is used when drawing a frequency polygon.
Choosing the right number of bins
The number of class intervals (bins) affects how informative a grouped frequency distribution looks. Too few bins compress all the data into a handful of wide classes and hide the shape of the distribution. Too many bins spread counts so thinly that every class has only one or two observations and the table loses its summary value. Sturges' rule, which is the default in this calculator, recommends k = 1 + 3.322 * log10(n) classes and works well for moderately large, roughly symmetric data sets. You can override it by entering a specific bin count - a common guideline is between 5 and 20 classes for most data sets. If the distribution is heavily skewed or has clear natural groupings, adjusting the bin count manually gives better results.
Quick guide: distribution shapes
| Shape | Mean vs. Median | Tail direction | Example |
|---|---|---|---|
| Symmetric (normal) | Mean = Median | Both equal | Heights of adults |
| Right-skewed | Mean > Median | Right (positive) | Household incomes |
| Left-skewed | Mean < Median | Left (negative) | Exam scores (easy test) |
| Bimodal | May be equal | Two peaks | Mixed population data |
| Uniform | Mean = Median | No tail | Random number generation |
Use mean vs. median to identify skew direction from a frequency distribution.
Frequently asked questions
What is relative frequency and how is it calculated?
Relative frequency is the proportion of observations that fall in a given class, calculated by dividing the class frequency by the total number of observations (n). For example, if 8 out of 40 values fall in a class, the relative frequency is 8/40 = 0.2 (or 20%). The relative frequencies across all classes always sum to exactly 1.0 (100%). Relative frequency is useful because it lets you compare distributions from data sets of different sizes on equal footing.
What is cumulative frequency used for?
Cumulative frequency is the running total of frequencies from the first class to the current class. It tells you how many observations fall at or below a particular value. For example, if the cumulative frequency at the third class is 25, then 25 data points are less than or equal to that class's upper boundary. Cumulative frequency is plotted as an ogive (S-shaped curve) and is used to estimate percentiles and medians from grouped data.
How do I choose between ungrouped and grouped modes?
Use the ungrouped mode for discrete data with a small number of distinct values - for example, exam grades from A to F, number of siblings, or Likert-scale survey responses. Use the grouped mode for continuous data or any data set where the range is large relative to the number of observations. A useful rule of thumb is to switch to grouped mode when the number of unique values exceeds about 15 to 20.
What is Sturges' rule and when should I override it?
Sturges' rule suggests k = 1 + 3.322 * log10(n) class intervals, where n is the sample size. For a sample of 100 it suggests about 7 or 8 classes; for 1,000 it suggests about 11. The rule tends to underestimate the ideal number of classes for skewed or multimodal distributions and for very large data sets (n over a few thousand). In those cases, use the Scott or Freedman-Diaconis rule, or simply experiment by increasing the bin count until the distribution shape becomes clear.
What does the midpoint of a class interval represent?
The midpoint is the average of the lower and upper class boundaries: midpoint = (lower bound + upper bound) / 2. It is used as a representative value for all observations in that class when calculating grouped mean, grouped variance, and when plotting a frequency polygon. In a frequency polygon, the midpoints of each class are plotted against the class frequencies and connected with straight lines.
Can I use this calculator for qualitative (categorical) data?
This calculator is designed for quantitative (numerical) data. For categorical data such as colours, names, or Yes/No responses, a simple tally or bar chart is more appropriate. Each category is its own class and the concept of class width, midpoint, and cumulative frequency in the ordered sense does not apply.