Frequency Distribution Calculator
Generate a complete frequency distribution table for any dataset. Choose ungrouped for discrete data or grouped for continuous data with custom class widths.
📊 What is a Frequency Distribution?
A frequency distribution is an organized summary of a dataset that shows how many times each value (or range of values) appears. It transforms a raw list of numbers into a structured table with columns for frequency, relative frequency, and cumulative frequency, making patterns in the data immediately visible. It is one of the most fundamental tools in descriptive statistics and is the basis for histograms, ogive curves, and many probability models.
There are two main types. An ungrouped frequency distribution lists every distinct value in the dataset and counts exact occurrences. It is ideal for discrete data: quiz scores with values of 0 to 10, survey responses on a 1-to-5 scale, the number of defects per product, or die roll outcomes. An grouped frequency distribution divides the data into class intervals (for example 50-60, 60-70, 70-80) and counts how many values fall in each interval. It is essential for continuous data such as heights, weights, temperatures, or incomes where individual values rarely repeat.
You use a frequency distribution any time you need to understand the shape of a dataset. A quality engineer studying defect counts per batch uses it to spot whether the distribution is symmetric or skewed. A teacher analyzing exam grades uses it to identify whether students cluster around a particular score range. A researcher reporting survey data uses relative frequencies to compare subgroups of different sizes. A financial analyst examining daily stock returns uses a grouped distribution to estimate the probability of returns falling in certain bands.
The four columns this calculator produces cover everything you need. Frequency (f) is the raw count. Relative frequency (rf) converts counts to proportions so you can compare datasets of different sizes. Cumulative frequency (CF) is the running total, which is used to find medians and percentiles. Cumulative relative frequency (CRF) is the running proportion and directly estimates the probability that a randomly chosen observation is at most a given value.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Die Rolls (Ungrouped, Discrete Data)
Record of 15 rolls of a standard six-sided die
Example 2 - Student Heights (Grouped, Continuous Data)
Heights (cm) of 12 students measured to the nearest centimetre, 5 classes
Example 3 - Exam Scores (Ungrouped, Discrete Data)
Scores out of 10 for 20 students on a short quiz
❓ Frequently Asked Questions
🔗 Related Calculators
What is a frequency distribution in statistics?
A frequency distribution is a table that shows how often each value or class of values occurs in a dataset. It organizes raw data into a structured summary showing frequency (count), relative frequency (proportion), and cumulative frequency (running total). It is the foundation for histograms, bar charts, and many descriptive statistics.
How do you calculate relative frequency?
Relative frequency equals the class frequency divided by the total number of observations: rf = f / N. For example, if 8 out of 40 students scored in the 70-80 range, the relative frequency is 8/40 = 0.20 or 20%. All relative frequencies in a distribution must sum to 1 (or 100%).
What is cumulative frequency and how is it used?
Cumulative frequency is the running total of frequencies from the lowest class to the current class. It tells you how many observations fall at or below a given value. For example, if the first three classes have frequencies 5, 8, and 12, the cumulative frequencies are 5, 13, and 25. It is used to find medians, quartiles, and percentiles.
What is the difference between ungrouped and grouped frequency distributions?
An ungrouped frequency distribution lists every unique value individually and counts how many times each appears. It is best for discrete data with few distinct values. A grouped frequency distribution organizes data into class intervals (e.g., 10-20, 20-30) and is better for continuous data or large datasets where individual values are rarely repeated.
How do you choose the number of classes for grouped frequency distribution?
A common rule of thumb is Sturges' Formula: k = 1 + 3.322 x log10(N), where N is the sample size. For N = 20 use about 5 classes; for N = 100 use about 7 to 8; for N = 1000 use about 11. The goal is enough classes to show the shape of the distribution without excessive empty classes.
How is class width calculated for a grouped frequency distribution?
Class width equals the data range divided by the number of classes: w = (max - min) / k. For example, if heights range from 150 cm to 190 cm and you choose 5 classes, the class width is (190 - 150) / 5 = 8 cm. This calculator computes class width automatically based on your data and chosen number of classes.
What is the midpoint of a class interval?
The midpoint (or class mark) is the average of the lower and upper class boundaries: midpoint = (lower + upper) / 2. For a class interval of 20-30, the midpoint is 25. Midpoints are used when estimating the mean and variance from a grouped frequency distribution.
What is the difference between a frequency distribution and a histogram?
A frequency distribution is a table showing frequencies for each value or class. A histogram is the graphical representation of that table, with class intervals on the horizontal axis and frequencies (or relative frequencies) on the vertical axis as bars. The two convey the same information in different forms.
Can a frequency distribution be used for categorical data?
Yes. A frequency distribution for categorical data (also called a frequency table) lists each category and its count. For example, a survey of preferred colors might show: Blue 45, Red 30, Green 25. Relative frequency would be 45%, 30%, and 25% respectively. This calculator handles numeric data; for categories, tally the counts manually.
Why do cumulative relative frequencies always end at 100%?
Because each relative frequency represents a proportion of the whole dataset. When you add all proportions together, you are counting every observation exactly once, so the total is N/N = 1.0 or 100%. If your last cumulative relative frequency is not 100%, it usually means the data was entered incorrectly or a class has been omitted.
What is an open-ended class interval and should I use it?
An open-ended class has no fixed lower or upper boundary, such as 'Under 20' or '100 and above'. They are used when extreme outliers would create very wide or almost-empty classes. This calculator uses equal-width closed classes based on the actual data range, which works well for most distributions without extreme outliers.
How is frequency distribution related to probability?
Relative frequency is a direct estimate of probability. If a die is rolled 60 times and lands on 3 exactly 12 times, the relative frequency of 3 is 12/60 = 0.20. By the law of large numbers, as sample size increases, relative frequencies converge to true probabilities. A relative frequency distribution approximates the probability distribution of the underlying population.