Types of Grouped Frequency Distribution: Open-End, Relative, and Cumulative

Raw data rarely makes sense on its own. A list of two hundred exam scores or a spreadsheet of household incomes only becomes useful once it is organised into classes. That is where grouped frequency distributions come in, but a plain frequency table does not always fit every dataset. Some data has extreme outliers that are hard to box into a fixed range, some datasets need to be compared despite unequal sizes, and some questions are really about “how many fall below this point” rather than “how many fall in this exact class.” This is why statisticians rely on three specialised variations: open-end, relative, and cumulative frequency distributions. Each solves a different practical problem, and together they cover most of what you will encounter in a data presentation unit.

Why the basic grouped table isn’t always enough 🔗

A standard grouped frequency distribution divides data into class intervals of equal width, such as 0-10, 10-20, and so on, and counts how many observations fall into each. This works well when the data is fairly evenly spread. Problems appear when a few values are much larger or smaller than the rest. Forcing every class to have a fixed width either creates a long tail of near-empty classes or forces analysts to guess an upper boundary that does not really exist in the data. The three distributions covered below are essentially adaptations built to handle these situations.

Open-end frequency distribution 🔗

An open-end frequency distribution is one where at least one class has no defined boundary on one side. The first class might be written as “below 10” instead of “0-10,” or the last class as “60 and above” instead of “60-70.” According to Statistics How To, this simply means one or more classes has no fixed lower or upper limit, which is useful when a small number of extreme values would otherwise force the creation of many sparse classes.

Where you see this in everyday data 🔗

Income tax slabs are a familiar example. Under the new tax regime for FY 2025-26, income is taxed progressively, but the highest bracket is simply defined as income above Rs. 24 lakh, with no upper limit specified. Trying to cap this bracket at a fixed number would be pointless since incomes above that threshold can vary enormously. The same logic applies to age data (“65 years and above”), family size (“four or more members”), or survey responses on income where very high earners are reluctant to disclose an exact figure.

The trade-off with open-end classes 🔗

Open-end classes solve the boundary problem but create a new one: several statistical measures, including the mean, require a defined class width and midpoint. Statology notes that when calculations like the median are needed, analysts typically use the class width of the nearest bounded class as a working assumption for the open-end class. This is a workable approximation, not an exact value, so open-end distributions are best used for descriptive summaries rather than precise calculations involving the extreme classes.

Relative frequency distribution 🔗

A relative frequency distribution replaces raw counts with proportions. Instead of stating that 12 students scored between 40 and 50, it states that this class represents 0.24, or 24 percent, of the total. The formula is straightforward: divide the frequency of each class by the total number of observations. As explained in this introductory statistics resource, a relative frequency is the ratio of how often a value occurs to the total number of observations, and it can be expressed as a fraction, decimal, or percentage.

Why proportions matter more than raw counts 🔗

Relative frequencies become essential the moment you want to compare two datasets of different sizes. Comparing the raw number of students who scored above 90 percent in a college of 500 students versus a college of 5,000 students tells you very little, since the larger college will almost always have a higher count. Converting both to relative frequencies (percentage of the total student body) makes the comparison meaningful. This is exactly why relative frequency tables are the standard format for cross-institution or cross-region academic and economic comparisons.

Notice that the relative frequencies always sum to 1 (or 100 percent). This is a useful self-check when building any relative frequency table.

Cumulative frequency distribution 🔗

A cumulative frequency distribution shows the running total of frequencies up to and including a given class, rather than the count within that class alone. It answers a very specific and commonly asked question: how many observations fall below (or above) a particular value?

Less-than and more-than types 🔗

There are two standard formats. A less-than cumulative frequency adds up all frequencies up to the upper boundary of a class, telling you how many observations are below that point. A more-than cumulative frequency works in reverse, summing frequencies from a given lower boundary to the end of the distribution. GeeksforGeeks illustrates this using cricket scores, converting a simple frequency table into both less-than and more-than cumulative tables, which can then be plotted as cumulative frequency curves, also known as ogives.

Practical use: finding the median and percentiles 🔗

Cumulative frequency tables are the standard tool for locating the median in grouped data, since the median class is identified as the one where the cumulative frequency first crosses half the total number of observations. The same logic extends to quartiles and percentiles, which is why cumulative frequency curves are widely used in board exam result analysis, competitive exam cut-off calculations, and university admission processes across India, where a student’s rank often depends on knowing how many candidates scored below a particular mark.

Cumulative relative frequency distribution 🔗

The cumulative relative frequency distribution combines the previous two ideas. Instead of a running total of raw counts, it shows a running total of proportions. According to this statistics resource, cumulative relative frequency is calculated by dividing the cumulative frequency of a class by the total number of observations, and by definition, the last class in any dataset will always show a cumulative relative frequency of 100 percent.

Reading a cumulative relative frequency table 🔗

The advantage here is interpretability. A statement like “83 percent of respondents spend less than 115 minutes on the internet daily” is immediately meaningful in a way that a raw cumulative count of, say, 25 out of 30 respondents is not, especially when comparing across surveys of different sizes. This measure is also what percentile-based reporting relies on: national exam authorities and large-scale surveys routinely convert scores into percentile ranks using exactly this method, since it standardises results regardless of how many people took the test.

Choosing the right type for your data 🔗

These three variations are not mutually exclusive; a single, well-constructed table can include an open-end class as well as columns for relative frequency, cumulative frequency, and cumulative relative frequency all at once. The choice depends on the question being asked. Use an open-end distribution when a few extreme values make a fixed boundary impractical. Use a relative frequency distribution when comparing datasets of different sizes. Use a cumulative frequency distribution when the question is about totals below or above a threshold, and a cumulative relative frequency distribution when that threshold needs to be expressed as a percentage or rank.

What do you think? If you were analysing monthly household expenditure data for a city with a small number of very high-income households, would an open-end class at the top of the distribution give a fairer picture than forcing a fixed upper limit? And when reporting exam results to a large batch of students, does a cumulative relative frequency table communicate their standing more clearly than a plain frequency count?

References

1. https://www.statisticshowto.com/open-ended-distribution/
2. https://www.bajajfinserv.in/insights/income-tax-slab
3. https://www.statology.org/open-ended-distribution/
4. https://ecampusontario.pressbooks.pub/introstats2ed/chapter/2-1-frequency-distributions/
5. https://www.geeksforgeeks.org/maths/frequency-distribution/
6. https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Inferential_Statistics_and_Probability_-_A_Holistic_Approach_(Geraghty)/02:_Displaying_and_Analyzing_Data_with_Graphs/2.05:_Graphs_of_Numeric_Data/2.5.05:_Cumulative_Frequency_and_Relative_Frequency