Quantitative Aptitude · Statistical Description of Data

Cumulative and Relative Frequency Distributions

Updated 1 October 2026 · Fact-checked

A cumulative frequency distribution shows running totals of frequencies: less-than totals add from the lowest class up, more-than totals add from the highest class down. Relative frequency is class frequency divided by total frequency. Multiply by 100 for percentage frequency. A bivariate table shows two variables together in rows and columns.

Understand Cumulative and Relative Frequency Distributions

A simple frequency distribution tells you how many observations fall in each class. Often the exam asks a different question: how many observations are below a value, above a value, or what share of the total a class holds. Cumulative and relative frequencies answer these.

Less-than cumulative frequency (lcf) at a class's upper limit is the number of observations below that limit. You get it by adding class frequencies from the first class downwards. More-than cumulative frequency (mcf) at a class's lower limit is the number of observations at or above that limit. You get it by adding from the last class upwards.

Relative frequency is the fraction of the total that falls in a class: f ÷ N. Percentage frequency is that fraction times 100. Relative frequencies of all classes add up to 1, and percentage frequencies add up to 100. You can also make cumulative versions of these, which end at 1 or 100%.

A bivariate frequency distribution classifies the same observations by two variables at once, for example marks and attendance. Each cell holds the joint frequency. Row totals and column totals are the marginal frequencies. Fixing one row or column gives a conditional distribution of the other variable.

Key formulas to remember

Relative frequency: Relative frequency of a class = f ÷ N; N = total frequency (Σf). Answer is a fraction between 0 and 1.
Percentage frequency: Percentage frequency = (f ÷ N) × 100; All percentage frequencies add up to 100.
Less-than cumulative frequency: lcf of a class = sum of frequencies of that class and all classes before it; Pair it with the upper limit of the class. The last lcf equals N.
More-than cumulative frequency: mcf of a class = sum of frequencies of that class and all classes after it; Pair it with the lower limit of the class. The first mcf equals N.
Recovering class frequency from lcf: f of a class = its lcf − lcf of the previous class; For the first class, f = its lcf.
Recovering class frequency from mcf: f of a class = its mcf − mcf of the next class; For the last class, f = its mcf.
Link between lcf and mcf: lcf at a limit + mcf at the same limit = N; Holds for continuous classes when both are read at the same boundary value.
Conditional share in a bivariate table: Share = joint cell frequency ÷ relevant row or column total; The denominator is the group named in the condition, not the grand total.

How to solve Cumulative and Relative Frequency Distributions questions

Use this order for any question on cumulative, relative or bivariate frequencies.

1Read the question and decide what is asked: a count, a share, or a percentage. Note whether it says less than, more than, or between.
2Find N, the total frequency. If only cumulative frequencies are given, N is the last lcf or the first mcf.
3If cumulative frequencies are given, convert them back to class frequencies by taking differences.
4For less-than questions, use the lcf at the stated upper limit. For more-than questions, use the mcf at the stated lower limit.
5For a share, divide the required frequency by N. Multiply by 100 if the options are percentages.
6In a bivariate table, locate the cell, row or column asked for. Choose the denominator from the condition in the question: 'of those who...' means that group's total.
7Check that your answer is sensible: a part cannot exceed its total, and relative frequencies must be at most 1.

Quickest way: Subtract from N and scan the options

When to use it: Use in the MCQ paper when a table is given and you need a count or a percentage fast.

Write N first. Almost every answer uses it.
For 'more than' or 'at least', compute N minus the lcf of the boundary. It is quicker than adding many classes.
For 'between', subtract two cumulative values instead of adding classes.
Convert shares to simple fractions such as 25/60 = 5/12 and compare with the options. You rarely need full decimals.
Eliminate options that equal a raw cumulative value or the wrong denominator. These are the usual traps.
If the table is large and the question needs several steps, mark it and move on. Wrong answers cost 0.25.

Common mistakes in Cumulative and Relative Frequency Distributions

Pairing the cumulative frequency with the lower limit for a less-than distribution.
Students see the class 20-30 and attach the total to 20.
Fix: Less-than totals belong to the upper limit: 'less than 30'. More-than totals belong to the lower limit: '20 and above'.
Taking the lcf as the class frequency.
Students read the cumulative column as if it were the frequency column.
Fix: Subtract the previous lcf to get the class frequency. Do this before computing any share.
Dividing by the wrong total in a bivariate table.
Students always use the grand total.
Fix: Look at the words 'of those who' or 'among'. The denominator is that row or column total.
Adding more-than frequencies from the top instead of from the bottom.
Students copy the method used for less-than totals.
Fix: Start from the last class and add upwards. The first mcf must equal N. If it does not, recheck.
Giving relative frequency as a percentage, or the reverse.
Students forget the ×100 step or apply it twice.
Fix: Read the options. If they are like 0.25, give a fraction. If they are like 25%, multiply by 100 once.
Ignoring whether a boundary is included in 'more than'.
Class limits are written as 20-30 and the wording is loose.
Fix: Use the table's own cumulative columns. Read the mcf at the lower limit of the class that starts at the stated value.

Worked examples

Example 1

The distribution of 50 observations is: 0-10: 5; 10-20: 8; 20-30: 12; 30-40: 15; 40-50: 10. How many observations are less than 30? (A) 12 (B) 25 (C) 37 (D) 40

Show the solution

N = 5 + 8 + 12 + 15 + 10 = 50.
Less-than cumulative frequencies: below 10 is 5; below 20 is 5 + 8 = 13; below 30 is 13 + 12 = 25.
The count below 30 is the lcf at upper limit 30, which is 25.
Check: observations of 30 or more = 15 + 10 = 25, and 25 + 25 = 50.

Answer: (B) 25

Example 2

The less-than cumulative frequencies are: less than 20: 8; less than 40: 20; less than 60: 45; less than 80: 60. What is the percentage frequency of the class 40-60, to two decimals? (A) 20% (B) 33.33% (C) 41.67% (D) 75%

Show the solution

N = 60, the last cumulative frequency.
Frequency of class 40-60 = lcf at 60 − lcf at 40 = 45 − 20 = 25.
Relative frequency = 25 ÷ 60 = 5/12.
Percentage frequency = 5/12 × 100 = 41.67%.
Options 33.33% (20/60) and 75% (45/60) come from using cumulative values directly.

Answer: (C) 41.67%

Example 3

In a group of 100 students, 30 males passed and 10 males failed. 40 females passed and 20 females failed. Of the students who passed, what percentage are female? (A) 40% (B) 57.14% (C) 66.67% (D) 75%

Show the solution

Total who passed = 30 + 40 = 70.
Females who passed = 40.
The condition is 'of those who passed', so the denominator is 70, not 100.
Percentage = 40 ÷ 70 × 100 = 57.14%.
40% uses the grand total 100. 66.67% is 40 ÷ 60, the pass rate among females. 75% is 30 ÷ 40, the pass rate among males.

Answer: (B) 57.14%

Exam tips

Questions are usually small tables with a count or percentage asked. Write N first and keep it in view.
Expect options built from the common traps: a raw cumulative value, the grand total used as denominator, or a missing ×100.
For 'more than' questions, N minus the lcf is often the fastest route.
In bivariate tables, underline the group named in the question before choosing the denominator.
Practise reading ogive-style data too. The less-than and more-than curves meet where both cumulative values are N ÷ 2.

Practice questions from Statistical Description of Data

Cumulative and Relative Frequency Distributions: frequently asked questions

What is the difference between less-than and more-than cumulative frequency?

Less-than cumulative frequency adds class frequencies from the lowest class upwards and is tied to upper limits. More-than cumulative frequency adds from the highest class downwards and is tied to lower limits. The last lcf and the first mcf both equal N.

How do I calculate relative frequency?

Divide the class frequency by the total frequency N. For a percentage, multiply by 100. All relative frequencies add up to 1.

How do I get class frequencies back from a cumulative table?

For less-than data, subtract the previous lcf from the current one. For more-than data, subtract the next mcf from the current one. The first lcf and the last mcf are already class frequencies.

What is a bivariate frequency distribution?

It is a table that classifies the same observations by two variables at once. Cells give joint frequencies, and the row and column totals give the marginal frequencies of each variable.