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Fundamentals of Business Mathematics and Statistics · Statistical Representation of Data

Frequency Distribution and Class Intervals Explained for CMA Foundation

Updated 10 October 2026 · Fact-checked

A frequency distribution groups raw data into classes and shows how many observations fall in each class. To build one, find the range, choose class width, list classes, tally the data and count frequencies. For inclusive classes, subtract 0.5 from lower limits and add 0.5 to upper limits to get exclusive classes.

Understand Frequency Distribution and Class Intervals

Raw data is a long list of numbers. It is hard to read. A frequency distribution puts the data in a table. One column shows the values or classes. The other shows the frequency, which is the number of times the value or class occurs.

There are two kinds. A discrete frequency distribution lists each separate value, such as the number of children in a family: 0, 1, 2, 3. A continuous frequency distribution groups values into class intervals, such as 10-20, 20-30. Use it when data is measured, like marks, weights or salaries.

Every class has a lower limit and an upper limit. The class width (or size) is the gap between the class boundaries. The mid-point (mid value) is the centre of the class. It stands in for all values of that class in later calculations of mean and standard deviation.

Classes can be written in two ways. In the exclusive method, the upper limit of one class is the lower limit of the next (10-20, 20-30). A value equal to the upper limit goes into the next class. So 20 belongs to 20-30. In the inclusive method, both limits belong to the class (10-19, 20-29). There is a small gap between classes. This gap is closed by finding class boundaries.

A cumulative frequency shows running totals. Less than cumulative frequency adds frequencies from the top of the table downwards, against upper boundaries. More than cumulative frequency adds from the bottom upwards, against lower boundaries. These are the base for ogives, median and quartiles.

Key formulas to remember

Class width (exclusive classes)
Class width = Upper limit − Lower limit
Use boundaries when classes are inclusive.
Mid-point (mid value)
Mid-point = (Lower limit + Upper limit) ÷ 2
For inclusive classes, using limits or boundaries gives the same answer.
Adjustment factor for inclusive classes
d = (Lower limit of next class − Upper limit of this class) ÷ 2
For integer data with gap 1, d = 0.5.
Class boundaries
Lower boundary = Lower limit − d; Upper boundary = Upper limit + d
This converts inclusive classes to exclusive classes.
Approximate number of classes
Number of classes ≈ Range ÷ Class width
Range = Largest value − Smallest value. Round up.
Less than cumulative frequency
Add frequencies from the first class onwards
Placed against the upper boundary of each class.
More than cumulative frequency
Add frequencies from the last class backwards
Placed against the lower boundary of each class.
Total frequency
N = Σf
The last less than cumulative frequency and the first more than cumulative frequency both equal N.

How to solve Frequency Distribution and Class Intervals questions

Use this order for any question on constructing or reading a frequency distribution.

  1. 1Read what is asked: class width, boundary, mid-point, missing frequency or cumulative frequency.
  2. 2Check the type of classes. If the upper limit of one class equals the lower limit of the next, they are exclusive. If there is a gap (19 and 20), they are inclusive.
  3. 3For inclusive classes, find d as half the gap and compute boundaries: lower limit − d, upper limit + d.
  4. 4Find class width as upper boundary − lower boundary.
  5. 5Find mid-point as the average of the lower and upper limits of the class.
  6. 6For cumulative frequency, add step by step. Less than: top to bottom against upper boundaries. More than: bottom to top against lower boundaries.
  7. 7Check that the last cumulative frequency equals Σf. Then match your answer to the options.

Quickest way: Gap-and-total shortcut

When to use it: Use it for MCQs on boundaries, mid-points and cumulative frequency.

  1. Look at the gap between classes. Gap 0 means exclusive, nothing to change. Gap 1 means subtract 0.5 and add 0.5.
  2. For class width, subtract the two limits of an exclusive class, or add 1 to the difference of inclusive limits (gap 1). Example: 10-19 has width 10.
  3. Mid-point is just the average of the two limits. No need for boundaries.
  4. For cumulative frequency, keep adding in your head and confirm with Σf at the end.
  5. For a missing frequency, use total N minus the known frequencies.

Common mistakes in Frequency Distribution and Class Intervals

  • Taking class width of 10-19 as 9.

    You subtract the limits directly, even though the class is inclusive.

    Fix: Convert to boundaries 9.5-19.5. Width is 10.

  • Placing a value equal to the upper limit in the same class in an exclusive series.

    It feels natural to keep the value in the class where it appears as the upper limit.

    Fix: In exclusive classes, the upper limit is excluded. So 20 goes into 20-30, not 10-20.

  • Adding 1 instead of 0.5 when converting to boundaries.

    You use the gap itself and not half of it.

    Fix: Adjustment = half the gap. For gap 1, subtract and add 0.5.

  • Mixing up less than and more than cumulative frequencies.

    Both are called cumulative, and the direction of adding is forgotten.

    Fix: Less than: start from the top and use upper boundaries. More than: start from the bottom and use lower boundaries.

  • Calculating the mid-point from the boundaries of the wrong class or forgetting to divide by 2.

    Rushing through arithmetic.

    Fix: Write (L + U) ÷ 2 for each class. Check that mid-points rise by the class width.

Worked examples

Example 1

The marks of students are grouped in classes 20-29, 30-39, 40-49, 50-59 with frequencies 5, 12, 18, 10. Find (a) the class boundaries of 30-39, (b) the class width, and (c) the mid-point of 40-49.

Show the solution
  1. The gap between 29 and 30 is 1, so d = 1 ÷ 2 = 0.5.
  2. Boundaries of 30-39: lower = 30 − 0.5 = 29.5; upper = 39 + 0.5 = 39.5.
  3. Class width = 39.5 − 29.5 = 10.
  4. Mid-point of 40-49 = (40 + 49) ÷ 2 = 89 ÷ 2 = 44.5.

Answer: Boundaries 29.5-39.5, class width 10, mid-point of 40-49 is 44.5.

Example 2

A firm's daily sales (in ₹ thousand) are: 0-10: 4, 10-20: 7, 20-30: 11, 30-40: 8. Find the less than cumulative frequency for 30 and the more than cumulative frequency for 20.

Show the solution
  1. Total frequency N = 4 + 7 + 11 + 8 = 30.
  2. Less than 30 means the classes 0-10, 10-20 and 20-30: 4 + 7 + 11 = 22.
  3. More than 20 means the classes 20-30 and 30-40: 11 + 8 = 19.
  4. Check: less than 20 is 4 + 7 = 11, and 11 + 19 = 30 = N. The totals match.

Answer: Less than 30: 22 days. More than 20: 19 days.

Exam tips

  • Always check the gap between classes first. It tells you the series type in two seconds.
  • Class width, boundaries and mid-point questions are quick marks. Do them first.
  • In cumulative frequency questions, confirm that the final total equals Σf. It catches addition slips.
  • Read carefully whether the question says less than or more than, and whether the series is inclusive or exclusive.
  • No negative marking, so attempt every question, and eliminate options that break the width or total check.

Practice questions from Statistical Representation of Data

Frequency Distribution and Class Intervals in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Frequency Distribution and Class Intervals: frequently asked questions

What is the difference between inclusive and exclusive class intervals?

In inclusive classes, both limits belong to the class, as in 10-19 and 20-29. In exclusive classes, the upper limit is not part of the class, as in 10-20 and 20-30. A value of 20 goes into 20-30.

How do I convert inclusive classes to exclusive classes?

Find half the gap between the upper limit of one class and the lower limit of the next. Subtract it from every lower limit and add it to every upper limit. For a gap of 1, classes 10-19 become 9.5-19.5.

What is the difference between class limits and class boundaries?

Class limits are the values written in the table. Class boundaries are the true dividing points between classes, with no gap. In exclusive classes they are the same. In inclusive classes they differ by the adjustment.

How do I find the mid-value of a class?

Add the lower and upper limits of the class and divide by 2. For 40-50 it is 45. For inclusive 40-49 it is 44.5. Boundaries give the same answer.