Quantitative Aptitude · Statistical Description of Data

Classification and Frequency Distribution for CA Foundation

Updated 1 October 2026

Classification groups raw data into classes. A frequency distribution lists each class with the number of observations in it, called its frequency. To solve questions, find the class width, convert inclusive classes to exclusive ones by adjusting limits by half the gap, then count tallies and compute mid-values.

Understand Classification and Frequency Distribution

Raw data is a long list of numbers, such as marks of 60 students. You cannot see a pattern in it. Classification puts similar items together so the pattern becomes visible.

For numerical data, you divide the range into classes (also called class intervals). The number of observations falling in a class is its frequency. A table of classes with their frequencies is a frequency distribution.

An ungrouped frequency distribution lists each distinct value with how often it occurs. It suits data with few distinct values, such as the number of children in a family. A grouped frequency distribution lists class intervals like 10-20, 20-30. It suits continuous data or data with many distinct values.

Every class has a lower limit and an upper limit. In the exclusive method, the upper limit of a class is not included in that class. In 10-20, the value 20 goes to the next class 20-30. In the inclusive method, both limits are included, as in 10-19, 20-29. Inclusive classes leave a small gap between one class and the next.

That gap is why we need class boundaries. Boundaries are the true limits that remove the gap, so the classes become continuous. Exclusive classes already have boundaries equal to their limits. Most later topics (histogram, median, mode) need continuous classes, so converting to boundaries is a core skill.

Key formulas to remember

Class width (class interval): Class width = Upper boundary − Lower boundary; For exclusive classes this is simply upper limit − lower limit.
Mid-value (class mark): Mid-value = (Lower limit + Upper limit) ÷ 2; Mid-value from limits and from boundaries is the same for both exclusive and inclusive classes.
Adjustment factor for inclusive classes: Adjustment = (Lower limit of next class − Upper limit of this class) ÷ 2; For integer classes like 10-19, 20-29 the adjustment is 0.5.
Class boundaries from inclusive limits: Lower boundary = Lower limit − adjustment; Upper boundary = Upper limit + adjustment; Apply to every class. 10-19 becomes 9.5-19.5.
Approximate number of classes (Sturges' rule): k ≈ 1 + 3.322 × log₁₀ N; A guide only, not a strict rule. N is the number of observations.
Class width from range: Class width ≈ Range ÷ Number of classes; Range = Largest value − Smallest value. Round up to a convenient number.
Total frequency: N = Σf; The sum of all class frequencies equals the number of observations.

How to solve Classification and Frequency Distribution questions

Use this order for any question on classes, limits, boundaries or building a distribution.

1Identify the type of classes: exclusive (10-20, 20-30) or inclusive (10-19, 20-29).
2If inclusive, find the gap between the upper limit of one class and the lower limit of the next. Half of it is the adjustment.
3Subtract the adjustment from every lower limit and add it to every upper limit to get class boundaries.
4Find the class width as upper boundary − lower boundary. Check that it is equal for all classes if the question says equal width.
5Find the mid-value as the average of the two limits (or the two boundaries).
6For raw data, find the range, choose class width and number of classes, then make classes that cover the smallest and largest values.
7Tally the data class by class, write the frequencies, and check that Σf equals the number of observations.

Quickest way: Half-the-gap shortcut and option elimination

When to use it: Use in MCQs that ask for boundaries, width or mid-value of a given class.

For inclusive integer classes, the adjustment is always 0.5. Subtract 0.5 from the lower limit and add 0.5 to the upper limit. Do not recompute the gap.
Class width equals the difference between two consecutive lower limits. For 10-19, 20-29 it is 20 − 10 = 10. This also works for inclusive classes.
Mid-value is the average of the limits. For a missing mid-value in an equal-width series, add the width to the previous mid-value.
Eliminate options that show a width equal to upper limit − lower limit for inclusive classes. That gives 9 instead of 10 and is a typical trap.
Skip questions that need tallying a long raw list if time is short. Come back to them last.

Common mistakes in Classification and Frequency Distribution

Taking class width of 10-19 as 19 − 10 = 9.
You subtract the limits directly and forget that the inclusive classes have a gap.
Fix: Convert to boundaries first (9.5-19.5), so width is 10. Or use the difference between consecutive lower limits.
Confusing class limits with class boundaries.
Both are numbers at the ends of a class, so they look alike.
Fix: Limits are the values printed in the table. Boundaries are the true continuous ends. They differ only for inclusive classes.
Adding or subtracting 1 instead of 0.5 when converting inclusive to exclusive.
You use the gap itself instead of half the gap.
Fix: Always use half the gap. Gap 1 gives 0.5. Gap 2 gives 1.
Placing a value equal to the upper limit in the wrong class under the exclusive method.
You assume both ends belong to the class.
Fix: In 10-20, the value 20 belongs to 20-30. Only the lower limit is included.
Computing the mid-value from the wrong class or without adjusting for open-ended classes.
You rush and use the unconverted class or the wrong pair of numbers.
Fix: Add the two limits of the same class and halve. For an open-ended class, assume it has the same width as the neighbouring class only if the question says so.
Frequencies that do not add up to N.
Tally marks are missed or counted twice during counting.
Fix: Strike through each value as you tally it, then check that Σf equals the number of observations.

Worked examples

Example 1

The classes of a distribution are 20-29, 30-39, 40-49, 50-59. The class boundaries of the class 30-39 are: (a) 29-39 (b) 29.5-39.5 (c) 30.5-38.5 (d) 30-40

Show the solution

The classes are inclusive, because the upper limit 29 is below the next lower limit 30.
Gap = 30 − 29 = 1, so adjustment = 1 ÷ 2 = 0.5.
Lower boundary = 30 − 0.5 = 29.5.
Upper boundary = 39 + 0.5 = 39.5.

Answer: (b) 29.5-39.5

Example 2

In a frequency distribution with inclusive classes 15-24, 25-34, 35-44, 45-54, the class width and the mid-value of the class 35-44 are respectively: (a) 9 and 39.5 (b) 10 and 39.5 (c) 10 and 40 (d) 9 and 40

Show the solution

Class width = difference between consecutive lower limits = 25 − 15 = 10.
Check with boundaries: the class 35-44 becomes 34.5-44.5, so width = 44.5 − 34.5 = 10.
Mid-value = (35 + 44) ÷ 2 = 79 ÷ 2 = 39.5.
Using boundaries gives (34.5 + 44.5) ÷ 2 = 39.5, which agrees.

Answer: (b) 10 and 39.5

Example 3

The marks of 20 students range from 12 to 47. You want 5 exclusive classes of equal width, starting at 12, with the last class including 47. The smallest convenient whole-number class width is: (a) 5 (b) 6 (c) 7 (d) 8

Show the solution

Range = 47 − 12 = 35.
Range ÷ number of classes = 35 ÷ 5 = 7. This gives only a minimum width, not the final width.
Classes are exclusive, so the upper limit of a class is not included in it. Five classes of width w cover 5w units starting from 12, so the last upper limit is 12 + 5w. For 47 to fall inside the last class, we need 12 + 5w > 47.
Solving, 5w > 35, so w > 7. The smallest whole number greater than 7 is 8.
Check width 7: the classes are 12-19, 19-26, 26-33, 33-40, 40-47. The last upper limit is exactly 47, which is excluded under the exclusive method, so 47 is left out. Width 7 fails.
Check width 8: the classes are 12-20, 20-28, 28-36, 36-44, 44-52. The last upper limit is 52, and 47 falls in 44-52.
Widths 5 and 6 span even less (25 and 30 units), so they fail too.

Answer: (d) 8

Exam tips

Look at the printed limits first. If the upper limit of one class is less than the lower limit of the next, the classes are inclusive and you must adjust.
Questions on boundaries, width and mid-value take under a minute if you use the 0.5 shortcut. Do them first.
Many later topics such as histogram, median and mode need continuous classes. Convert at the start, not midway.
With 0.25 negative marking, guess only after you have eliminated at least two options.
Watch for missing-class or missing-frequency questions: use Σf = N to find the unknown.

Practice questions from Statistical Description of Data

Classification and Frequency Distribution: frequently asked questions

What is the difference between class limits and class boundaries?

Class limits are the values written in the table, such as 10-19. Class boundaries are the true continuous ends, such as 9.5-19.5. For exclusive classes, limits and boundaries are the same.

How do I convert inclusive to exclusive class intervals?

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 10-19, 20-29 the gap is 1, so the classes become 9.5-19.5, 19.5-29.5.

What is the difference between ungrouped and grouped frequency distribution?

An ungrouped distribution lists each distinct value with its frequency. A grouped distribution puts values into class intervals and gives a frequency for each class. Grouped form is used when there are many distinct values.

How do I construct a frequency distribution from raw data?

Find the range, decide the number of classes and a convenient width, and write the classes so they cover all values. Tally each observation into its class, count the tallies, and check that the frequencies add up to the total number of observations.