Quantitative Aptitude · Measures of Central Tendency and Dispersion

Mode: Formula, Empirical Relation and Graphical Method for CA Foundation

Updated 1 October 2026 · Fact-checked

The mode is the value that occurs most often in a data set. For grouped data, find the modal class (highest frequency), then use Mode = L + (f1 − f0) ÷ (2f1 − f0 − f2) × h. If the mean and median are given, use the empirical relation Mode = 3 Median − 2 Mean.

Understand Mode

The mode is the value that appears most often in a set of data. It answers the question: which value is the most common? Shoe size, the most sold price point and the most frequent marks in a test are all modes.

For ungrouped data, count how often each value occurs. The value with the highest count is the mode. A data set can have no mode (all values occur equally often), one mode (unimodal), two modes (bimodal) or more (multimodal).

For grouped data, you cannot see single values, only class intervals. The class with the highest frequency is the modal class. The mode lies inside it, but its exact position depends on the frequencies of the classes just before and just after. If the previous class has a low frequency and the next class has a high one, the mode is pulled toward the upper end of the modal class. The formula captures this pull.

The empirical relation links the three averages for a moderately skewed distribution: Mode = 3 Median − 2 Mean. It is an approximation, not an exact law. Exams use it when one of the three values is missing.

The graphical method uses a histogram. You find the tallest rectangle, join diagonals with its neighbours, and read the mode from the x-axis. It gives the same value as the formula when the classes have equal width.

Key formulas to remember

Mode for ungrouped data: Mode = value with the highest frequency; There can be no mode, one mode or several modes.
Mode for grouped data: Mode = L + (f1 − f0) ÷ (2f1 − f0 − f2) × h; L = lower limit of modal class, f1 = frequency of modal class, f0 = frequency of class before it, f2 = frequency of class after it, h = class width. Classes must be continuous (exclusive) with equal width.
Empirical relation: Mode = 3 Median − 2 Mean; Approximate, for moderately skewed distributions.
Empirical relation (alternate form): Mean − Mode = 3 (Mean − Median); Same relation rearranged. Median = (Mode + 2 Mean) ÷ 3 and Mean = (3 Median − Mode) ÷ 2.
Inclusive to exclusive classes: Adjustment = (next lower limit − previous upper limit) ÷ 2; Subtract it from every lower limit and add it to every upper limit. For 10-19, 20-29 the adjustment is 0.5, so the classes become 9.5-19.5, 19.5-29.5.

How to solve Mode questions

Use this method for any mode question, whether the data is raw, grouped or given through other averages.

1Read what is given. If you get raw values, count frequencies. If you get a frequency table, go to grouped data. If mean and median are given, use the empirical relation.
2For raw data, pick the value with the highest frequency. State if there are two modes.
3For grouped data, check that the classes are continuous. If they are inclusive (10-19, 20-29), convert them to exclusive (9.5-19.5, 19.5-29.5).
4Find the modal class: the class with the highest frequency. Check that class widths are equal.
5Write L, h, f1, f0 and f2 from the table. If the modal class is first, f0 = 0. If it is last, f2 = 0.
6Substitute in Mode = L + (f1 − f0) ÷ (2f1 − f0 − f2) × h and compute the denominator first.
7Check that the answer lies inside the modal class. If it does not, you have an error.
8Compare with the four options and mark the closest exact match.

Quickest way: Shortcut: locate the modal class, then eliminate options

When to use it: Use in MCQs on grouped data or when mean and median are given.

Find the modal class first. Any option outside that class is wrong. This often leaves one or two options.
Look at the neighbours. If f0 is larger than f2, the mode is in the lower half of the class. If f2 is larger, it is in the upper half. If equal, it is the midpoint.
Compute only the fraction (f1 − f0) ÷ (2f1 − f0 − f2), multiply by h, and add to L.
For empirical questions, plug directly into Mode = 3 Median − 2 Mean. Do not rearrange unless the mode is given.
If the class conversion looks long, skip and return later. Negative marking is 0.25 per wrong answer, so guess only after eliminating options.

Common mistakes in Mode

Using the highest frequency number as the mode of a grouped table
Students confuse the frequency with the value.
Fix: The highest frequency only identifies the modal class. Then apply the formula to get the mode.
Not converting inclusive classes to exclusive classes
The formula is applied directly to classes like 10-19, 20-29.
Fix: Subtract 0.5 from lower limits and add 0.5 to upper limits first. Then L is 29.5, not 30, for the class 30-39.
Swapping f0 and f2 in the formula
Both look like 'neighbouring frequencies'.
Fix: f0 is the class before (lower) and f2 is the class after (higher). Only f0 appears in the numerator.
Forgetting to multiply by h
Students stop after finding the fraction.
Fix: The fraction is only a proportion of the class width. Always multiply by h and add to L.
Treating Mode = 3 Median − 2 Mean as exact for every data set
The relation is memorised without conditions.
Fix: Use it as an approximation for moderately skewed data, and when the question tells you to use it.
Reading the mode wrongly from a histogram
Students drop the perpendicular from the top of the tallest bar.
Fix: Join the top-right corner of the modal bar to the top-right corner of the previous bar, and the top-left corner of the modal bar to the top-left corner of the next bar. Drop the perpendicular from the intersection of these lines.

Worked examples

Example 1

For a moderately skewed distribution, the mean is 40 and the median is 38. The approximate mode is: (a) 34 (b) 36 (c) 42 (d) 44

Show the solution

Use Mode = 3 Median − 2 Mean.
3 × 38 = 114.
2 × 40 = 80.
Mode = 114 − 80 = 34.

Answer: (a) 34

Example 2

The frequency distribution is: 0-10: 5, 10-20: 8, 20-30: 15, 30-40: 9, 40-50: 3. The mode is approximately: (a) 24.62 (b) 25.38 (c) 26.67 (d) 27.50

Show the solution

The highest frequency is 15, so the modal class is 20-30.
L = 20, h = 10, f1 = 15, f0 = 8, f2 = 9.
Numerator: f1 − f0 = 15 − 8 = 7.
Denominator: 2f1 − f0 − f2 = 30 − 8 − 9 = 13.
Mode = 20 + (7 ÷ 13) × 10 = 20 + 5.38 = 25.38.
This lies within 20-30, as it should.

Answer: (b) 25.38

Example 3

Classes and frequencies: 10-19: 4, 20-29: 10, 30-39: 16, 40-49: 6. The mode is: (a) 32.50 (b) 33.25 (c) 34.50 (d) 35.00

Show the solution

The classes are inclusive. The gap between 19 and 20 is 1, so adjustment = 0.5.
The new classes are 9.5-19.5, 19.5-29.5, 29.5-39.5, 39.5-49.5. Width h = 10.
The highest frequency is 16, so the modal class is 29.5-39.5.
L = 29.5, f1 = 16, f0 = 10, f2 = 6.
Numerator: 16 − 10 = 6.
Denominator: 2 × 16 − 10 − 6 = 16.
Mode = 29.5 + (6 ÷ 16) × 10 = 29.5 + 3.75 = 33.25.

Answer: (b) 33.25

Exam tips

Check first whether classes are inclusive or exclusive. Examiners often set inclusive classes to trap students who skip conversion.
Always verify that your answer lies inside the modal class. This catches most calculation slips in seconds.
In empirical-relation questions, read carefully which two values are given, then rearrange once and compute.
Expect conceptual MCQs too: the mode is not affected by extreme values, and it may not be unique or may not exist.
For graphical questions, remember that the mode is found from the histogram, while the median is found from the ogive.

Practice questions from Measures of Central Tendency and Dispersion

Mode: frequently asked questions

What is the mode formula for grouped data?

Mode = L + (f1 − f0) ÷ (2f1 − f0 − f2) × h. Here L is the lower limit of the modal class, f1 its frequency, f0 and f2 the frequencies of the previous and next classes, and h the class width.

What is the empirical relation between mean, median and mode?

Mode = 3 Median − 2 Mean. It is an approximation that holds for moderately skewed distributions. You can rearrange it as Mean − Mode = 3 (Mean − Median).

How do you find the mode using a histogram?

Draw the histogram and pick the tallest rectangle. Join its top-right corner to the top-right corner of the previous rectangle, and its top-left corner to the top-left corner of the next rectangle. Drop a perpendicular from where the two lines cross to the x-axis. That point is the mode.

Can a data set have more than one mode?

Yes. If two values share the highest frequency, the data is bimodal. With more than two, it is multimodal. If every value occurs equally often, there is no mode.

What if the modal class is the first or last class?

If it is the first class, take f0 = 0. If it is the last class, take f2 = 0. Then apply the same formula.