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Fundamentals of Business Mathematics and Statistics · Measures of Central Tendency and Dispersion

Mode and Empirical Relation between Mean, Median and Mode

Updated 10 October 2026 · Fact-checked

The mode is the value that occurs most often. For grouped data, find the modal class (highest frequency) and use Mode = L + [(f1 − f0) ÷ (2f1 − f0 − f2)] × h. For moderately skewed data, the empirical relation is Mode = 3 Median − 2 Mean.

Understand Mode and Empirical Relation

The mode is the most frequently occurring value in a data set. If a shop sells shirts in sizes 38, 40, 40, 42, 40 and 44, the mode is 40. It tells you the most popular or typical item, which is why businesses use it for sizes, brands and common prices.

A data set can have one mode (unimodal), two modes (bimodal), more than two (multimodal), or no mode at all if every value occurs the same number of times. Unlike the mean, the mode is not affected by extreme values.

For grouped data you cannot see single values, only classes. So you first find the modal class, the class with the highest frequency. The mode lies inside it. The formula then pulls the mode towards the neighbouring class that has the higher frequency. If the class before has a larger frequency than the class after, the mode sits in the lower part of the modal class, and the other way round.

Mean, median and mode are linked in data that is moderately skewed. Experience shows that Mean − Mode is about three times Mean − Median. This gives the empirical relation Mode = 3 Median − 2 Mean. It is an approximation, not an exact law. In a perfectly symmetric distribution all three are equal.

Key formulas to remember

Mode for grouped data
Mode = L + [(f1 − f0) ÷ (2f1 − f0 − f2)] × h
L = lower limit of the modal class, f1 = frequency of the modal class, f0 = frequency of the class before it, f2 = frequency of the class after it, h = class width. Classes must be continuous and of equal width.
Equivalent form with differences
Mode = L + [Δ1 ÷ (Δ1 + Δ2)] × h
Δ1 = f1 − f0 and Δ2 = f1 − f2. Same result as the formula above.
Empirical relation
Mode = 3 Median − 2 Mean
Holds approximately for moderately skewed distributions. Not exact.
Mean from mode and median
Mean = (3 Median − Mode) ÷ 2
Rearranged form for when the mean is the missing value.
Median from mean and mode
Median = (Mode + 2 Mean) ÷ 3
Rearranged form for when the median is the missing value.
Mode of ungrouped data
Mode = value with the highest frequency
Count how often each value appears. Ties give more than one mode.

How to solve Mode and Empirical Relation questions

Use this method for any question on mode or the relation between the three averages.

  1. 1Read what is given. If you are given a list of values or a frequency table, you must calculate the mode. If you are given two of mean, median and mode, use the empirical relation.
  2. 2For ungrouped data, count the frequency of each value. The value with the highest count is the mode. If two values tie, the data is bimodal.
  3. 3For grouped data, check that classes are continuous. If they are like 10–19, 20–29, convert them to 9.5–19.5, 19.5–29.5 first.
  4. 4Identify the modal class: the class with the highest frequency.
  5. 5Note f1 (modal class), f0 (class just before) and f2 (class just after). Also note L and h.
  6. 6Put the values in Mode = L + [(f1 − f0) ÷ (2f1 − f0 − f2)] × h. Work out the denominator first.
  7. 7For the empirical relation, write Mode = 3 Median − 2 Mean and rearrange for the missing value before substituting.
  8. 8Check that the mode lies inside the modal class. If it does not, you have swapped f0 and f2 or misread L.

Quickest way: Spot the modal class, then fill the formula

When to use it: Use for MCQs on grouped data when the options are close and you need to avoid slips.

  1. Underline the highest frequency in the table. That row gives L, f1 and h in seconds.
  2. Write f0 above and f2 below it. Compute f1 − f0 (top) and 2f1 − f0 − f2 (bottom).
  3. Divide the top by the bottom, multiply by h, and add to L.
  4. Sanity check: the answer must be between L and L + h. This often eliminates two options at once.
  5. For relation questions, just remember 3 Median − 2 Mean. If the mean is missing, use (3 × Median − Mode) ÷ 2.

Common mistakes in Mode and Empirical Relation

  • Swapping f0 and f2 in the formula

    Students take the 'next' class as f0 instead of the 'previous' class.

    Fix: f0 is the class before the modal class (lower values). f2 is the class after it. The numerator is always f1 − f0.

  • Using the formula with inclusive classes such as 10–19, 20–29

    Students take L = 20 directly without making the classes continuous.

    Fix: Convert to exact limits first. Here the class 20–29 becomes 19.5–29.5, so L = 19.5 and h = 10.

  • Choosing the modal class as the one with the highest midpoint or the middle class

    Confusion with the median class, which is found using cumulative frequency.

    Fix: The modal class is simply the class with the highest frequency. No cumulative frequency is needed.

  • Using the formula when class widths are unequal

    Students apply it without checking the table.

    Fix: The standard formula assumes equal widths. If widths differ, the frequencies must first be adjusted. In Foundation questions the widths are normally equal, so check and note this.

  • Writing Mode = 3 Mean − 2 Median

    The two averages get reversed in memory.

    Fix: Remember: mode is 3 times the median minus 2 times the mean. Check with equal values: if all three are 10, 3 × 10 − 2 × 10 = 10.

  • Treating the empirical relation as exact for any data

    It is written like an equation, so it looks like a law.

    Fix: It holds approximately for moderately skewed data. In the exam, use it when the question asks for it or says the distribution is moderately skewed.

Worked examples

Example 1

Find the mode of this distribution. Class: 0–10, 10–20, 20–30, 30–40, 40–50. Frequency: 4, 9, 15, 6, 2.

Show the solution
  1. The highest frequency is 15, so the modal class is 20–30.
  2. L = 20, h = 10, f1 = 15, f0 = 9, f2 = 6.
  3. Numerator: f1 − f0 = 15 − 9 = 6.
  4. Denominator: 2f1 − f0 − f2 = 30 − 9 − 6 = 15.
  5. Mode = 20 + (6 ÷ 15) × 10 = 20 + 4 = 24.
  6. Check: 24 lies between 20 and 30, so it is valid.

Answer: Mode = 24

Example 2

For a moderately skewed distribution, the mode is 48 and the median is 54. Find the arithmetic mean.

Show the solution
  1. Use Mode = 3 Median − 2 Mean.
  2. Substitute: 48 = 3 × 54 − 2 × Mean.
  3. 48 = 162 − 2 × Mean.
  4. 2 × Mean = 162 − 48 = 114.
  5. Mean = 114 ÷ 2 = 57.
  6. Check: mean 57 > median 54 > mode 48, which fits a positively skewed distribution.

Answer: Mean = 57

Exam tips

  • Questions often give mean and median and ask for the mode, or the reverse. Rearrange the relation before you substitute to avoid sign errors.
  • In a grouped-data MCQ, the answer must lie inside the modal class. Use this to reject wrong options quickly.
  • Check class limits first. Inclusive classes (10–19, 20–29) need a correction of 0.5 at each end before you use L and h.
  • For a positively skewed distribution, mean > median > mode. For a negatively skewed one, the order reverses. This helps in conceptual questions.
  • There is no negative marking, so always mark an answer. If time is short, narrow down using the modal class range.

Practice questions from Measures of Central Tendency and Dispersion

Mode and Empirical Relation in other exams

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

Mode and Empirical Relation: frequently asked questions

What is the mode formula for grouped data in CMA Foundation?

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 classes before and after it, and h the class width.

How do I find the modal class?

Look at the frequency column and pick the class with the highest frequency. That class is the modal class. You do not need cumulative frequencies for this.

Is Mode = 3 Median − 2 Mean always true?

No. It is an empirical relation that holds approximately for moderately skewed distributions. In a symmetric distribution all three averages are equal, and the relation still gives the same value.

Can a data set have more than one mode?

Yes. If two values share the highest frequency, the data is bimodal. If more than two do, it is multimodal. If all values occur equally often, there is no mode.