EconomicsISC Class 11 · Measures of Central Value

Median, Mode and the Relationship among Averages (ISC Class 11 Economics)

Updated 17 October 2026 · Fact-checked

The median is the middle value of the series arranged in order; for a continuous series median = L + (N/2 - cf)/f x h. The mode is the most frequent value; for a continuous series mode = L + (f1 - f0)/(2f1 - f0 - f2) x h. In a moderately skewed distribution, Mode = 3 Median - 2 Mean; if mean > median > mode the distribution is positively skewed, and if mean < median < mode it is negatively skewed.

Notice the mean is pulled toward the long tail, with the median between mean and mode; when the curve is symmetrical all three coincide.

Illustration drawn for QuestionBuddy.

Understand Median, Mode and the Relationship among Mean, Median and Mode

Median. The positional average that divides the series into two equal parts.

*Individual series:* arrange in order; median = ((n + 1)/2)th item. For an even n take the mean of the two middle items.
*Discrete series:* find cumulative frequencies; median = size of the ((N + 1)/2)th item. Find the cumulative frequency that first reaches or exceeds (N + 1)/2 and read off the corresponding value.
*Continuous series:* median class = class with cf first at or above N/2; Median = L + [(N/2 - cf)/f] x h, where L is the lower limit of the median class, cf the cumulative frequency of the previous class, f the frequency and h the width.

Mode. The most frequently occurring value.

*Individual/discrete:* the value with the highest frequency.
*Continuous:* modal class = class with the highest frequency; Mode = L + [(f1 - f0)/(2f1 - f0 - f2)] x h, where f1 is the modal class frequency, f0 the frequency before it and f2 after it.

Relationship (empirical). For a moderately skewed distribution: Mode = 3 Median - 2 Mean. In a symmetrical distribution: Mean = Median = Mode.

Skewness (idea). *Positive:* a long tail to the right; mean > median > mode. *Negative:* tail to the left; mean < median < mode.

Merits and demerits. Median: not affected by extreme values, can be found in open-ended series; but ignores the size of items. Mode: easy to see; but may not be unique or well defined.

Other positional values. Quartiles divide the series into four parts (used in the next chapter).

Median, Mode and the Relationship among Mean, Median and Mode: key facts

Median (continuous): Median = L + [(N/2 - cf) / f] x h | cf = cumulative frequency before the median class; Median class: cf >= N/2.
Mode (continuous): Mode = L + [(f1 - f0) / (2 f1 - f0 - f2)] x h | f1 = modal frequency; Modal class: highest f.
Relationship: Mode = 3 Median - 2 Mean | Mean > Median > Mode means positive skew; Empirical.

How to answer Median, Mode and the Relationship among Mean, Median and Mode questions

Locate the class first and then apply the formula.

1Arrange the data and compute cumulative frequencies.
2For the median find N/2 and the median class; for the mode find the highest frequency.
3Substitute L, cf, f, h (or f1, f0, f2).
4Compare the three averages to describe skewness.

Median, Mode and the Relationship among Mean, Median and Mode in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Median is robust

Suppose the monthly incomes of five families are Rs 15,000, 16,000, 18,000, 20,000 and 1,50,000.

Which average describes a typical family better, mean or median?

Try it first, then show the solutionSolution

Mean = 2,19,000 / 5 = Rs 43,800.
Median = Rs 18,000 (middle value).

Answer: The median, Rs 18,000, because one rich family inflates the mean.

Memory hook: Median ignores extreme values.

Tricks to remember Median, Mode and the Relationship among Mean, Median and Mode

Short memory aids. Say each one aloud once, then use it on a question the same day.

N over 2
Median class is where cf first reaches N/2.
Use it when: Median.
f1 minus f0 over 2f1 minus f0 minus f2
Mode formula.
Use it when: Mode.
Mean follows the tail
Mean is pulled towards the long tail.
Use it when: Skewness.

Median, Mode and the Relationship among Mean, Median and Mode: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Odd and even · 1 mark · MCQBoard level

The median of 5, 8, 9, 12, 15, 18, 22 is:
(a) 9 (b) 12 (c) 15 (d) 13.5

Try it first, then show the solutionSolution

7 items; the 4th is 12.

Answer: (b) 12

Example 2: Median and mode of a discrete series · 4 marks · Short answerBoard level

For x = 2, 4, 6, 8, 10 with f = 3, 5, 8, 6, 3, find the median and the mode. Also find the median if the series 5, 8, 9, 12, 15, 18, 22, 30 is given. (4 marks)

Try it first, then show the solutionSolution

cf: 3, 8, 16, 22, 25; N/2 = 12.5; the first cf at or above 12.5 is 16, so median = 6.
Mode = 6 (highest frequency 8).
Eight items: median = (12 + 15) / 2 = 13.5.

Answer: Median 6; mode 6; median of the eight items 13.5.

Example 3: Median and mode of a continuous series · 6 marks · Long answerHarder board level

Classes 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 4, 8, 15, 9, 4. Find the median and the mode, and compare with the mean 25.25. (6 marks)

Try it first, then show the solutionSolution

cf: 4, 12, 27, 36, 40; N/2 = 20, so the median class is 20-30 (L = 20, cf = 12, f = 15, h = 10).
Median = 20 + (20 - 12)/15 x 10 = 20 + 5.33 = 25.33.
Modal class 20-30: f1 = 15, f0 = 8, f2 = 9; Mode = 20 + (15 - 8)/(30 - 8 - 9) x 10 = 20 + 7/13 x 10 = 25.38.
Mean 25.25, median 25.33, mode 25.38 are almost equal: nearly symmetrical.

Answer: Median 25.33; mode 25.38; mean 25.25: the distribution is almost symmetrical.

Example 4: Using the empirical relation · 4 marks · Short answerHarder board level

In a moderately skewed distribution the mean is 52 and the median is 50. Find the mode and the type of skewness. If instead the mean is 40 and the mode is 46, find the median. (4 marks)

Try it first, then show the solutionSolution

Mode = 3 x 50 - 2 x 52 = 150 - 104 = 46; mean > median > mode: positive skew.
Mode = 3 Median - 2 Mean: 46 = 3 Median - 80, so Median = 42.
Here mean < median < mode: negative skew.

Answer: Mode 46, positively skewed; in the second case median 42, negatively skewed.

Example 5: Which average? · 5 marks · Long answerHOTS / competency

The wages of 7 workers (Rs per day) are 300, 320, 320, 340, 360, 380 and 1,500 (a supervisor). Find the mean, median and mode and say which best represents the typical worker's wage. (5 marks)

Try it first, then show the solutionSolution

Mean = 3,520 / 7 = Rs 502.86.
Median (4th item of the ordered list) = Rs 340.
Mode = Rs 320 (occurs twice).
The mean is pulled up by 1,500; the median (340) or mode (320) is more typical.
Mean > median > mode: positively skewed.

Answer: Mean Rs 502.86; median Rs 340; mode Rs 320; the median is the best measure of the typical wage.

Where students lose marks in Median, Mode and the Relationship among Mean, Median and Mode

The slips examiners see most often, and what to write instead.

Mistake: Using the class with the highest cf as the median class
Wrong identification.
Wrong working
40-50
Right working
First class with cf at least N/2
Do this: Find N/2 first.
Usually costs: 2 marks.
Mistake: Using the wrong cf in the median formula
Using the cf of the median class.
Wrong working
cf = 27
Right working
cf = 12
Do this: cf is of the class before.
Usually costs: 2 marks.
Mistake: Mode from the middle class always
Assuming symmetry.
Wrong working
Mode = mid-point
Right working
Use f1, f0, f2
Do this: Apply the formula.
Usually costs: 1-2 marks.

Exam tips

Write N/2 and mark the median class.
List L, cf, f, h before substituting.
Use the relation to check mode and median results.

Practice questions on Median, Mode and the Relationship among Mean, Median and Mode

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
A distribution has mean 45, median 47 and mode 51. It is:
(a) positively skewed (b) negatively skewed (c) symmetrical (d) not defined
Hint
Mean is the smallest.
Answer
(b) negatively skewed
Q2 · 2 marks · Short answerBoard level
Find the median of 15, 22, 9, 31, 27, 18. (2 marks)
Hint
Arrange first.
Answer
Ordered: 9, 15, 18, 22, 27, 31; median = (18 + 22) / 2 = 20.
Q3 · 6 marks · Long answerHarder board level
Classes 10-20, 20-30, 30-40, 40-50, 50-60 with frequencies 6, 14, 24, 10, 6. Find the median, the mode and the mean. (6 marks)
Hint
N = 60.
Answer
cf: 6, 20, 44, 54, 60; N/2 = 30; median class 30-40: 30 + (30 - 20)/24 x 10 = 34.17. Mode: modal class 30-40, f1 = 24, f0 = 14, f2 = 10: 30 + (24 - 14)/(48 - 14 - 10) x 10 = 34.17. Mean: Sum fm = 90 + 350 + 840 + 450 + 330 = 2,060; 2,060 / 60 = 34.33.
Q4 · 4 marks · Short answerHarder board level
The mean of a distribution is 72 and the mode is 64. Find the median using the empirical relation and the type of skewness. A second series has mean 30, median 28: find the mode and the skewness. (4 marks)
Hint
Mode = 3 Median - 2 Mean.
Answer
64 = 3M - 144, so Median = 69.33; mean > median > mode: positive skew. Second: mode = 84 - 60 = 24; mean > median > mode: positive skew.
Q5 · 4 marks · Short answerHOTS / competency
A shopkeeper stocks shoe sizes by the average size sold. Sales by size are: 6 (20 pairs), 7 (35), 8 (25), 9 (10), 10 (10). Find the mean, median and mode and say which size he should stock most. (4 marks)
Hint
N = 100.
Answer
Mean = (120 + 245 + 200 + 90 + 100) / 100 = 7.55. Median: cf 20, 55; N/2 = 50, so size 7. Mode = 7 (highest frequency 35). He should stock size 7 most, since the mode shows the most wanted size.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Median class?Show answer
Where cf first reaches N/2.
2.Mode formula?Show answer
L + (f1 - f0)/(2f1 - f0 - f2) x h.
3.Empirical relation?Show answer
Mode = 3 Median - 2 Mean.
4.Positive skew order?Show answer
Mean > median > mode.
5.Best average with outliers?Show answer
Median.

Median, Mode and the Relationship among Mean, Median and Mode: frequently asked questions

Can a series have two modes?

Yes (bimodal).

Is the median affected by extreme values?

No.

What is N?

Sum of frequencies.

Is the relation exact?

No, approximate.