EconomicsCBSE Class 11 · Measures of Central Tendency

Median, Quartiles, Percentiles and Mode

Updated 17 October 2026 · Fact-checked

Median is the middle value of an ordered series; for a continuous series median = l + ((N/2 − cf)/f) x h. Quartiles divide the data into four equal parts (Q1 at N/4, Q3 at 3N/4). Mode is the most frequent value; for continuous data mode = l + ((f1 − f0)/(2f1 − f0 − f2)) x h. Mean, median and mode are related by the empirical formula mode = 3 median − 2 mean.

Median and mode both lie in the tallest class 20-30, while Q1 and Q3 cut the data into quarters on either side.

Illustration drawn for QuestionBuddy.

Understand Median, Quartiles and Mode

Median. The middle value when data are arranged in order; half the items lie below and half above. Unaffected by extreme values.

Individual series (odd n): value at position (n + 1) ÷ 2. Even n: average of the two middle values. Example: 4, 7, 9, 12, 15, 21 → median = (9 + 12) ÷ 2 = 10.5.
Continuous series: cumulative frequency (cf); median class contains N ÷ 2; Median = l + [(N/2 − cf) ÷ f] × h, where l = lower limit of the median class, cf = cumulative frequency of the class before it, f = its frequency, h = width.

Quartiles and percentiles. Q₁ (lower quartile) at N ÷ 4, Q₃ (upper quartile) at 3N ÷ 4, Q₂ = median; percentiles divide into 100 parts (P₉₀ at 90N ÷ 100). Same formula with N/4, 3N/4 or kN/100 in place of N/2.

Mode. The most frequently occurring value.

Individual or discrete series: value with the highest frequency (inspection, or grouping).
Continuous series: modal class has the highest frequency; Mode = l + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h, where f₁ = frequency of modal class, f₀ = frequency of the preceding class, f₂ = frequency of the succeeding class.

Worked data. Classes 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 15, 12, 10 (N = 50); cf = 5, 13, 28, 40, 50.

Median: N/2 = 25 falls in 20-30 (cf before = 13): 20 + (25 − 13)/15 × 10 = 20 + 8 = 28.
Q₁: N/4 = 12.5 in 10-20: 10 + (12.5 − 5)/8 × 10 = 19.375.
Q₃: 3N/4 = 37.5 in 30-40: 30 + (37.5 − 28)/12 × 10 = 37.92 (approximately).
P₉₀: 45 falls in 40-50: 40 + (45 − 40)/10 × 10 = 45.
Mode: modal class 20-30 (f₁ = 15, f₀ = 8, f₂ = 12): 20 + (15 − 8)/(30 − 8 − 12) × 10 = 20 + 7/10 × 10 = 27.
Mean (from the previous page) = 27.8: mean, median and mode are close (27.8, 28, 27), so the distribution is nearly symmetrical.

Empirical relation for moderately skewed data: Mode = 3 Median − 2 Mean (= 84 − 55.6 = 28.4 here, close to the calculated mode 27).

Choosing an average. Mean: for further calculations, symmetric data. Median: skewed data or open classes (e.g. income). Mode: most typical value (e.g. most common shoe size).

Draw: locate the median on the ogive at N/2.

Median, Quartiles and Mode: key facts

Median (continuous): l + ((N/2 - cf) / f) x h; Median class contains N/2
Quartiles: Q1: N/4
Q3: 3N/4
same formula; Percentile kN/100
Mode (continuous): l + ((f1 - f0) / (2 f1 - f0 - f2)) x h; Modal class
Empirical: Mode = 3 Median - 2 Mean; Moderate skew

How to answer Median, Quartiles and Mode questions

Questions give a frequency table and ask for median, quartiles or mode.

1Make the cumulative frequency column and find N/2.
2Identify the class and substitute in the formula.
3For mode, find the modal class and its neighbours.
4Check by the empirical relation or the mean.
5For even individual series, average the middle two.

Median, Quartiles 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.

Shoe shop

Suppose a shop sells 50 pairs of shoes with sizes grouped and the owner wants the size to stock most.

Which average does the owner need?

Try it first, then show the solutionSolution

The owner wants the most commonly sold size, i.e. the mode.
The mean might be a size nobody wears.
The median helps divide customers into smaller and larger halves.

Answer: The mode.

Memory hook: Most common means mode.

Tricks to remember Median, Quartiles and Mode

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

N by 2 for the middle
Median at N/2; Q1 at N/4; Q3 at 3N/4.
Use it when: Positions
f1 minus f0 over twice f1 minus both
Mode formula denominator 2f1 - f0 - f2.
Use it when: Mode
Mode = 3 Med - 2 Mean
Empirical rule.
Use it when: Relation

Median, Quartiles and Mode: solved examples

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

Example 1: Individual median · 2 marks · Very short answerBoard level

Find the median of 4, 7, 9, 12, 15, 21 and of 4, 7, 9, 12, 15.

Try it first, then show the solutionSolution

Six items: median = (9 + 12)/2 = 10.5 (1 mark).
Five items: middle value = 9 (1 mark).

Answer: 10.5; 9.

Example 2: Median of a continuous series · 4 marks · Short answerBoard level

Find the median for classes 0-10, 10-20, 20-30, 30-40, 40-50 with f = 5, 8, 15, 12, 10.

Try it first, then show the solutionSolution

cf = 5, 13, 28, 40, 50; N/2 = 25 (1 mark).
Median class 20-30 with l = 20, cf = 13, f = 15, h = 10 (1 mark).
Median = 20 + (25 - 13)/15 x 10 (1 mark).
= 20 + 8 = 28 (1 mark).

Answer: 28.

Example 3: Mode · 3 marks · Short answerHarder board level

Find the mode of the same data and compare with the mean 27.8 and median 28.

Try it first, then show the solutionSolution

Modal class 20-30: f1 = 15, f0 = 8, f2 = 12 (1 mark).
Mode = 20 + (15 - 8)/(30 - 8 - 12) x 10 = 20 + 7 = 27 (1 mark).
Mean 27.8, median 28, mode 27 are close, so the distribution is nearly symmetric (1 mark).

Answer: Mode 27.

Example 4: Quartiles · 5 marks · Long answerHarder board level

For the same data find Q1, Q3 and the quartile range Q3 - Q1.

Try it first, then show the solutionSolution

Q1: N/4 = 12.5 in 10-20 (cf 5, f 8) (1 mark).
Q1 = 10 + (12.5 - 5)/8 x 10 = 19.375 (1 mark).
Q3: 3N/4 = 37.5 in 30-40 (cf 28, f 12) (1 mark).
Q3 = 30 + (37.5 - 28)/12 x 10 = 37.92 (1 mark).
Q3 - Q1 = 37.92 - 19.38 = 18.54 (1 mark).

Answer: Q1 19.38; Q3 37.92; difference 18.54.

Example 5: Empirical check and choice · 5 marks · Long answerHOTS / competency

Using mean 27.8 and median 28, find the empirical mode, compare with 27, and explain which average best describes household incomes in a village with a few very rich families.

Try it first, then show the solutionSolution

Mode = 3 x 28 - 2 x 27.8 (1 mark).
= 84 - 55.6 = 28.4 (1 mark).
28.4 is close to the calculated mode 27, since skewness is slight (1 mark).
For incomes with a few very rich families the mean is pulled upward by extreme values (1 mark).
The median is a better typical value because it is unaffected by extremes (1 mark).

Answer: 28.4; median suits skewed incomes.

Where students lose marks in Median, Quartiles and Mode

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

Mistake: Median class by N
Used N instead of N/2.
Wrong working
Find the class containing N.
Right working
Find the class containing N/2.
Do this: Half.
Usually costs: 1 mark.
Mistake: Wrong cf
Took cumulative frequency of the median class.
Wrong working
cf of median class = 28.
Right working
cf of the preceding class = 13.
Do this: Previous.
Usually costs: 1 mark.
Mistake: Mode as highest value
Frequency vs value.
Wrong working
Mode is the largest value.
Right working
Mode is the most frequent value.
Do this: Frequency.
Usually costs: 1 mark.

Exam tips

Always tabulate cumulative frequencies.
Name the median/modal class and the values of l, cf, f, h.
Check with the empirical relation.
Say which average fits the situation.

Practice questions on Median, Quartiles 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
The median divides a distribution into
(a) two equal parts (b) four equal parts (c) ten equal parts (d) hundred equal parts
Hint
Middle.
Answer
(a).
Q2 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Median is preferred for income data.
Reason (R): It is not affected by extreme values.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Skewed.
Answer
(a).
Q3 · 2 marks · Very short answerHarder board level
Find the 90th percentile for classes 0-10 to 40-50 with f = 5, 8, 15, 12, 10.
Hint
0.9 N.
Answer
0.9 x 50 = 45 falls in 40-50 (cf 40, f 10): 40 + (45 - 40)/10 x 10 = 45.
Q4 · 4 marks · Case-basedHarder board level
Read and answer.
Suppose a series has mean 40 and mode 34.
(a) Estimate the median by the empirical relation. (2)
(b) State whether the distribution is symmetric or skewed. (1)
(c) Which average would you prefer for open-ended classes? (1)
Hint
Mode = 3 Med - 2 Mean.
Answer
(a) 34 = 3 x Median - 80 gives Median = 114 / 3 = 38. (b) Mean is greater than median than mode, so positively skewed. (c) The median (or mode), since the mean needs closed classes.
Q5 · 3 marks · Short answerHOTS / competency
Suppose the median of a series is 28, the first quartile 19.4 and the third quartile 37.9. Compare the distances of Q1 and Q3 from the median and say what it shows about symmetry.
Hint
Skew check.
Answer
Median - Q1 = 8.6; Q3 - median = 9.9. The upper part is a little longer, so the distribution is slightly skewed to the right but nearly symmetric.

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 position?Show answer
N/2.
2.Q3 position?Show answer
3N/4.
3.Mode meaning?Show answer
Most frequent value.
4.f0 means?Show answer
Frequency of the preceding class.
5.Empirical relation?Show answer
Mode = 3 Median - 2 Mean.
6.Average for skewed data?Show answer
Median.

Median, Quartiles and Mode: frequently asked questions

What is the median?

Middle value of ordered data.

What is a quartile?

A value dividing data into four equal parts.

What is the mode?

The most frequent value.

When is mode useful?

To know the most common size or value.

Do all three averages match?

Only for symmetric distributions.