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

Median, Quartiles, Deciles and Percentiles Explained

Updated 10 October 2026 · Fact-checked

The median is the middle value of data arranged in order. Quartiles, deciles and percentiles split ordered data into 4, 10 and 100 equal parts. For grouped data, find the cumulative frequency, locate the class holding the required position, then apply L + [(position − cf) ÷ f] × h.

Understand Median, Quartiles, Deciles and Percentiles

The median is the middle item when data is arranged in ascending or descending order. Half the items lie below it and half above. Unlike the mean, extreme values do not pull it. That is why it suits incomes, house prices and other skewed data. The mode is the most frequent value, while the median is the central position. They answer different questions.

Quartiles, deciles and percentiles are positional values. They work like the median but cut the data at other points. Quartiles (Q1, Q2, Q3) divide the data into four equal parts. Deciles (D1 to D9) divide it into ten parts. Percentiles (P1 to P99) divide it into a hundred parts. So Q2 = D5 = P50 = median, Q1 = P25 and Q3 = P75.

For raw data, sort first, then count to the required position. For a discrete frequency table, build the cumulative frequency (cf) column and find the first value whose cf is at least the position. For grouped data, the position falls inside a class. You then assume items are spread evenly in that class and interpolate with the formula.

You can also find these values graphically. Draw a less-than ogive (cumulative frequency curve). Mark N/2 on the vertical axis, go across to the curve, then drop down to the horizontal axis. That reading is the median. Use N/4 or 3N/4 for quartiles.

Key formulas to remember

Median of raw data: n odd: median = value of ((n + 1) ÷ 2)th item; n even: average of the (n ÷ 2)th and (n ÷ 2 + 1)th items; Sort the data first. Never take the middle of unsorted data.
Quartiles, deciles, percentiles of raw or discrete data: Qi = i(N + 1) ÷ 4 th item; Di = i(N + 1) ÷ 10 th item; Pi = i(N + 1) ÷ 100 th item; This gives the position, not the value. For a frequency table, read the value whose cf first reaches or exceeds that position.
Median of grouped data: Median = L + [(N ÷ 2 − cf) ÷ f] × h; L = lower limit of the median class, cf = cumulative frequency before it, f = its frequency, h = class width, N = total frequency.
Quartile of grouped data: Qi = L + [(iN ÷ 4 − cf) ÷ f] × h, for i = 1, 2, 3; Locate the class containing the iN/4 th item using the cf column.
Decile of grouped data: Di = L + [(iN ÷ 10 − cf) ÷ f] × h, for i = 1 to 9; Same method; only the position changes.
Percentile of grouped data: Pi = L + [(iN ÷ 100 − cf) ÷ f] × h, for i = 1 to 99; Same method; only the position changes.
Equivalent positional values: Q2 = D5 = P50 = Median; Q1 = P25; Q3 = P75; Useful for checking and for saving time.

How to solve Median, Quartiles, Deciles and Percentiles questions

Use this method for any question on the median or other positional values.

1Check the data type: raw list, discrete frequency table, or grouped classes.
2Arrange raw data in order. For frequency tables, write the cumulative frequency (cf) column and find N, the total frequency.
3For grouped data, make sure classes are continuous (exclusive). If they are like 10–19, 20–29, first adjust the limits by 0.5 on each side.
4Work out the position: N/2 for the median, iN/4 for quartiles, iN/10 for deciles, iN/100 for percentiles. For raw or discrete data, you may use (N + 1) in place of N as shown in the formulas.
5Find the class (or value) whose cf first reaches or exceeds that position. This is the median, quartile, decile or percentile class.
6Pick out L, cf (the one before the class, not its own), f and h, and put them into the formula.
7Calculate step by step and check that the answer lies inside the chosen class.
8If a graph is asked for, plot the less-than ogive and read off the value at the required cumulative frequency.

Quickest way: Cumulative frequency and position check

When to use it: Use for grouped-data MCQs where time is short and four options are given.

Write only the cf column and compute the position (N/2, N/4 and so on).
Spot the class that contains the position. Eliminate any option outside that class at once.
Substitute into the formula. Often (position − cf) ÷ f simplifies to a neat fraction.
If options still remain, check that the answer is not lower than L or higher than the upper limit.
For Q2, D5 or P50 questions, treat them as the median and skip extra work.

Common mistakes in Median, Quartiles, Deciles and Percentiles

Using the cumulative frequency of the median class instead of the class before it.
The word cf is read as the cf at the class you found.
Fix: cf in the formula is always the cf of the previous class. Underline it before substituting.
Not sorting raw data before picking the middle.
Students rush and take the middle of the list as given.
Fix: Always write the data in ascending order first, then count to the position.
Ignoring that classes must be continuous.
Inclusive classes like 10–19 and 20–29 are used directly with L = 20.
Fix: Convert to 9.5–19.5, 19.5–29.5 and so on before applying the formula.
Using N/2 for the median class but then forgetting to change the position for Q1, Q3, deciles and percentiles.
Students memorise one formula and apply it every time.
Fix: Write the position first: iN/4, iN/10 or iN/100. Only the position changes.
Mixing up the median with the mode or the mean.
All three are called averages.
Fix: Median is the middle by position, mode is the most frequent, and mean is the sum divided by the count.
Choosing the median class by the first frequency that is larger than N/2, not by cf.
The frequency column is read instead of the cf column.
Fix: Always use the cf column: the class where cf first reaches or exceeds the position.

Worked examples

Example 1

The marks of 40 students are: 0–10: 4, 10–20: 8, 20–30: 10, 30–40: 10, 40–50: 8. Find the median, the first quartile and the third quartile.

Show the solution

Cumulative frequencies: 4, 12, 22, 32, 40. So N = 40.
Median: N/2 = 20. The first cf that is at least 20 is 22, so the median class is 20–30. L = 20, cf before = 12, f = 10, h = 10.
Median = 20 + [(20 − 12) ÷ 10] × 10 = 20 + 8 = 28.
Q1: N/4 = 10. The first cf at least 10 is 12, so the class is 10–20. L = 10, cf before = 4, f = 8, h = 10.
Q1 = 10 + [(10 − 4) ÷ 8] × 10 = 10 + 7.5 = 17.5.
Q3: 3N/4 = 30. The first cf at least 30 is 32, so the class is 30–40. L = 30, cf before = 22, f = 10, h = 10.
Q3 = 30 + [(30 − 22) ÷ 10] × 10 = 30 + 8 = 38.

Answer: Median = 28, Q1 = 17.5, Q3 = 38.

Example 2

A shop records the number of items per bill: 2 items on 3 bills, 4 items on 7 bills, 6 items on 10 bills, 8 items on 6 bills, 10 items on 4 bills. Find the median, Q1, Q3 and P90.

Show the solution

N = 3 + 7 + 10 + 6 + 4 = 30. Cumulative frequencies: 3, 10, 20, 26, 30.
Median position = (N + 1) ÷ 2 = 15.5. The first cf at least 15.5 is 20, which belongs to 6 items.
Q1 position = (30 + 1) ÷ 4 = 7.75. The first cf at least 7.75 is 10, which belongs to 4 items.
Q3 position = 3 × 31 ÷ 4 = 23.25. The first cf at least 23.25 is 26, which belongs to 8 items.
P90 position = 90 × 31 ÷ 100 = 27.9. The first cf at least 27.9 is 30, which belongs to 10 items.

Answer: Median = 6, Q1 = 4, Q3 = 8, P90 = 10.

Exam tips

Most questions are grouped-data calculations. Practise the formula until you can write L, cf, f and h from the table in under a minute.
Use options to your advantage: once you know the class, any option outside the class limits is wrong.
Remember Q2 = D5 = P50 = median. A question asking for any of these needs only the median calculation.
Watch the class form. If limits are inclusive (10–19), convert to exclusive first, or your answer will be off by 0.5.
Graphical questions usually ask where the median is read on a less-than ogive. Remember it is read at N/2 on the cumulative frequency axis.

Practice questions from Measures of Central Tendency and Dispersion

Median, Quartiles, Deciles and Percentiles in other exams

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

CA FoundationMedian, Quartiles, Deciles and Percentiles

Median, Quartiles, Deciles and Percentiles: frequently asked questions

What is the median formula for grouped data?

Median = L + [(N ÷ 2 − cf) ÷ f] × h. Here L is the lower limit of the median class, cf is the cumulative frequency of the class before it, f is the frequency of the median class, h is the class width and N is the total frequency.

How do I find quartiles from a frequency distribution?

Make the cumulative frequency column and work out iN/4 for Qi. Find the class where cf first reaches that position, then use L + [(iN/4 − cf) ÷ f] × h. For a discrete table with single values, read the value whose cf first reaches the position.

What is the difference between median and mode?

The median is the middle value of ordered data, so it depends on position. The mode is the value that occurs most often. In a skewed set of data they can be very different.

How are deciles and percentiles different from quartiles?

They use the same method but divide data into more parts. Quartiles give four parts, deciles ten and percentiles a hundred. Only the position changes: iN/4, iN/10 or iN/100.