Quantitative Aptitude · Measures of Central Tendency and Dispersion

Median, Quartiles, Deciles and Percentiles for CA Foundation

Updated 1 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 ungrouped data, find the item at position i(n+1) ÷ parts. For grouped data, find the class where cumulative frequency first reaches iN ÷ parts, then apply L + (iN ÷ parts − cf) ÷ f × h.

Understand Median, Quartiles, Deciles and Percentiles

The median is the middle value when data is sorted. Half the items are below it and half are above it. Because it depends on position, not on size, extreme values do not disturb it. That is why it suits skewed data such as incomes.

Partition values extend the same idea. Quartiles (Q1, Q2, Q3) cut the sorted data into four equal parts. Deciles (D1 to D9) cut it into ten parts. Percentiles (P1 to P99) cut it into a hundred parts. So Q2, D5 and P50 are all the median.

For ungrouped data you sort the values and count to a position. For grouped data you do not have individual values. You build a cumulative frequency column, find the class that holds the required position, and then estimate inside that class by interpolation. The formula assumes the items are spread evenly inside the class.

An ogive is the graph of cumulative frequency. Plot a less-than ogive and read across from the position N/2 on the vertical axis to the curve, then down to the horizontal axis. That value is the median. The same method gives any quartile, decile or percentile. If you draw both less-than and more-than ogives, they cross at the median.

Median versus mode: the median is a positional average (the middle item). The mode is the most frequent value. The median always exists for ordered data. A data set may have no mode or several modes.

Key formulas to remember

Median, ungrouped data: Median = value of the (n + 1) ÷ 2 th item in sorted data; If n is even, the position ends in .5. Take the average of the two middle items.
Quartiles, ungrouped data: Qi = value of the i(n + 1) ÷ 4 th item, i = 1, 2, 3; If the position is fractional, take the lower item plus that fraction of the gap to the next item.
Deciles and percentiles, ungrouped data: Di = value of the i(n + 1) ÷ 10 th item; Pi = value of the i(n + 1) ÷ 100 th item; Sort the data first. Use the same interpolation for fractional positions.
Median, grouped data: Median = L + [(N ÷ 2 − cf) ÷ f] × h; L = lower boundary of median class, N = total frequency, cf = cumulative frequency of the class before it, f = frequency of the median class, h = class width. Classes must be continuous.
Quartile, decile, percentile, grouped data: Qi = L + [(iN ÷ 4 − cf) ÷ f] × h; Di = L + [(iN ÷ 10 − cf) ÷ f] × h; Pi = L + [(iN ÷ 100 − cf) ÷ f] × h; Same formula as the median. Only the target position changes. L, cf, f and h refer to the class that contains that position.
Equivalences: Q2 = D5 = P50 = Median; Q1 = P25; Q3 = P75; Di = P(10i); Use these to convert one question into another type.

How to solve Median, Quartiles, Deciles and Percentiles questions

This method works for ungrouped data, grouped data and ogive questions.

1Identify what is asked: median, a quartile, a decile or a percentile. Note the number of parts (2, 4, 10 or 100) and the index i.
2For ungrouped data, sort the values in ascending order and count n.
3For ungrouped data, find the position i(n + 1) ÷ parts. If it is a whole number, read that item. If it ends in a fraction, interpolate between the two neighbouring items.
4For grouped data, make sure classes are continuous. If they are inclusive (such as 10–19, 20–29), subtract 0.5 from lower limits and add 0.5 to upper limits.
5Build the less-than cumulative frequency column and find N.
6Compute the target position iN ÷ parts. Pick the first class whose cumulative frequency is at least this position. That is your class.
7Read L, cf (of the previous class), f and h. Substitute into L + [(target − cf) ÷ f] × h.
8Check that the answer lies inside the chosen class. For an ogive question, read the value across from the target position on the cumulative frequency axis.

Quickest way: Locate the class first, then eliminate options

When to use it: Grouped-data MCQs where the four options are numbers. This saves time and avoids wrong-answer penalties.

Write the cumulative frequency column quickly and find N.
Find the target position, such as N/2 or 3N/4, and the class that holds it.
Cross out every option that lies outside that class. This often leaves one or two options.
If two remain, do the one-line interpolation. Compute (target − cf) ÷ f first, then multiply by h and add L.
Skip the question if the table has many classes and you have not yet done easier ones. A wrong answer costs 0.25 marks.

Common mistakes in Median, Quartiles, Deciles and Percentiles

Not sorting ungrouped data before finding the median or a quartile.
Students count positions in the order the data is given.
Fix: Always rewrite the data in ascending order first. Do it even if the data looks nearly sorted.
Using (n + 1) in the grouped formula, or using N ÷ 2 as a position in ungrouped data.
The two cases look similar and students mix up the rules.
Fix: Ungrouped data uses i(n + 1) ÷ parts. Grouped data uses iN ÷ parts inside the interpolation formula.
Using the class frequency instead of the cumulative frequency as cf.
Both columns are in the table, and the letters f and cf look alike.
Fix: cf is the cumulative frequency of the class just before the median class. f is the frequency of the median class itself.
Choosing the wrong class by picking the last class whose cumulative frequency is below the target.
Students stop one row early or one row late.
Fix: Pick the first class whose cumulative frequency is greater than or equal to the target position. Then check that the answer falls inside that class.
Ignoring class boundaries when classes are inclusive, such as 10–19 and 20–29.
Students take L directly from the table.
Fix: Convert to continuous classes by subtracting 0.5 from lower limits and adding 0.5 to upper limits. Then h is the width of the boundaries.
Mixing up decile and percentile targets, for example using 8N ÷ 100 for D8.
Students forget that decile uses 10 parts and percentile uses 100.
Fix: Remember Di = P(10i). For D8 use 8N ÷ 10, which equals P80 at 80N ÷ 100.

Worked examples

Example 1

The sorted data 5, 8, 12, 15, 17, 20, 24, 28, 30 are given. The third quartile Q3 is: (a) 24 (b) 26 (c) 27 (d) 28

Show the solution

Data is already sorted and n = 9.
Position of Q3 = 3(n + 1) ÷ 4 = 3 × 10 ÷ 4 = 7.5.
The 7th item is 24 and the 8th item is 28.
Q3 = 24 + 0.5 × (28 − 24) = 24 + 2 = 26.

Answer: (b) 26

Example 2

Find the median of this distribution. Class 0–10: frequency 6; 10–20: 10; 20–30: 20; 30–40: 16; 40–50: 8. Options: (a) 23 (b) 27 (c) 30 (d) 36

Show the solution

N = 6 + 10 + 20 + 16 + 8 = 60.
Cumulative frequencies are 6, 16, 36, 52, 60.
N ÷ 2 = 30. The first cumulative frequency that is at least 30 is 36, so the median class is 20–30.
L = 20, cf = 16, f = 20, h = 10.
Median = 20 + [(30 − 16) ÷ 20] × 10 = 20 + 0.7 × 10 = 20 + 7 = 27.
Check: 27 lies within 20–30. Option 23 also lies inside 20–30, but it is not what the interpolation gives (27). Option 30 is the upper limit of the class and 36 is a cumulative frequency, not a median.

Answer: (b) 27

Example 3

For the same distribution (0–10: 6; 10–20: 10; 20–30: 20; 30–40: 16; 40–50: 8), the 8th decile D8 is: (a) 35.625 (b) 36 (c) 37.5 (d) 40

Show the solution

N = 60. Target position = 8N ÷ 10 = 48.
Cumulative frequencies are 6, 16, 36, 52, 60. The first one at least 48 is 52, so the class is 30–40.
L = 30, cf = 36, f = 16, h = 10.
D8 = 30 + [(48 − 36) ÷ 16] × 10 = 30 + 0.75 × 10 = 30 + 7.5 = 37.5.
Check: 37.5 lies within 30–40. Option (a) is Q3 of this data, not D8.

Answer: (c) 37.5

Exam tips

Look at the options before calculating. In grouped data, finding the class that holds the target position usually removes two or three options.
Read the question for the word used: quartile, decile or percentile. Write the target position (such as 3N ÷ 4 or 8N ÷ 10) at the top before touching the table.
Check whether the classes are inclusive or continuous. Questions sometimes use inclusive classes to test boundary correction.
For ogive questions, remember that the median is read from N ÷ 2 on the cumulative frequency axis. The point where the less-than and more-than ogives cross also gives it.
Do the cumulative frequency column carefully and check that the last value equals N. One addition slip spoils the whole answer.

Practice questions from Measures of Central Tendency and Dispersion

Median, Quartiles, Deciles and Percentiles: frequently asked questions

What is the median formula for grouped data in CA Foundation?

Median = L + [(N ÷ 2 − cf) ÷ f] × h. Here L is the lower boundary of the median class, cf is the cumulative frequency before that class, f is its frequency and h is its width. The median class is the first class whose cumulative frequency is at least N ÷ 2.

How do I calculate quartiles, deciles and percentiles?

Use the same method with a different target position. For ungrouped sorted data, the position is i(n + 1) ÷ 4, 10 or 100. For grouped data, the target is iN ÷ 4, 10 or 100, and you interpolate in the class that contains it.

How does the ogive method find the median?

Draw the less-than cumulative frequency curve. Mark N ÷ 2 on the vertical axis, move across to the curve, then drop down to the horizontal axis. That value is the median. The less-than and more-than ogives also intersect at the median.

What is the difference between median and mode?

The median is the middle value of ordered data, so half the items lie on each side. The mode is the value that occurs most often. The median is not affected by extreme values, and a data set can have more than one mode or none.