Quartiles Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Quartiles.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Quartiles divide an ordered data set into four equal parts: Q1 is the 25th percentile, Q2 is the median (50th), and Q3 is the 75th percentile.

Q1 = 25th percentile, Q2 = median (50th), Q3 = 75th percentile.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Quartiles split data into quarters: 25\% below Q1, 25\% between Q1-Q2, etc.

Common stuck point: Different methods for calculating quartiles can give slightly different results.

Sense of Study hint: Sort the data, then find the median to split it into halves. Q1 is the median of the lower half; Q3 is the median of the upper half.

Worked Examples

Example 1

easy
Find the quartiles Q_1, Q_2, and Q_3 for the data set: \{4, 7, 9, 11, 14, 18, 22, 25, 30\}.

Solution

  1. 1
    The data is already sorted; n = 9
  2. 2
    Q_2 (median): middle value at position 5 → Q_2 = 14
  3. 3
    Lower half (below median): \{4, 7, 9, 11\}; Q_1 = median of lower half = \frac{7+9}{2} = 8
  4. 4
    Upper half (above median): \{18, 22, 25, 30\}; Q_3 = median of upper half = \frac{22+25}{2} = 23.5

Answer

Q_1 = 8, Q_2 = 14, Q_3 = 23.5
Quartiles divide ordered data into four equal parts. Q1 is the 25th percentile, Q2 (median) is the 50th percentile, and Q3 is the 75th percentile. For even-sized halves, average the two middle values.

Example 2

medium
A dataset of 10 values (sorted): \{2, 5, 8, 12, 15, 18, 21, 25, 30, 40\}. Find all quartiles and the five-number summary.

Example 3

medium
Find Q_1, Q_2, and Q_3 for the dataset: 4, 7, 8, 12, 15, 18, 21, 25, 30.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find Q_1 and Q_3 for: \{10, 20, 30, 40, 50, 60, 70\}.

Example 2

hard
A student scores at the 75th percentile on a standardized test with scores \{45, 52, 60, 67, 72, 78, 83, 89, 95, 98\}. Confirm that Q_3 equals the 75th percentile score and find what percent of students scored below Q_1.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

median