Quartiles Formula

The Formula

Q_1 = \text{median of lower half}, Q_3 = \text{median of upper half}

When to use: Q1 = 25th percentile, Q2 = median (50th), Q3 = 75th percentile.

Quick Example

Data: 1, 2, 3, 4, 5, 6, 7, 8. Q1 = 2.5, Q2 = 4.5, Q3 = 6.5.

Notation

Q_1 (25th percentile), Q_2 (median, 50th percentile), Q_3 (75th percentile)

What This Formula Means

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.

Formal View

Q_p is the value where P(X \leq Q_p) = p; specifically Q_1 = Q_{0.25}, Q_2 = Q_{0.50}, Q_3 = Q_{0.75}

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.

Common Mistakes

  • Forgetting to sort the data before computing quartiles
  • Including the median in both halves when finding Q1 and Q3 — different methods handle this differently, leading to confusion
  • Confusing quartiles (values that split data into fourths) with quarters (25\%) — quartiles are specific data values, not percentages

Why This Formula Matters

Quartiles provide a robust summary of distribution shape and spread, and they define the IQR — the standard measure of variability used in box plots and outlier detection.

Frequently Asked Questions

What is the Quartiles formula?

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.

How do you use the Quartiles formula?

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

What do the symbols mean in the Quartiles formula?

Q_1 (25th percentile), Q_2 (median, 50th percentile), Q_3 (75th percentile)

Why is the Quartiles formula important in Math?

Quartiles provide a robust summary of distribution shape and spread, and they define the IQR — the standard measure of variability used in box plots and outlier detection.

What do students get wrong about Quartiles?

Different methods for calculating quartiles can give slightly different results.

What should I learn before the Quartiles formula?

Before studying the Quartiles formula, you should understand: median.

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