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.

The Formula

Q1=median of lower halfQ_1 = \text{median of lower half}, Q3=median of upper halfQ_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=2.5Q1 = 2.5, Q2=4.5Q2 = 4.5, Q3=6.5Q3 = 6.5.

Notation

Q1Q_1 (25th percentile), Q2Q_2 (median, 50th percentile), Q3Q_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

QpQ_p is the value where P(X≤Qp)=pP(X \leq Q_p) = p; specifically Q1=Q0.25Q_1 = Q_{0.25}, Q2=Q0.50Q_2 = Q_{0.50}, Q3=Q0.75Q_3 = Q_{0.75}

Worked Examples

Example 1

easy
Find the quartiles Q1Q_1, Q2Q_2, and Q3Q_3 for the data set: {4,7,9,11,14,18,22,25,30}\{4, 7, 9, 11, 14, 18, 22, 25, 30\}.

Answer

Q1=8Q_1 = 8, Q2=14Q_2 = 14, Q3=23.5Q_3 = 23.5

First step

1
The data is already sorted; n=9n = 9

Full solution

  1. 2
    Q2Q_2 (median): middle value at position 5 → Q2=14Q_2 = 14
  2. 3
    Lower half (below median): {4,7,9,11}\{4, 7, 9, 11\}; Q1Q_1 = median of lower half =7+92=8= \frac{7+9}{2} = 8
  3. 4
    Upper half (above median): {18,22,25,30}\{18, 22, 25, 30\}; Q3Q_3 = median of upper half =22+252=23.5= \frac{22+25}{2} = 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}\{2, 5, 8, 12, 15, 18, 21, 25, 30, 40\}. Find all quartiles and the five-number summary.

Example 3

medium
Find Q1Q_1, Q2Q_2, and Q3Q_3 for the dataset: 4,7,8,12,15,18,21,25,304, 7, 8, 12, 15, 18, 21, 25, 30.

Common Mistakes

  • Splitting by equal value width instead of equal count — quartiles divide the data into equal numbers of points, not equal ranges.
  • Forgetting to order the data first — quartiles, like the median, require a sorted list.
  • Confusing Q2 with the mean — Q2 is the median (the 50th-percentile rank), which may differ from the average.

Why This Formula Matters

Quartiles are the percentile workhorses of middle-school stats: they build the five-number summary, the box plot, and the IQR, and they describe spread in an outlier-resistant way the mean and SD cannot. They turn 'where does this value rank?' into a clean answer. Recognizing it by "Am I splitting ordered data into four groups that each contain the same fraction of the values?" — rather than by familiar numbers — is what lets a student tell it apart from median (q2) and interquartile range (iqr) and percentile (general) in a mixed problem set.

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?

Q1Q_1 (25th percentile), Q2Q_2 (median, 50th percentile), Q3Q_3 (75th percentile)

Why is the Quartiles formula important in Math?

Quartiles are the percentile workhorses of middle-school stats: they build the five-number summary, the box plot, and the IQR, and they describe spread in an outlier-resistant way the mean and SD cannot. They turn 'where does this value rank?' into a clean answer. Recognizing it by "Am I splitting ordered data into four groups that each contain the same fraction of the values?" — rather than by familiar numbers — is what lets a student tell it apart from median (q2) and interquartile range (iqr) and percentile (general) in a mixed problem set.

What do students get wrong about Quartiles?

The procedure for quartiles is the easy part; the trap is splitting by equal value width instead of equal count. Asking "Am I splitting ordered data into four groups that each contain the same fraction of the values?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Quartiles formula?

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

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