Quartiles for Ungrouped Data

Three Boundaries across an Ordered Dataset

The median marks the middle of an ordered dataset at $50\%$. Quartiles extend that idea by marking three positions across the ordered data.

Those positions divide the distribution by rank into four parts, each representing about one quarter of the observations. With a finite dataset or tied values, the parts need not contain four visibly distinct sets of equal size.

There are three quartile boundaries:

1. Lower Quartile ($Q_1$): The boundary at approximately the $25\%$ rank.
2. Middle Quartile ($Q_2$): The median, at approximately the $50\%$ rank.
3. Upper Quartile ($Q_3$): The boundary at approximately the $75\%$ rank.

Together, $Q_1$, $Q_2$, and $Q_3$ describe the lower quarter, center, and upper quarter of an ordered distribution.

How to Find the Position of Quartiles

How do we find the rank positions of $Q_1$, $Q_2$, and $Q_3$ in ordered data?

Assume we have $n$ data points that we have sorted from smallest to largest.

Quartiles can be defined by several valid ranking conventions, so software packages may return slightly different values for the same small dataset. The calculations below consistently use the interpolated position $p(n+1)$, where $p$ is $\frac14$, $\frac12$, or $\frac34$. A position below $1$ is clamped to the minimum and a position above $n$ to the maximum. For a valid comparison, calculate every result with the same convention.

Lower Quartile

For the lower quartile:

• If the result is a whole number, for example $5$, then $Q_1$ is the value at position $5$.
• If the result has a decimal, for example $5.25$, then $Q_1$ lies between positions $5$ and $6$. We will calculate its interpolated value later. For now, we are only locating it.

Example:

Suppose we have $20$ data points ($n=20$).

The position of $Q_1$ is the data point at $\frac{1}{4}(20+1)$, which is data point $\frac{21}{4}$ or $5.25$.

This means $Q_1$ lies between positions $5$ and $6$.

Middle Quartile

The middle quartile is the median:

The rules are the same as for $Q_1$:

• If the result is a whole number, say $10$, $Q_2$ is the value of data point $10$.
• If the result has a decimal, say $10.5$, $Q_2$ is between data points $10$ and $11$.

Example ($n=20$):

The position of $Q_2$ is the data point at $\frac{1}{2}(20+1)$, which is data point $\frac{21}{2}$ or $10.5$.

This means $Q_2$ (the median) is between data points $10$ and $11$.

Upper Quartile

The formula is similar again:

The same interpolation rule applies:

• If the result is a whole number, say $15$, $Q_3$ is the value of data point $15$.
• If the result has a decimal, say $15.75$, $Q_3$ is between data points $15$ and $16$.

Example ($n=20$):

The position of $Q_3$ is the data point at $\frac{3}{4}(20+1)$, which is data point $\frac{63}{4}$ or $15.75$.

A position of $15.75$ places $Q_3$ between data points $15$ and $16$.

Worked exercise

Try to find the position of $Q_1$, $Q_2$, and $Q_3$ from the math test scores of these $7$ children:

Scores: $7, 5, 8, 6, 9, 7, 10$

Try the problem yourself first. Then compare your reasoning with the complete solution below.

Step 1: Sort the data first!

Sorted data: $5, 6, 7, 7, 8, 9, 10$

Number of data points: $n = 7$

Step 2: Find the quartile positions using the formulas

• Position of $Q_1$:

  $$\begin{aligned}\frac{1}{4}(n+1) &= \frac{1}{4}(7+1) \\ &= \frac{8}{4} \\ &= 2\end{aligned}$$

  The result is a whole number ($2$), so $Q_1$ is the $2$nd data point.

• Position of $Q_2$ (Median):

  $$\begin{aligned}\frac{1}{2}(n+1) &= \frac{1}{2}(7+1) \\ &= \frac{8}{2} \\ &= 4\end{aligned}$$

  The result is a whole number ($4$), so $Q_2$ is the $4$th data point.

• Position of $Q_3$:

  $$\begin{aligned}\frac{3}{4}(n+1) &= \frac{3}{4}(7+1) \\ &= \frac{24}{4} \\ &= 6\end{aligned}$$

  The result is a whole number ($6$), so $Q_3$ is the $6$th data point.

Step 3: Determine the quartile values

Using the sorted data from Step 1:

• $Q_1$ is $2$nd data point = $6$
• $Q_2$ is $4$th data point = $7$
• $Q_3$ is $6$th data point = $9$

The Fourth Quartile

Quartiles divide ordered data into four parts. $Q_1$, $Q_2$, and $Q_3$ mark the boundaries at $25\%$, $50\%$, and $75\%$. The outer boundary at $100\%$ is the maximum value.

The maximum is sometimes labeled $Q_4$, and the minimum is sometimes labeled $Q_0$. Most statistical summaries name the minimum and maximum directly and reserve quartile notation for $Q_1$, $Q_2$, and $Q_3$.