Quartiles for Ungrouped Data

What Are Quartiles?

The median is like a ruler that divides data into two equal parts, right in the middle ($50\%$). Well, there's another friend of the median, called quartiles.

If the median divides data into two, quartiles are even better; they divide the sorted data into four equal parts! Imagine you have a chocolate bar, and you break it into four equal pieces. Quartiles are the breaking points.

There are three quartile breaking points:

1. Lower Quartile ($Q_1$): This is the first break. It separates the smallest $25\%$ of the data from the rest. Like the first quarter of the chocolate.
2. Middle Quartile ($Q_2$): This is the median! It's exactly in the middle, dividing the data in half ($50\%$ left, $50\%$ right). Like the break in the middle of the chocolate.
3. Upper Quartile ($Q_3$): This is the last break. It separates the smallest $75\%$ of the data from the largest $25\%$. Like the boundary after three-quarters of the chocolate.

So, $Q_1$, $Q_2$, and $Q_3$ divide our data into four small groups with the same number of data points ($25\%$ each).

How to Find the Position of Quartiles

Okay, now how do we know the position (rank) of $Q_1$, $Q_2$, and $Q_3$ in our ordered data?

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

Lower Quartile

The formula is simple:

• If the result is a whole number, for example $5$, then $Q_1$ is the value of the $5^{\text{th}}$ data point.
• If the result has a decimal, for example $5.25$, then $Q_1$ lies between the $5^{\text{th}}$ and $6^{\text{th}}$ data points. (There's a way to calculate its value later, but for now, we're just finding the position).

Simple 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 $Q_1$ is between the $5^{\text{th}}$ and $6^{\text{th}}$ data points.

Middle Quartile

This is the median, so the formula is:

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 $11$.

Simple 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 rules are exactly the same:

• 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 $16$.

Simple 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 $15.75$.

This means $Q_3$ is between data points $15$ and $16$.

Exercise

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

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

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 $Q_1$:

  $\text{Data point at }\frac{1}{4}(n+1) = \text{Data point at }\frac{1}{4}(7+1) = \text{Data point at }\frac{8}{4} = \text{Data point at }2$

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

• Position of $Q_2$ (Median):

  $\text{Data point at }\frac{1}{2}(n+1) = \text{Data point at }\frac{1}{2}(7+1) = \text{Data point at }\frac{8}{2} = \text{Data point at }4$

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

• Position of $Q_3$:

  $\text{Data point at }\frac{3}{4}(n+1) = \text{Data point at }\frac{3}{4}(7+1) = \text{Data point at }\frac{24}{4} = \text{Data point at }6$

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

Step $3$: Determine the quartile values

Look at the sorted data: $5, 6, 7, 7, 8, 9, 10$

• $Q_1$ 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

You might be wondering, "If there's $Q_1$, $Q_2$, and $Q_3$, is there a $Q_4$?"

Technically, the concept of quartiles divides the data into four parts. $Q_1$ is the boundary for the first $25\%$, $Q_2$ (the median) is the $50\%$ boundary, and $Q_3$ is the $75\%$ boundary. The final boundary, which encompasses $100\%$ of the data, is actually the maximum value of the dataset.

So, while we could refer to the maximum value as $Q_4$, in statistical analysis, we don't typically use the term $Q_4$ explicitly. The main focus is on $Q_1$, $Q_2$, and $Q_3$ because they provide important information about the spread and center of the data in the lower, middle, and upper sections. The minimum value is sometimes called $Q_0$, but like $Q_4$, it's less commonly used than $Q_1$, $Q_2$, and $Q_3$.