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.

Q1 (Lower Quartile)

The formula is simple:

• If the result is a whole number, for example 5, then $Q_1$ is the value of the 5th data point.
• If the result has a decimal, for example $5.25$, then $Q_1$ lies between the 5th and 6th 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 ($n=20$).

Position of $Q_1$ = Data point at $\frac{1}{4}(20+1)$ = Data point at $\frac{21}{4}$ = Data point at 5.25.

This means $Q_1$ is between the 5th and 6th data points.

Q2 (Median or 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 the 10th data point.
• If the result has a decimal, say 10.5, $Q_2$ is between the 10th and 11th data points.

Simple Example ($n=20$):

Position of $Q_2$ = Data point at $\frac{1}{2}(20+1)$ = Data point at $\frac{21}{2}$ = Data point at 10.5.

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

Q3 (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 the 15th data point.
• If the result has a decimal, say 15.75, $Q_3$ is between the 15th and 16th data points.

Simple Example ($n=20$):

Position of $Q_3$ = Data point at $\frac{3}{4}(20+1)$ = Data point at $\frac{63}{4}$ = Data point at 15.75.

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

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 2nd 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 4th 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 6th data point.

Step 3: Determine the quartile values

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

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

The Fourth Quartile (Q4)

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 $Q_3$.