DAT Quantitative Reasoning: Quartiles and the Interquartile Range

DAT High-Yield Concept

Quartile Intervals: Quartiles split ordered data into four equal parts; each quartile band contains approximately 25% of the values.
Odd Sample Exclusion Rule: When the sample count $N$ is odd, leave out the median value itself from both the lower and upper halves when calculating $Q_1$ and $Q_3$ .
Interquartile Range Formula: $IQR = Q_3 - Q_1$
Outlier Resistance: The $IQR$ measures central spread while ignoring extreme values, unlike the total Range ( $\text{Max} - \text{Min}$ ).

Determining Quartiles

To compute quartiles for any dataset on the DAT, follow a three-step method:

Order the Data: Arrange all observations in ascending order from smallest to largest.
Split at the Median ( $Q_2$ ): Find the median value that splits the ordered dataset into two equal halves. If $N$ is odd, $Q_2$ is the single middle number. If $N$ is even, $Q_2$ is the arithmetic average of the two middle numbers.
Find $Q_1$ and $Q_3$ : The first quartile ( $Q_1$ ) is the median of the lower half, and the third quartile ( $Q_3$ ) is the median of the upper half. Crucial Rule: When $N$ is odd, the median value itself is left out of both halves when finding $Q_1$ and $Q_3$ .

Quartile Bands: Quartiles divide data into four sequential bands. Approximately 25% (or at least 25% in discrete sample counts) of observations fall into each quartile band: below $Q_1$ , between $Q_1$ and $Q_2$ , between $Q_2$ and $Q_3$ , and above $Q_3$ .

Worked Example 1: Finding Quartiles with an Odd Sample Count

Problem: Determine $Q_1$ , $Q_2$ , and $Q_3$ for the dataset: $3, 15, 5, 22, 12, 9, 18$ .

Step 1: Order the data ( $N = 7$ ).

$3, 5, 9, 12, 15, 18, 22$

Step 2: Find the median ( $Q_2$ ).

Since $N = 7$ is odd, the 4th value is the median: $Q_2 = 12$ .

Step 3: Split into halves excluding $Q_2$ .

Lower half: $\{3, 5, 9\}$ → $Q_1 = 5$ .

Upper half: $\{15, 18, 22\}$ → $Q_3 = 18$ .

DAT Exam Trap: A common trap is including the median ( $12$ ) in both halves when $N$ is odd. Including $12$ in the lower half $\{3, 5, 9, 12\}$ falsely gives $Q_1 = \frac{5 + 9}{2} = 7$ , which is incorrect.

Interquartile Range (IQR) vs. Total Range

The Interquartile Range (IQR) measures the spread of the middle 50% of the dataset:

$IQR = Q_3 - Q_1$

Because the IQR depends only on $Q_1$ and $Q_3$ , it measures spread while ignoring extreme values (outliers). In contrast, the total range ( $\text{Range} = \text{Max} - \text{Min}$ ) uses only the extreme boundary values and is easily distorted. (Note: While boxplot interpretation is covered in depth in its own lesson, visually, the interquartile range corresponds to the width of the box).

Worked Example 2: Calculating IQR and Range with Outliers

Problem: Calculate the Range and IQR for: $4, 7, 10, 14, 18, 20, 25, 90$ .

Step 1: Find $Q_2$ ( $N = 8$ , even).

$Q_2 = \frac{14 + 18}{2} = 16$

Step 2: Find $Q_1$ and $Q_3$ .

Lower half: $\{4, 7, 10, 14\}$ → $Q_1 = \frac{7 + 10}{2} = 8.5$ .

Upper half: $\{18, 20, 25, 90\}$ → $Q_3 = \frac{20 + 25}{2} = 22.5$ .

Step 3: Compute IQR and Range.

$IQR = Q_3 - Q_1 = 22.5 - 8.5 = 14$

$\text{Range} = 90 - 4 = 86$

Insight: The extreme outlier ( $90$ ) inflates the total range to $86$ , whereas the $IQR = 14$ accurately measures the central spread without being distorted by the extreme value.

Feature	Total Range	Interquartile Range (IQR)
Formula	$\text{Max} - \text{Min}$	$Q_3 - Q_1$
Data Span	100% of data values	Middle ~50% of values
Sensitivity to Outliers	Highly sensitive (heavily skewed by extreme points)	Resistant (ignores extreme values)

DAT Mnemonic — O-M-L-U:

Order data from smallest to largest
Median first ( $Q_2$ )
Lower half median = $Q_1$ (exclude $Q_2$ if $N$ is odd)
Upper half median = $Q_3$ (exclude $Q_2$ if $N$ is odd)

Critical Thinking Questions

If a dataset of 15 patient scores has its maximum score increased by 50 points, how will this change affect the total Range versus the Interquartile Range?
Why is the IQR preferred over the total Range when evaluating clinical trial data that contains severe extreme values?
Under what dataset condition would $Q_1$ equal $Q_3$ , and what would be the resulting value of the IQR?

References

Sources returned by Google Search grounding during lesson verification.

Quartile - Wikipedia — en.wikipedia.org
Quartiles & Quantiles | Calculation, Definition & Interpretation - Scribbr — scribbr.com

Quartiles and the Interquartile Range

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