Quartile Positions (Discrete Data)

Statement

Quartiles split an ordered data set into four equal parts. For discrete data, approximate positions of the quartiles can be found using:

\[ Q_1 \text{ near } \frac{N+1}{4}, \quad \text{Median near } \frac{N+1}{2}, \quad Q_3 \text{ near } \frac{3(N+1)}{4} \]

Here, \(N\) is the number of data values. These positions are then rounded or interpreted as falling between two values to identify the actual quartiles.

Why it’s true

• Quartiles divide data into four quarters, each containing roughly 25% of the values.
• The median splits the dataset into two halves, located around the \((N+1)/2\)-th position.
• Similarly, \(Q_1\) is about one quarter of the way into the dataset, and \(Q_3\) is three quarters of the way in.

Recipe (how to use it)

1. Order the data from smallest to largest.
2. Find \(N\), the total number of values.
3. Compute the positions: \(Q_1 \approx (N+1)/4\), median \(\approx (N+1)/2\), \(Q_3 \approx 3(N+1)/4\).
4. Identify the corresponding data value(s) at those positions.

Spotting it

Whenever a question asks for quartiles, median, or interquartile range (IQR) for a raw data set, this is the method to use.

Common pairings

• Finding the interquartile range (IQR = Q3 − Q1).
• Comparing data distributions with box plots.
• Using median and quartiles to describe skewness.

Mini examples

1. Data: 2, 4, 7, 9, 11. \(N=5\). Median at (5+1)/2 = 3rd → 7.
2. Data: 3, 5, 8, 10, 12, 14, 15. \(N=7\). \(Q_1\) at (7+1)/4 = 2nd → 5.

Pitfalls

• Forgetting to order the data first.
• Mixing up positions (e.g., using N/2 instead of (N+1)/2).
• Confusion when positions are not whole numbers — interpolate or choose the closest value consistently.

Exam strategy

• Always list data in order before applying quartile positions.
• Double-check calculations of positions using (N+1).
• Show clearly which data value corresponds to each quartile.

Summary

The quartile position formulas give a quick method to locate quartiles and the median in discrete datasets. They are central in statistics questions on spread and comparison, especially when constructing box plots and calculating the interquartile range.