Quartiles

Quartiles & Related Descriptive Statistics

What Are Quartiles?

• Definition: Quartiles split an ordered data set into four equal parts so that 25 %, 50 %, and 75 % of observations fall below $Q_1$, $Q_2$ (median), and $Q_3$ respectively.
• Together with the minimum and maximum, the three quartiles form the five-number summary that underpins box-and-whisker plots.

Computing Quartiles (Inclusive Method)

• Quartile position: $\text{Pos}(Q_k)=(n+1)\frac{k}{4},k=1,2,3$ where $n$ is the sample size; interpolate if the position is not an integer.
• Most spreadsheet packages (e.g., Excel QUARTILE.INC, Google Sheets QUARTILE) and MATLAB’s quantile default to this inclusive rule.

Interquartile Range (IQR)

• Equation: $\text{IQR}=Q_3-Q_1$
• Measures spread while ignoring extreme tails.

Detecting Outliers (Tukey Rule)

• Mild outlier fences: $Q_1-1.5\text{IQR}\text{and}{Q}_3+1.5\text{IQR}$
• Extreme outlier fences (optional): $Q_1-3\text{IQR},Q_3+3\text{IQR}$

Box-and-Whisker Checklist

1. Draw a rectangular box from $Q_1$ to $Q_3$ with a vertical line at the median $Q_2$.
2. Extend whiskers to the most extreme non-outlier points; mark outliers individually beyond the 1.5 IQR fences.

Formula Sheet & When to Use It

1. Quartile Position (inclusive): $\text{Pos}(Q_k)=(n+1)\frac{k}{4}$
2. Interquartile Range: $\text{IQR}=Q_3-Q_1$
3. Outlier Fences (Tukey): $[Q_1-1.5\text{IQR},Q_3+1.5\text{IQR}]$
4. Extreme Outlier Fences: $[Q_1-3\text{IQR},Q_3+3\text{IQR}]$
5. Five-Number Summary for Box Plots: $\{\min ,Q_1,Q_2,Q_3,\max \}$