5.2 Quartiles, Percentiles & Boxplots

5.2 Quartiles, Percentiles & Boxplots

While mean and standard deviation provide powerful summaries for symmetric distributions, positional metrics such as percentiles, quartiles, and boxplots excel at describing positional standing and non-symmetric or skewed data sets. The GRE tests your mastery of quartile calculations, five-number summaries, box-and-whisker plot interpretation, and quantitative outlier identification.

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Understanding Percentiles

The $p$-th percentile of a data set is a value such that at least $p\%$ of the data points are less than or equal to that value, and $(100-p)\%$ of the data points are greater than or equal to it.

• Distinction between Percentile and Percentage Score: Scoring in the 85th percentile does not mean you answered 85% of questions correctly. It means your score was equal to or higher than 85% of all test takers in the comparison group.
• Positioning: In an ordered list of $n$ elements, the percentile rank pinpoints a relative position. For GRE purposes, percentiles are evaluated on ordered sets.

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Quartiles & Interquartile Range (IQR)

Quartiles divide an ordered data set into four equal parts, each containing $25\%$ of the observations.

                      Data Partitioning by Quartiles
  |------- 25% -------|------- 25% -------|------- 25% -------|------- 25% -------|
 Min                 Q1                  Q2                 Q3                 Max
                                      (Median)
                      |<------------ IQR ------------>|

Computing Quartiles

To calculate $Q_1$, $Q_2$, and $Q_3$ for a data set of $n$ numbers:

1. Sort the data set in ascending order.
2. Find the median ($Q_2$).
3. If $n$ is odd, omit $Q_2$ from both halves. If $n$ is even, split the set evenly into a lower half and an upper half.
4. $Q_1$ is the median of the lower half; $Q_3$ is the median of the upper half.

The Interquartile Range (IQR)

The Interquartile Range measures the spread of the middle $50\%$ of the data:

$$\text{IQR} = Q_3 - Q_1$$

Unlike the total range ($\text{Max} - \text{Min}$), the IQR is resistant to extreme outliers because it ignores the lowest 25% and highest 25% of values.

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The Five-Number Summary & Boxplots

The Five-Number Summary consists of five key benchmarks that completely describe the distribution of a set:

$$\text{Five-Number Summary} = \{\text{Minimum}, Q_1, Q_2, Q_3, \text{Maximum}\}$$

Anatomy of a Boxplot (Box-and-Whisker Plot)

A boxplot visually represents the five-number summary on a number line:

                                  BOXPLOT
                 +-------------------+-------------------+
  |--------------|                   |                   |--------------|
 Min            Q1                Median (Q2)           Q3             Max
                 |<----------------- Box ---------------->|
  |<-- Whisker ->|                                       |<-- Whisker ->|

• The Box: Extends from $Q_1$ to $Q_3$. The width of the box is the IQR.
• The Vertical Line Inside: Marks the Median ($Q_2$).
• Whiskers: Extend from $Q_1$ down to the Minimum (or smallest non-outlier value) and from $Q_3$ up to the Maximum (or largest non-outlier value).

Interpreting Distribution Skewness from Boxplots

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Formal Outlier Detection Rule (1.5 × IQR Rule)

On the GRE, an observation $x$ is formally defined as a statistical outlier if it lies beyond $1.5$ times the interquartile range past the quartiles:

$$\text{High Outlier Threshold} = Q_3 + 1.5 \times \text{IQR}$$ $$\text{Low Outlier Threshold} = Q_1 - 1.5 \times \text{IQR}$$

Worked Step-by-Step Outlier Example: A data set has $Q_1 = 45$ and $Q_3 = 75$.

1. Calculate IQR: $\text{IQR} = 75 - 45 = 30$.
2. Multiply IQR by 1.5: $1.5 \times 30 = 45$.
3. Determine High Outlier Cutoff: $75 + 45 = 120$.
4. Determine Low Outlier Cutoff: $45 - 45 = 0$.
5. Any data point strictly greater than $120$ or strictly less than $0$ is an outlier.

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GRE Exam Tips & Boxplot Pitfalls

1. Equal Data Quarters: Each of the 4 sections of a boxplot (Min to $Q_1$, $Q_1$ to $Q_2$, $Q_2$ to $Q_3$, $Q_3$ to Max) contains exactly 25% of the data points, regardless of how wide or narrow that section visually appears on the plot.
2. Visual Trap: A longer whisker does not mean more data points are in that region! It only means the points in that quarter are more spread out (higher variance).