Quartiles & Percentiles

Quartiles: These are the three specific values that divide a sorted data set into four equal parts, each containing $25\%$ of the observations. The first quartile ($Q_1$) marks the 25th percentile, the second quartile ($Q_2$) is the median or 50th percentile, and the third quartile ($Q_3$) represents the 75th percentile.

Percentiles: A percentile is a measure indicating the value below which a given percentage of observations in a group of data falls. For example, the 90th percentile ($P_{90}$) is the value such that $90\%$ of the data points are less than or equal to it.

Interquartile Range (IQR): This is the difference between the third and first quartiles ($IQR = Q_3 - Q_1$). It measures the spread of the middle $50\%$ of the data and is a critical tool for identifying the variability of the core distribution.

Data Preparation: The first and most critical step is to arrange the data in ascending order. Calculations performed on unsorted data will result in incorrect positional values.

Finding the Index: To find the position $L$ of the $k$-th percentile in a data set of size $n$, use the formula $L = \frac{k}{100}(n+1)$. If $L$ is a whole number, the percentile is the value at that position; if $L$ is a fraction, interpolation between the two surrounding values is often required.

The 1.5 x IQR Rule: To identify potential outliers, calculate the lower bound as $Q_1 - 1.5(IQR)$ and the upper bound as $Q_3 + 1.5(IQR)$. Any data point falling outside these boundaries is statistically considered an outlier.

Always Sort First: The most common error in exams is calculating quartiles directly from a provided list without sorting it from smallest to largest. Always double-check that your sequence is strictly increasing before identifying positions.

Verify the Median: Before finding $Q_1$ and $Q_3$, always find the median ($Q_2$) first. $Q_1$ is effectively the median of the lower half of the data, and $Q_3$ is the median of the upper half; this hierarchical approach prevents calculation errors.

Check for Even/Odd $n$: Be mindful of whether the data set has an even or odd number of values. If $n$ is odd, the median is a specific data point; if $n$ is even, the median is the average of the two middle points, which affects how you define the 'lower' and 'upper' halves.

Inclusive vs. Exclusive Median: There is often confusion about whether to include the median in the lower and upper halves when $n$ is odd. Most standard methods exclude the median when calculating $Q_1$ and $Q_3$ to ensure the quarters are truly equal in size.

Confusing Percentile with Percentage: A student scoring in the 90th percentile does not necessarily have a $90\%$ score on the exam. It simply means they performed better than $90\%$ of the other test-takers, regardless of the actual raw score.

Interpolation Errors: When the calculated position $L$ is not an integer, students often round to the nearest whole number. In rigorous statistics, you should take a weighted average of the values at the two surrounding positions.

Box-and-Whisker Plots: These visual tools are built entirely on the five-number summary: Minimum, $Q_1$, Median, $Q_3$, and Maximum. They allow for immediate visual comparison of the center and spread of multiple data sets.

Standardized Scores: Percentiles are the foundation of reporting for exams like the SAT or GRE, where raw scores are less meaningful than a candidate's standing relative to the global pool of applicants.

Skewness Detection: By comparing the distance between $Q_1$ and $Q_2$ to the distance between $Q_2$ and $Q_3$, one can determine if a distribution is positively skewed (right-skewed) or negatively skewed (left-skewed).