Descriptive Statistics: Percentiles, Quartiles, and the Five-Number Summary

Percentiles and Data Partitioning

• Percentile Definition and Interpretation:

  • A percentile indicates the percentage of total data points that fall at or below a given numeric threshold.

  • For example, if $10\%$ of test-takers score $504$ points or less on a standardized test, then $504$ corresponds to the $10\text{th}$ percentile.

  • In very large datasets (such as $10,000,000$ total test scores), including or excluding the exact cutoff score at the top has a negligible effect on the proportion, remaining approximately $10\%$.

  • $80\text{th}$ Percentile: The value that cuts off the lowest $80\%$ of all data values, meaning $80\%$ of the scores are equal to or less than that value.

• The $50\text{th}$ Percentile (Halving the Dataset):

  • The primary division of a dataset begins by bisecting the data into two equal halves.

  • In a theoretical dataset of $100,000,000$ SAT scores, the $50\text{th}$ percentile is found by navigating to the $50,000,000\text{th}$ score and taking the midpoint between that value and the next score.

  • For a hypothetical $50\text{th}$ percentile value of $1148$ points, exactly $50,000,000$ scores lie at or below $1148$, and $50,000,000$ scores lie above $1148$

• Subdividing Halves into Quarters:

  • $25\text{th}$ Percentile: Obtained by bisecting the lower $50\%$ of the data. For a hypothetical score of $1024$, exactly $25\%$ of all test-takers scored $1024$ points or less.

  • $75\text{th}$ Percentile: Obtained by bisecting the upper $50\%$ of the data, representing the threshold where $75\%$ of all scores fall at or below that value (e.g., $1900$ points).

• Four Equal Groups (Quarters):

  • Partitioning data using the $25\text{th}$, $50\text{th}$, and $75\text{th}$ percentiles creates four distinct blocks, each containing exactly $25\%$ of the total observations:

    1. Lowest value to $25\text{th}$ percentile: $25\%$ of data.

    2. $25\text{th}$ percentile to $50\text{th}$ percentile: $25\%$ of data ($50\% - 25\% = 25\%$

    3. $50\text{th}$ percentile to $75\text{th}$ percentile: $25\%$ of data.

    4. $75\text{th}$ percentile to Highest value: $25\%$ of data.

  • This four-part structure is analogous to monetary currency, where four quarters of $25\%$ equal one full dollar.

Quartiles and Terminology

• Quartile Classifications:

  • When ordering values sequentially from minimum ($Min\,X$) to maximum ($Max\,X$) along a continuous scale, specific percentiles are designated as quartiles because they partition data into four quarters.

  • First Quartile ($Q_1$):

    • Also termed the Lower Quartile.

    • It is the first quartile encountered when moving from smallest to largest values.

    • Symbolized as $Q_1$ (or $Q1$).

    • $Q_1$ is equivalent to the $25\text{th}$ percentile.

  • Third Quartile ($Q_3$):

    • Also termed the Upper Quartile.

    • Symbolized as $Q_3$ (or $Q3$).

    • $Q_3$ is equivalent to the $75\text{th}$ percentile.

  • Second Quartile ($Q_2$) / Median ($MED$):

    • The $50\text{th}$ percentile corresponds logically to the symbol $Q_2$ (Second Quartile).

    • While $Q_2$ is technically valid, it is rarely used in practice.

    • The historical development of statistics established the concept of splitting data in half prior to formalized quartile systems, permanently adopting the term Median (abbreviated $MED$

    • Highway Analogy: The statistical median functions like a highway median strip—a physical divider in the middle of opposing traffic lanes that splits the roadway into equal halves.

The Five-Number Summary

• Definition and Composition:

  • The Five-Number Summary is a descriptive statistical summary that provides a structural overview of a dataset using five specific values arranged from minimum to maximum.

  • Percentile Representation:

    1. $Min\,X$ (Minimum value)

    2. $25\text{th}$ Percentile

    3. $50\text{th}$ Percentile

    4. $75\text{th}$ Percentile

    5. $Max\,X$ (Maximum value)

  • Standard Notation Representation:

    1. $Min\,X$

    2. $Q_1$

    3. Median ($MED$

    4. $Q_3$

    5. $Max\,X$

  • The minimum ($Min\,X$) and maximum ($Max\,X$) act as boundary endcaps for the dataset.

• Exclusions from the Five-Number Summary:

  • The Five-Number Summary consists strictly of these five positional numbers.

  • It explicitly excludes:

    • Range

    • Interquartile Range ($IQR$

    • Average / Mean

    • Standard Deviation

• Measure of Center:

  • Out of all values within the Five-Number Summary, the Median is the unique value that serves as a measure of center because it cuts the overall dataset directly in half.

Questions & Discussion

• Question: What percent of the SAT scores are below the twenty-fifth percentile?

  • Response: $25\%$

• Question: What percent of the SAT scores are below the fiftieth percentile?

  • Response: $50\%$

• Question: What percent of the SAT scores are between the twenty-fifth percentile and the fiftieth percentile?

  • Response: $25\%$ (calculated as $50\% - 25\% = 25\%$

• Question: What percent of the SAT scores are between the fiftieth percentile and the seventy-fifth percentile?

  • Response: $25\%$

• Question: What percent is left in the highest section above the seventy-fifth percentile?

  • Response: $25\%$

• Question: Based on the naming pattern of $Q_1$ and $Q_3$, what symbol would be expected for the fiftieth percentile?

  • Response: $Q_2$

• Question: Looking exclusively at the numbers in the Five-Number Summary, which value represents a measure of center?

  • Response: The median.