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Five-Number Summary and Outliers

Jul 12, 2025

Overview

This lecture covers how to find and interpret the five-number summary and use it to identify potential outliers in skewed data sets using the concept of fences and the interquartile range (IQR).

Five-Number Summary

  • The five-number summary includes: minimum, Q1 (first quartile), median, Q3 (third quartile), and maximum.
  • To compute it, enter data into your calculator, use "Stat" > "Calc" > "1-VarStat," and scroll to find the five-number summary.
  • Example: For quiz scores 28, 24, 27, 30, 19, 20, the five-number summary is 19, 20, 25.5, 28, 30.

Outliers and Fences

  • Outliers are data values significantly outside the rest of the data.
  • Z-scores and the empirical rule work for symmetric data; for skewed data or with outliers, use fences based on quartiles.
  • The interquartile range (IQR) is calculated as Q3 - Q1.
  • The lower fence is Q1 - 1.5 * IQR; values below this are potential outliers.
  • The upper fence is Q3 + 1.5 * IQR; values above this are potential outliers.

Identifying Outliers: Example

  • Given Q1 = 1.61, Q3 = 2.68, minimum = 1.01, maximum = 6.81, median = 2.65.
  • IQR = 2.68 - 1.61 = 1.07.
  • Lower fence: 1.61 - 1.5 * 1.07 = 0.005; no data values fall below this, so no lower outliers.
  • Upper fence: 2.68 + 1.5 * 1.07 = 4.285; values 5.22 and 6.81 are above this, so they are outliers.

Key Terms & Definitions

  • Five-number summary — Set of five descriptive statistics: minimum, Q1, median, Q3, maximum.
  • Quartile (Q1/Q3) — Values splitting data into quarters; Q1 is 25th percentile, Q3 is 75th percentile.
  • Median — Middle value of an ordered data set.
  • Interquartile range (IQR) — Difference between Q3 and Q1, measuring data spread.
  • Fence — Boundaries (Q1 - 1.5IQR and Q3 + 1.5IQR) used to identify outliers.
  • Outlier — Data value outside the fences, considered unusually high or low.

Action Items / Next Steps

  • Practice finding the five-number summary and fences using your calculator for different data sets.
  • Identify any outliers using the lower and upper fence methods.
  • Review section 3.5 for more details on outlier detection.

Full transcript