2.4: Identifying Outliers with Boxplots
- Page ID
- 58865
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A box-and-whisker plot, or simply a boxplot, is a compact visual that displays a full distribution using just five numbers. It helps you quickly see how data is spread out, where it centers, and whether any potential outliers exist.
What makes the boxplot powerful is that it combines elements you already know — the five-number summary and interquartile range (IQR) — into an easy-to-scan visualization that's useful for comparing groups or identifying odd values.
What a Boxplot Shows
- Box: spans from Q1 to Q3 (interquartile range). It captures the middle 50% of the dataset.
- Line inside the box: the median (Q2), showing the “typical” center of the data.
- Whiskers: extend to the smallest and largest values within the expected range (i.e., not outliers).
- Dots or stars (often): mark potential outliers beyond \(1.5 \times \text{IQR}\) from the quartiles.
Together, these parts create a visual “map” of your dataset’s shape. One glance at a boxplot can tell you: is the data skewed? Is it tightly grouped? Are there surprising highs or lows?
Example: Walkthrough with Real Data
Let’s walk through the full process using an original dataset. We’ll compute all the important summary statistics, identify any outliers, and construct our boxplot step-by-step. This hands-on example will give you a model to apply with your own project data.
Data Set: [INSERT TITLE OR THEME]
Here’s the dataset we’ll use:
[INSERT DATA — please provide numeric values, ideally 15–25 values with at least one clear outlier]
Please provide the full dataset you’d like to use here, and I’ll plug it in and compute:
- Minimum
- Q1 (25th percentile)
- Median
- Q3 (75th percentile)
- Maximum
- IQR
- Any outliers (based on IQR rule)
- Mean and standard deviation (for extra context)
Calculated Summary Statistics
| Statistic | Value |
|---|---|
| Minimum | [...] |
| Q1 | [...] |
| Median | [...] |
| Q3 | [...] |
| Maximum | [...] |
| IQR | [...] |
| Mean | [...] |
| Standard Deviation | [...] |
Building the Box-and-Whisker Plot
- Draw a number line that covers the full range of your data.
- Mark the five-number summary: Min, Q1, Median, Q3, Max.
- Draw a box from Q1 to Q3 and place a line at the median.
- Extend whiskers to the lowest and highest values that are not outliers.
- If any points fall beyond 1.5 × IQR from Q1 or Q3, plot them as outliers (dots or x-marks).
[PLACEHOLDER – Insert boxplot figure here once stats are calculated]
Boxplot tip: If the median is closer to the bottom or top of the box, the data might be skewed. Skewness also shows if one whisker is much longer than the other.
Related Videos
What's Coming Next
In Chapter 3, we’ll formally introduce the world of data visualizations — histograms, bar graphs, scatterplots, and more. This box-and-whisker plot is your first step into that world. You can use it to describe full distributions, compare groups, or amplify your project’s story with a clean, simple summary.