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Wize AP Statistics Textbook > Presenting Quantitative Data

Boxplots and Five-Number Summary

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Boxplots and Five Number Summary

Five Number Summary

The five number summary provides a quick summary of the center and variability of a data set.

MIN, Q1, MEDIAN, Q3, MAX\text{MIN},\ Q_1,\ \text{MEDIAN},\ Q_3,\ \text{MAX}

Watch Out!
All of the above numbers describe the position of their data. We do not use the mean in boxplots.


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Boxplot

A boxplot is used to display how quantitative data is distributed, and gives you an idea of the spread and skewness of the data. It also lets you check for outliers (extreme values).

A boxplot includes the 5-number summary and, if any, outliers:
  • small outliers (if any)
  • lowest non-outlier
  • Q1
  • Median
  • Q3
  • highest non-outlier
  • large outliers (if any)


Wize Tip
The quartiles (Q1Q_1, Median, and Q3Q_3) divide the dataset into 4 equal parts, each containing 25% of the data. Percentiles are used to compute the quartiles.

Notes
  • Outliers are shown as separate dots.
  • The “whiskers” extend up to the largest observation and down to the smallest observation that are not identified as outliers.
  • Boxplots are also useful when comparing two or more distributions.
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Outliers

A value is an outlier if it is outside the inner fence:

xi>Q3+1.5(IQR)\boxed{x_i>Q_3+1.5\left(IQR\right)}

Or

xi<Q1−1.5(IQR)\boxed{x_i<Q_1-1.5\left(IQR\right)}

Wize Concept
The Interquartile Range (IQR) = Q3−Q1Q_3-Q_1


For extreme outliers outside the outer fence, use 3 instead of 1.5 in the equations:

xi>Q3+3(IQR)\boxed{x_i>Q_3+3\left(IQR\right)}

Or

xi<Q1−3(IQR)\boxed{x_i<Q_1-3\left(IQR\right)}

Watch Out!
Do not confuse between upper/lower fences vs. the line to which the whiskers extend up to on each end.

I randomly sample 8 wallets see how much cash is inside each of them. Results:

5 15 25 25 40 65 85 200

Find Q1 and Q3.

Practice: Boxplot

House prices (in thousands) for two neighborhoods, Twin Pines and Sadlands, are shown in boxplots.

(Review: Measure of Center and Spread)



Five number summary:


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(i) Which of the following is true?