Box Plots and Five-Number Summary

Min, Q1, Median, Q3, Max & IQR

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A box plot is a compact graph that summarizes how a data set is distributed. It shows the center, spread, and possible extreme values without plotting every single point. The five-number summary behind a box plot gives the minimum, first quartile, median, third quartile, and maximum. These ideas matter because they help students compare data sets quickly and spot skew or unusual values.

To build a box plot, you first sort the data and find the five-number summary. The box runs from $Q_1$ to $Q_3$ , the line inside the box marks the median, and the whiskers extend toward the smallest and largest non-outlier values. The width of the box represents the interquartile range, which contains the middle 50 percent of the data. By looking at the lengths of the box halves and whiskers, you can infer symmetry, skewness, and the presence of outliers.

Key Facts

Five-number summary = minimum, $Q_1$ , median, $Q_3$ , maximum.
Interquartile range: $IQR = Q_3 - Q_1$ .
The box in a box plot extends from $Q_1$ to $Q_3$ , and the line inside the box is the median.
The middle 50% of the data lies between $Q_1$ and $Q_3$ .
Lower outlier fence = $Q_1 - 1.5(IQR)$ , upper outlier fence = $Q_3 + 1.5(IQR)$ .
Whiskers usually extend to the smallest and largest data values that are not outliers.

Vocabulary

Median: The median is the middle value of an ordered data set, or the average of the two middle values if there are an even number of data points.
Quartile: A quartile is a value that divides ordered data into four equal parts.
Interquartile Range: The interquartile range is the distance from $Q_1$ to $Q_3$ and measures the spread of the middle half of the data.
Whisker: A whisker is the line segment in a box plot that extends from the box to a non-outlier minimum or maximum value.
Outlier: An outlier is a data value that lies unusually far from the rest of the data, often beyond 1.5 times the IQR from a quartile.

Common Mistakes to Avoid

Using unsorted data to find quartiles, which is wrong because the five-number summary must be based on values in order from least to greatest.
Assuming the whiskers always go to the absolute minimum and maximum, which is wrong when outliers are shown separately and whiskers stop at the most extreme non-outlier values.
Confusing the median with the mean, which is wrong because a box plot marks the median as the center line inside the box, not the arithmetic average.
Computing IQR as maximum - minimum, which is wrong because IQR measures only the spread of the middle 50 percent and must be calculated as $Q_3 - Q_1$ .

Practice Questions

1 Find the five-number summary and IQR for the data set 3, 5, 7, 8, 9, 12, 13, 15, 18.
2 A data set has $Q_1 = 22$ and $Q_3 = 34$ . Find the IQR and the lower and upper outlier fences using the $1.5 \times \text{IQR}$ rule.
3 Two box plots have the same median, but one has a much wider box and longer right whisker. What does this tell you about the spread and possible skewness of that data set?