Five-Number Summary & Box Plot Reference Cheat Sheet

A printable reference covering minimum, quartiles, median, maximum, interquartile range, outliers, and box plots for grades 7-12.

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A five-number summary describes a data set using the minimum, first quartile, median, third quartile, and maximum. This cheat sheet helps students organize data, find quartiles, and build box plots accurately. These skills are useful for comparing groups, spotting spread, and identifying unusual values in real-world data. The most important ideas are position, center, spread, and outliers. The median divides ordered data into two halves, while quartiles divide data into four parts. The interquartile range measures the spread of the middle half of the data using $IQR = Q_3 - Q_1$ . A box plot shows these values visually using a box, a median line, whiskers, and sometimes separate outlier points.

Key Facts

A five-number summary is written as $\min, Q_1, Q_2, Q_3, \max$ , where $Q_2$ is the median.
The range of a data set is $\text{range} = \max - \min$ .
The interquartile range is $IQR = Q_3 - Q_1$ , which measures the spread of the middle $50\%$ of the data.
The lower outlier fence is $Q_1 - 1.5(IQR)$ , and values below it are possible outliers.
The upper outlier fence is $Q_3 + 1.5(IQR)$ , and values above it are possible outliers.
In a box plot, the box runs from $Q_1$ to $Q_3$ , and the line inside the box marks the median $Q_2$ .
For a modified box plot, whiskers extend to the smallest and largest non-outlier values, not necessarily to the minimum and maximum.
A longer box or whisker means the data are more spread out over that part of the distribution.

Vocabulary

Five-number summary: A summary of a data set using the minimum, first quartile, median, third quartile, and maximum.
Median: The middle value of an ordered data set, also called $Q_2$ .
Quartile: A value that divides ordered data into four parts with about $25\%$ of the data in each part.
Interquartile range: The spread of the middle half of the data, found by $IQR = Q_3 - Q_1$ .
Box plot: A graph that displays the five-number summary with a box from $Q_1$ to $Q_3$ , a median line, and whiskers.
Outlier: A data value that is unusually far from the rest of the data, often checked using the $1.5(IQR)$ rule.

Common Mistakes to Avoid

Not ordering the data first, which is wrong because the median and quartiles must be found from values arranged from least to greatest.
Including the median in both halves when the method says to exclude it, which can change $Q_1$ and $Q_3$ for data sets with an odd number of values.
Using $\max - \min$ when asked for $IQR$ , which is wrong because $IQR$ measures only the middle $50\%$ of the data.
Drawing whiskers to outliers on a modified box plot, which is wrong because outliers should be plotted as separate points.
Thinking a box plot shows every data value, which is wrong because it summarizes position and spread rather than listing each value.

Practice Questions

1 Find the five-number summary for the data set $4, 7, 9, 10, 12, 15, 18, 20, 21$ .
2 For a data set with $Q_1 = 12$ and $Q_3 = 28$ , find $IQR$ , the lower fence, and the upper fence.
3 A modified box plot has $Q_1 = 35$ , median $= 42$ , $Q_3 = 50$ , minimum non-outlier $= 30$ , and maximum non-outlier $= 63$ . Describe the box, median line, and whiskers.
4 Two box plots have the same median, but one has a much larger $IQR$ . Explain what this means about the two data sets.

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