Interpret box-and-whisker plots

A box-and-whisker plot represents a data set through its five-number summary: minimum, lower quartile, median, upper quartile, and maximum. Interpretation includes reading values from the scale, understanding that quartiles divide the data into approximate quarters, calculating or comparing range and interquartile range, and using median and spread to compare distributions while recognizing possible skewness; formal outlier tests, modified box plots, and advanced statistical inference are not included.

Detailed Explanation: Interpret box-and-whisker plots

A box-and-whisker plot shows a data set using five important values:

Minimum: smallest value
Lower quartile ( $Q_1$ ): about 25% of the data are at or below this value
Median: middle value; about 50% of the data are at or below it
Upper quartile ( $Q_3$ ): about 75% of the data are at or below this value
Maximum: largest value

The box shows the middle 50% of the data, and the whiskers extend to the minimum and maximum.

Worked example

Suppose a box-and-whisker plot has these values when read from its scale:

Left whisker: $4$
Left side of the box: $7$
Line inside the box: $10$
Right side of the box: $14$
Right whisker: $18$

Step 1: Identify the five-number summary

The five-number summary is

\text{minimum}=4,\quad Q_1=7,\quad \text{median}=10,\quad Q_3=14,\quad \text{maximum}=18.

So, the smallest value is $4$ , and the largest value is $18$ .

Step 2: Find the range

The range describes the spread from the minimum to the maximum:

\text{Range}=\text{maximum}-\text{minimum}

\text{Range}=18-4=14

The data values spread across $14$ units.

Step 3: Find the interquartile range

The interquartile range, or IQR, describes the spread of the middle 50% of the data:

\text{IQR}=Q_3-Q_1

\text{IQR}=14-7=7

The middle half of the data is spread across $7$ units, from $7$ to $14$ .

Step 4: Interpret the quartiles

Because $Q_1=7$ , about one-quarter of the data values are $7$ or less.

Because the median is $10$ , about half of the data values are $10$ or less, and about half are $10$ or greater.

Because $Q_3=14$ , about three-quarters of the data values are $14$ or less.

Step 5: Notice the shape of the distribution

The distance from the minimum to the median is

10-4=6,

while the distance from the median to the maximum is

18-10=8.

The upper side is a little more spread out than the lower side, so the data may be slightly skewed to the right. However, a box plot does not always show a perfectly clear shape.

When comparing box plots, compare:

Medians to see which data set generally has larger values.
Ranges to compare the total spread.
IQRs to compare the spread of the middle 50% of the data.

Always read each value carefully from the plot’s scale before calculating.

Learn by doing: Interpret box-and-whisker plots

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Statistics - Quartiles - Data Set (No Outliers) to Box Plot

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