How to read a box-and-whisker plot

A box-and-whisker plot, also called a box plot, compresses a numerical data set into a picture of its center and spread. It is useful when comparing distributions because the median, middle half, and outer values can be read at a glance. Start by reading the axis scale carefully; the drawing has meaning only in relation to that scale.

The five-number summary

A conventional box plot is built from five values: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. Q1 marks the point at or below which roughly one quarter of ordered observations fall; Q3 marks the corresponding three-quarter point. The median divides the ordered data into lower and upper halves. The box begins at Q1 and ends at Q3, with a line inside the box at the median.

Whiskers extend from the box toward the extremes. Some school-level plots extend to the minimum and maximum. Other plots use a modified convention: whiskers reach the most extreme observations within 1.5 times the interquartile range, and observations farther away are marked individually as potential outliers. Always read the legend or directions instead of assuming which convention is being used.

Find the interquartile range

The interquartile range (IQR) is the width of the box in data units: IQR = Q3 − Q1. It describes the spread of the middle half of the observations and is less affected by extreme values than the total range. The total range, when the whiskers show minimum and maximum, is maximum − minimum.

Compare two distributions

Compare medians to discuss typical values, then compare IQRs to discuss the spread of the middle half. If the boxes overlap, that alone does not prove the distributions are statistically indistinguishable. The position and length of each whisker, the overall scale, and any marked points still matter. A box plot does not show every individual value, the mean, sample size, or the shape of the distribution in full.

Symmetry and skew

A median near the center of the box and whiskers of similar lengths can be consistent with a roughly balanced distribution. A longer upper whisker or a median closer to Q1 may suggest greater spread among higher values; a longer lower whisker or median closer to Q3 may suggest greater spread among lower values. These are visual clues, not a precise measurement of skew or proof of a particular distribution shape.

Quartile conventions can differ

For a plotted data set, software, calculator, or textbook may use different conventions to compute quartiles, especially with an even or small number of observations. In a question, follow the stated method or read the quartiles from the graph. Do not recompute Q1 or Q3 using a different convention unless the problem asks you to calculate them.

A reliable reading sequence

1. Read the axis units and scale intervals.
2. Locate Q1, the median, and Q3 at the box edges and internal line.
3. Subtract Q1 from Q3 to calculate the IQR.
4. Read whisker endpoints and identify whether points are plotted separately.
5. Compare centers using medians and spread using IQRs, then describe the evidence precisely.

A strong answer says exactly what the plot supports: for example, “Group A has a higher median and a wider middle 50% than Group B.” Avoid calling the median an average, treating every point beyond a whisker as an error, or inferring values that are not displayed.

Read the five-number summary

A box-and-whisker plot summarizes a distribution with the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. The left and right edges of the box mark Q1 and Q3; the line inside marks the median. The whiskers extend to the minimum and maximum in a basic plot, though some conventions use whiskers to the most extreme non-outlier values. Read the graph’s labels or note to know which convention applies.

The interquartile range is IQR = Q3 − Q1 and measures the spread of the middle 50% of the data. If Q1 = 12 and Q3 = 20, IQR = 8. The full range is maximum minus minimum and includes the tails. A longer box means the middle half is more spread out, while a median line near one edge can signal asymmetry within the box.

Compare centers and spreads

To compare two groups, compare their medians for typical central values and their IQRs for middle spread. One plot may have a higher median but smaller IQR. Do not infer that every value in one group exceeds every value in another unless the plots show nonoverlapping ranges. Box plots summarize distributions but do not show individual observations or sample size unless separately marked.

Skew can be suggested by unequal whisker lengths, an off-center median, or one side of the box being longer. A longer right whisker suggests a longer upper tail; a longer left whisker suggests a longer lower tail. These visual cues are approximate and should be described as evidence of shape, not a precise measure of skew.

Check outliers and quartile conventions

Some box plots mark outliers as separate points and draw whiskers only to values within 1.5 IQR of the quartiles. Under that convention, values below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR) are flagged for investigation. A flagged point is not automatically an error; it may be a valid extreme observation. Other introductory plots may simply draw whiskers to the minimum and maximum.

Different methods can place quartiles slightly differently, especially for small data sets. If exact calculations are required, follow the method specified in the question or course materials. When interpreting a provided plot, rely on the labeled positions rather than recomputing from missing raw data.

• Read Q1, median, Q3, and whisker endpoints from the axis.
• IQR is Q3 − Q1 and covers the middle half.
• Compare medians for center and IQRs for middle spread.
• Interpret skew from relative whisker and box lengths cautiously.
• Check whether whiskers show extremes or exclude flagged outliers.