A box plot (box-and-whisker plot) shows the shape of a data set in one picture. The box spans the middle half of the data, from the first quartile Q1 to the third quartile Q3, and the line inside the box marks the median. The whiskers stretch to the smallest and largest values that are not outliers, and any outliers are drawn as separate points.
Because it is built from quartiles rather than the mean and standard deviation, a box plot is robust: one extreme value does not change the box. That makes it ideal for comparing groups, spotting skewness (a median off-centre in the box, or one whisker much longer than the other) and flagging unusual values before you run other statistics.
How to use the Box Plot Calculator
- Data set
- Outlier fence multiplier
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How it works
Sort the data, find the quartiles by linear interpolation (the method used by Excel’s QUARTILE.INC), compute the fences k × IQR beyond the box (k = 1.5 by default, 3 for “extreme” outliers), mark everything outside the fences as an outlier and draw the whiskers to the most extreme remaining values.
Worked example
For the data 2, 5, 7, 8, 9, 11, 12, 13, 15, 17, 19, 21, 45: Q1 = 8, median = 12 and Q3 = 17, so IQR = 9. The fences are 8 − 13.5 = −5.5 and 17 + 13.5 = 30.5, so 45 is an outlier. The upper whisker stops at 21, the largest value inside the fence.
Formula
IQR = Q3 − Q1 · Lower fence = Q1 − 1.5 × IQR · Upper fence = Q3 + 1.5 × IQR · whiskers reach the most extreme values inside the fences
Worked example
Inputs — Data set: 2, 5, 7, 8, 9, 11, 12, 13, 15, 17, 19, 21, 45; Outlier fence multiplier: 1.5 × IQR (Tukey – standard).
Results — Median (Q2) = 12; First quartile (Q1) = 8; Third quartile (Q3) = 17; Interquartile range (IQR) = 9; Minimum = 2; Maximum = 45.
Key terms
- Five-number summary
- Minimum, Q1, median, Q3 and maximum.
- Interquartile range (IQR)
- Q3 − Q1, the spread of the middle half of the data.
- Fence
- The cut-off beyond which a value counts as an outlier.
Frequently asked questions
What does a box plot show?
The median, the quartiles (the middle 50% of the data in the box), the range of typical values (whiskers) and any outliers as separate points.
How are outliers defined?
Values more than 1.5 × IQR below Q1 or above Q3 (Tukey's rule). Choose 3 × IQR to flag only extreme outliers.
Why do different tools draw slightly different boxes?
There are several ways to compute quartiles. This calculator interpolates between ranked values, as Excel's QUARTILE.INC and R's default do.
How do I read skewness from a box plot?
If the median sits closer to Q1 and the right whisker is longer, the data are right-skewed; the mirror image means left-skewed.
Is the mean shown?
Yes – the dot on the plot is the mean, so you can see how far outliers pull it away from the median.
How many values do I need?
At least four, but box plots are most informative with 10 or more values.