About

A box plot (box-and-whisker diagram) encodes five critical order statistics into a single glyph: the minimum, first quartile Q₁, median Q₂, third quartile Q₃, and maximum. Points beyond Q₁ − 1.5 ⋅ IQR or Q₃ + 1.5 ⋅ IQR are flagged as outliers. Misidentifying outliers corrupts regression models, inflates variance estimates, and produces misleading confidence intervals. This tool computes the full five-number summary, detects outliers using Tukey's fence method, and renders a scaled box plot on canvas. It approximates quartiles via linear interpolation (the inclusive method used by most statistics textbooks). Note: for datasets with fewer than 4 observations, quartile definitions become ambiguous.

Formulas

The interquartile range and outlier fences are computed as follows:

IQR = Q₃ − Q₁

LF = Q₁ − 1.5 × IQR

UF = Q₃ + 1.5 × IQR

Quartiles are calculated using linear interpolation. For a sorted dataset of n values, the p-th percentile index is:

i = p × (n − 1)

If i is not an integer, interpolate between x_floor(i) and x_ceil(i):

Q = x_floor(i) + (i − floor(i)) × (x_ceil(i) − x_floor(i))

Sample standard deviation uses Bessel's correction:

s = √n∑i=1 (x_i − x)²n − 1

Where Q₁ = 25th percentile, Q₃ = 75th percentile, LF = lower fence, UF = upper fence, n = sample size, x = sample mean.

Reference Data

Statistic	Symbol	Description	Sensitivity to Outliers
Minimum	x_min	Smallest observed value	Extremely sensitive
First Quartile	Q₁	25th percentile; splits lowest 25% of data	Resistant
Median	Q₂	50th percentile; central value	Resistant
Third Quartile	Q₃	75th percentile; splits lowest 75% of data	Resistant
Maximum	x_max	Largest observed value	Extremely sensitive
Interquartile Range	IQR	Q₃ − Q₁	Resistant
Lower Fence	LF	Q₁ − 1.5 × IQR	Resistant
Upper Fence	UF	Q₃ + 1.5 × IQR	Resistant
Mean	x	Arithmetic average of all values	Highly sensitive
Standard Deviation	s	Square root of sample variance	Highly sensitive
Skewness	g₁	Measure of distribution asymmetry	Sensitive
Kurtosis	g₂	Measure of tail heaviness (excess)	Sensitive
Range	R	x_max − x_min	Extremely sensitive
Lower Whisker	-	Smallest datum ≥ LF	Resistant
Upper Whisker	-	Largest datum ≤ UF	Resistant
Mild Outlier	-	Beyond 1.5 × IQR from box	-
Extreme Outlier	-	Beyond 3.0 × IQR from box	-

Frequently Asked Questions

This tool uses the inclusive (interpolation) method, which computes the index as i = p × (n − 1) and linearly interpolates between adjacent sorted values. This matches Excel's PERCENTILE.INC function and is the most widely taught method in introductory statistics courses. Results may differ slightly from tools using the exclusive method or Hyndman-Fan Type 7.

A mild outlier falls between 1.5 × IQR and 3.0 × IQR beyond the quartile boundary. An extreme outlier exceeds 3.0 × IQR. On the plot, mild outliers appear as open circles and extreme outliers as filled red circles. Extreme outliers often indicate measurement errors or genuinely rare events and warrant investigation before removal.

A minimum of 4 data points is needed to compute distinct values for Q₁, Q₂, and Q₃. With fewer points, the quartile boundaries collapse and the box plot loses interpretive value. For reliable inference, most statisticians recommend at least 20 - 30 observations.

If the median line sits closer to Q₁ with a longer upper whisker, the distribution is right-skewed (positive skewness). If the median sits closer to Q₃ with a longer lower whisker, it is left-skewed. A symmetric distribution shows roughly equal whisker lengths and the median centered in the box. The computed skewness coefficient g₁ quantifies this: values near 0 indicate symmetry, positive values indicate right skew.

Yes. You can enter up to 5 datasets, each in its own input area. The tool renders all box plots on a shared scale so you can visually compare medians, spreads, and outlier patterns across groups. This is the primary use case for box plots in ANOVA-style comparisons and quality control (e.g., comparing batch measurements).

Whiskers extend only to the most extreme data point that falls within the fences (Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR). Any data points beyond these fences are plotted individually as outliers. This is Tukey's convention and prevents outliers from compressing the box into an unreadable sliver.