About

A box-and-whisker plot compresses a dataset into five statistics: minimum, first quartile (Q₁), median (Q₂), third quartile (Q₃), and maximum. Outliers are flagged using Tukey’s fences at Q₁ − 1.5⋅IQR and Q₃ + 1.5⋅IQR. Misidentifying outliers distorts regression models, inflates variance estimates, and corrupts hypothesis tests. This tool computes interpolated quartiles consistent with Excel’s QUARTILE.INC function and renders publication-grade plots on an HTML Canvas. It supports up to 10 simultaneous datasets for direct visual comparison. Note: quartile methods vary across software (R alone offers 9 types). This tool uses the inclusive linear-interpolation method (Type 7), which is the most widely adopted default.

Formulas

The interquartile range is the core measure of spread in a box plot:

IQR = Q₃ − Q₁

Tukey’s fences define outlier boundaries:

F_L = Q₁ − 1.5 × IQR

F_U = Q₃ + 1.5 × IQR

Whiskers extend to the most extreme data points within fences (not the fence values themselves). Any observation x_i where x_i < F_L or x_i > F_U is plotted individually as an outlier.

Interpolated quartile at position p (where p = 0.25 or 0.75):

k = p × (n − 1)

Q = x_floor(k) + (k − floor(k)) × (x_floor(k)+1 − x_floor(k))

Where x is the sorted dataset of n observations, k is the fractional index, and floor truncates to the integer part. This is the Type 7 interpolation method used by Excel, Python (numpy default), and most statistical software.

Reference Data

Statistic	Symbol	Definition	Sensitivity to Outliers
Minimum	x_min	Smallest observed value (or whisker lower bound)	Extreme
First Quartile	Q₁	25th percentile - splits lowest 25% of data	Low
Median	Q₂	50th percentile - middle value of sorted data	Very Low
Third Quartile	Q₃	75th percentile - splits highest 25% of data	Low
Maximum	x_max	Largest observed value (or whisker upper bound)	Extreme
Interquartile Range	IQR	Q₃ − Q₁	Very Low
Lower Fence	F_L	Q₁ − 1.5 × IQR	Low
Upper Fence	F_U	Q₃ + 1.5 × IQR	Low
Mean	x	Arithmetic average of all values	High
Standard Deviation	σ	Square root of variance (spread measure)	High
Mild Outlier	-	Between 1.5×IQR and 3×IQR from box	-
Extreme Outlier	-	Beyond 3×IQR from box	-
Skewness Indicator	-	If median closer to Q₁ → right-skewed; closer to Q₃ → left-skewed	Moderate
Range	R	x_max − x_min	Extreme
Sample Size	n	Count of observations in dataset	None

Frequently Asked Questions

There are at least 9 recognized methods for computing quartiles (Hyndman & Fan, 1996). The differences arise from how the fractional index is handled - some methods average adjacent ranked values, others interpolate linearly, and some use exclusive vs. inclusive endpoint conventions. This tool uses Type 7 (linear interpolation on the order statistic), which matches Excel's QUARTILE.INC, NumPy's default, and Google Sheets. For small datasets (n < 20), method choice can shift Q1 and Q3 noticeably, so always report which method you used.

Technically you need at least 5 values to produce all five summary statistics. However, with fewer than about 15-20 observations, the quartile estimates are unstable and outlier detection becomes unreliable - a single value can flip between outlier and non-outlier status. For robust analysis, aim for n ≥ 20. With very small samples, consider a dot plot or strip chart instead.

Skewness is visible in three ways: (1) the median line is not centered inside the box - if it sits closer to Q1, the distribution is right-skewed; (2) whisker lengths are asymmetric - a longer upper whisker indicates right skew; (3) outliers cluster on one side. A perfectly symmetric distribution would have the median centered in the box with equal whisker lengths. Comparing the mean marker (if shown) to the median also reveals skewness: mean > median suggests right skew.

Whiskers extend only to the most extreme data points that fall within Tukey's fences (Q1 − 1.5×IQR and Q3 + 1.5×IQR). Points beyond the fences are plotted individually as outliers. If no outliers exist, whiskers do reach the true min and max. This design prevents extreme values from compressing the box into an unreadable sliver, which would happen if whiskers always went to the full data range.

Yes. A box plot summarizes data into five numbers and cannot reveal bimodality or multiple peaks. Two very different distributions - one unimodal and one bimodal - can produce nearly identical box plots. If you suspect multimodality (e.g., mixing two populations), supplement the box plot with a histogram, kernel density estimate, or violin plot. This tool shows individual outlier points, which may hint at clustering, but cannot substitute for a full distributional view.

Box plots handle negative values, zeros, and mixed-sign data without any issue. The five-number summary and IQR calculations are based on rank ordering, which is sign-agnostic. The axis will automatically scale to accommodate the full range. One caveat: if you later apply a log transformation to the data, zeros and negatives will produce undefined results - address those before transforming.