Example 1 — No outliers (Type 7)
Data: \(1,3,3,6,7,8,9\) (\(n=7\)). \(Q_2=6\). \(Q_1=Q(0.25)=\frac{3+3}{2}=3\). \(Q_3=Q(0.75)=\frac{7+8}{2}=7.5\). \(\mathrm{IQR}=4.5\). Fences: \(-3.75\) and \(14.25\). Whiskers: \(1\) and \(9\). No outliers.
Box Plot Calculator
Paste a list of numbers to compute the five-number summary and draw a box plot.
Quartiles use the Type-7 interpolation (like R): p = 1 + (n−1)·q.
Separate with commas, spaces, or new lines.
For displayed values only.
Tukey fences:
Q1 − k·IQR, Q3 + k·IQR.Box Plot
The box spans
Q1 to Q3 with a median line. Whiskers reach to data within fences.
Helping Notes
- Remove non-numeric items or labels before calculating.
- Type-7 quartiles (R default) can differ slightly from other definitions (Excel inclusive median, Tukey hinges, etc.).
- Try changing the fence factor to 3.0 for a “mild outlier” tolerant plot.
Result
Equation Preview
Quartiles use linear interpolation at positions
p = 1 + (n−1)·q for q ∈ {0.25, 0.5, 0.75}.
Five-Number Summary & Outliers
Whiskers extend to the most extreme values within the fences. Points beyond are flagged as outliers.
What is a Box Plot Calculator?
A Box Plot Calculator is a statistics tool that summarizes a dataset using five-number summary statistics: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum (often of non-outlier values). It visualizes spread and skewness, highlights potential outliers, and compares groups at a glance. By entering raw numbers (integers, decimals, fractions), the calculator sorts the data, computes quartiles, derives the interquartile range (IQR), and determines whisker endpoints and outlier points. Because box plots are nonparametric, they are robust to skewed data and make no distribution assumptions. All formulas are written in LaTeX and render responsively with MathJax or math.js.
About the Box Plot Calculator
Let the sorted sample be \(x_{(1)} \le \cdots \le x_{(n)}\). The median (Q2) is the middle value (or average of the two middles). Quartiles can be defined in several consistent ways; by default, this calculator uses the widely adopted “Type 7” quantile (R/NumPy default) with linear interpolation, and can also show Tukey’s hinges for teaching purposes. The interquartile range is \( \mathrm{IQR} = Q_3 - Q_1 \). Whiskers typically extend to the most extreme non-outlier values, where outliers are defined with the 1.5×IQR rule. Extreme outliers may be flagged with 3×IQR.
How to Use this Box Plot Calculator
- Paste or type your dataset (comma, space, or newline separated). The tool sorts and validates numeric entries.
- Choose quartile method (default: Type 7). Optionally enable Tukey’s hinges for comparison.
- Review the five-number summary, IQR, fences, whisker endpoints, and any identified outliers.
- Compare multiple groups by entering separate datasets; the calculator will compute per-group summaries.
- Export the computed statistics or copy the LaTeX expressions for documentation and lesson plans.
Core Formulas (LaTeX for MathJax/math.js)
Median (Q2): \[ Q_2 = \begin{cases} x_{\left(\frac{n+1}{2}\right)}, & n \text{ odd}\\[4pt] \dfrac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2}, & n \text{ even} \end{cases} \]
Type 7 quantile (default) for \(p\in[0,1]\): \[ h = 1 + (n-1)p,\quad k=\lfloor h \rfloor,\quad \gamma = h-k,\quad Q(p) = (1-\gamma)\,x_{(k)} + \gamma\,x_{(k+1)}. \]
Interquartile range: \[ \mathrm{IQR} = Q_3 - Q_1. \]
Outlier fences (Tukey): \[ \text{Lower Fence } = Q_1 - 1.5\,\mathrm{IQR}, \qquad \text{Upper Fence } = Q_3 + 1.5\,\mathrm{IQR}. \]
Whiskers: \[ \text{Lower whisker} = \min\{x_i \mid x_i \ge \text{Lower Fence}\},\quad \text{Upper whisker} = \max\{x_i \mid x_i \le \text{Upper Fence}\}. \]
Examples (Illustrative)
Example 2 — With an outlier
Data: \(10,12,12,13,13,14,50\). \(Q_2=13\). \(Q_1=12\). \(Q_3=13.5\). \(\mathrm{IQR}=1.5\). Fences: \(9.75\) and \(15.75\). Whiskers: \(10\) and \(14\). Outlier: \(50\).
Example 3 — Even \(n\) with interpolation
Data: \(2,4,6,8,10,12\) (\(n=6\)). \(Q_2=\frac{6+8}{2}=7\). \(Q_1=4.5\). \(Q_3=9.5\). \(\mathrm{IQR}=5\). Fences: \(-3\) and \(17\). Whiskers: \(2\) and \(12\). No outliers.
FAQs
What is a box plot used for?
To visualize center, spread, skewness, and potential outliers of a dataset via the five-number summary.
Which quartile method does this calculator use?
Type 7 (default) with linear interpolation; you can also display Tukey’s hinges for comparison in teaching contexts.
What are whiskers in a box plot?
Lines from the box to the most extreme non-outlier values, bounded by the 1.5×IQR fences.
How are outliers defined?
Values below \(Q_1-1.5\,\mathrm{IQR}\) or above \(Q_3+1.5\,\mathrm{IQR}\). Extreme outliers may use 3×IQR.
Does the calculator handle duplicate values?
Yes. Duplicates are valid; interpolation still yields correct quartiles.
Can I compare multiple groups?
Yes—enter separate datasets; the tool computes summaries for side-by-side comparison.
What if my data contain non-numeric entries?
They’re ignored or flagged; only numeric values are used in calculations.
How do quartiles differ from percentiles?
Quartiles are specific percentiles: \(Q_1=P_{25}, Q_2=P_{50}, Q_3=P_{75}\); computed using the chosen quantile method.
When should I use 3×IQR instead of 1.5×IQR?
Use 3×IQR to flag extreme outliers in heavy-tailed distributions; 1.5×IQR is the standard default.
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