5 Number Summary Calculator

What is a 5 Number Summary Calculator?

A 5 Number Summary Calculator extracts five robust descriptors from numerical data: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. These values summarize center and spread without assuming a particular distribution, making them ideal for exploratory data analysis, dashboards, and teaching. With ordered data $x_{(1)}\le \cdots \le x_{(n)}$, the five-number set is $$\{\min=x_{(1)},\ Q_1,\ \mathrm{Median}=Q_2,\ Q_3,\ \max=x_{(n)}\}.$$ The calculator also reports the interquartile range (IQR) and identifies potential outliers using Tukey’s fences, providing a quick picture of variability and unusual values.

$$\textbf{Median (Q2)}:\quad Q_2=\begin{cases} x_{(\frac{n+1}{2})}, & n\ \text{odd}\\[4pt] \dfrac{x_{(\frac{n}{2})}+x_{(\frac{n}{2}+1)}}{2}, & n\ \text{even} \end{cases}$$

About the 5 Number Summary Calculator

The calculator accepts comma-, space-, or newline-separated values, cleans non-numeric entries, sorts, and then computes quartiles using a consistent convention. Two common approaches are: (i) “median-of-halves” (split around $Q_2$) and (ii) percentile-based methods. A widely used percentile rule (Tukey/Hazen-style) is $$\text{rank}(p)=1+p\,(n-1),\qquad Q_p=\text{linear interpolate at rank}.$$ After determining $Q_1$ and $Q_3$, it computes $$\mathrm{IQR}=Q_3-Q_1,$$ and flags potential outliers with Tukey’s fences: $$\text{Lower fence}=Q_1-1.5\,\mathrm{IQR},\qquad \text{Upper fence}=Q_3+1.5\,\mathrm{IQR}.$$ Formulas render responsively with MathJax; arithmetic can be handled precisely with math.js when you embed this content in your page.

How to Use this 5 Number Summary Calculator

Paste your data values (separated by commas, spaces, or line breaks).
Select the quartile convention (median-of-halves or percentile interpolation) if your workflow requires consistency with a standard.
Run the calculation to obtain $\min, Q_1, Q_2, Q_3, \max$, plus $\mathrm{IQR}$ and Tukey outlier fences.
Interpret: higher $\mathrm{IQR}$ means more spread; outliers beyond fences may warrant investigation, trimming, or robust modeling.

Examples (using the same formulas)

Example 1: Data: $7,\, 9,\, 13,\, 15,\, 18$. Sorted is identical. $n=5$ (odd).
$Q_2=x_{(3)}=13.$ Lower half $\{7,9\}\Rightarrow Q_1=\dfrac{7+9}{2}=8.$ Upper half $\{15,18\}\Rightarrow Q_3=\dfrac{15+18}{2}=16.5.$
$\min=7,\ \max=18,\ \mathrm{IQR}=16.5-8=8.5.$
Fences: $8-1.5(8.5)=-4.75,\ \ 16.5+1.5(8.5)=29.75$ (no outliers).

Example 2: Data: $4,\, 6,\, 6,\, 7,\, 12,\, 19$. Sorted is identical. $n=6$ (even).
$Q_2=\dfrac{x_{(3)}+x_{(4)}}{2}=\dfrac{6+7}{2}=6.5.$ Lower half $\{4,6,6\}\Rightarrow Q_1=6.$ Upper half $\{7,12,19\}\Rightarrow Q_3=12.$
$\min=4,\ \max=19,\ \mathrm{IQR}=12-6=6.$ Fences: $6-1.5(6)=-3,\ \ 12+1.5(6)=21$ (no outliers).

Example 3 (percentile interpolation concept): With $n$ values and desired quartile $p\in\{0.25,0.5,0.75\}$, compute $\text{rank}(p)=1+p(n-1)$ and linearly interpolate between surrounding ordered values to obtain $Q_p$.

FAQs

Q1: What are the five numbers in the 5 number summary?
Minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

Q2: Which quartile method should I use?
Choose the method used by your course, textbook, or software (median-of-halves vs. percentile interpolation) for consistent results.

Q3: How are outliers detected?
Using Tukey’s fences: below $Q_1-1.5\,\mathrm{IQR}$ or above $Q_3+1.5\,\mathrm{IQR}$ are flagged as potential outliers.

Q4: Can this handle repeated values or small samples?
Yes. Quartiles and IQR are well-defined for ties and small $n$; interpretation remains robust and distribution-agnostic.