What Is a Five Number Summary?

A five number summary is a concise set of five descriptive statistics that together capture the full spread of any numerical dataset. The five values are the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. Together they describe where data starts, how it clusters, and where it ends.

Introduced by the statistician John Tukey in his 1977 book Exploratory Data Analysis, the five number summary is the foundation of the box-and-whisker plot — one of the most widely used charts in statistics, data science, and scientific research.

Students in high school and college statistics courses, data analysts, researchers, and quality engineers all use the five number summary as a fast, robust way to understand any dataset without assuming it follows a normal distribution.

Source: Tukey, J.W. (1977). Exploratory Data Analysis. Addison-Wesley, Reading, MA.

How the Five Number Summary Formula Works

Step-by-Step Method (Inclusive / Tukey)

1. Sort the dataset in ascending order.
2. Minimum = first value.
3. Maximum = last value.
4. Median (Q2) = middle value (or average of two middle values if n is even).
5. Q1 = median of the lower half (including the median if n is odd).
6. Q3 = median of the upper half (including the median if n is odd).

Worked example: Dataset: {2, 5, 7, 10, 14, 18, 22, 25, 30}
Sorted: 2, 5, 7, 10, 14, 18, 22, 25, 30 (n=9)
Min=2 · Q1=median(2,5,7,10,14)=7 · Median=14 · Q3=median(14,18,22,25,30)=22 · Max=30
IQR = Q3−Q1 = 22−7 = 15

Outlier fences: Lower = Q1 − 1.5×IQR = 7 − 22.5 = −15.5 · Upper = Q3 + 1.5×IQR = 22 + 22.5 = 44.5 · No outliers in this set.

Source: Moore, D.S. & McCabe, G.P. (2006). Introduction to the Practice of Statistics, 5th ed. W.H. Freeman, New York.

How to Use This Five Number Summary Calculator

Step 1 – Enter your dataset. Type or paste numbers into the input field. You can separate values with commas, spaces, semicolons, or new lines. The parser handles mixed separators automatically.

Step 2 – Choose your quartile method. Select "Inclusive (Tukey)" for most school and textbook problems. Use "Exclusive" if your instructor or software (e.g., some R functions) specifies it.

Step 3 – Set decimal precision. Choose how many decimal places you need in your results. Default is 2 for most coursework.

Step 4 – Click Calculate. Instantly see your five number summary cards, IQR, outlier detection, extended stats, sorted list, and box plot chart.

Step 5 – Review the box plot. The visual box plot shows the data distribution at a glance. The box spans Q1 to Q3, the line marks the median, and whiskers reach the fences. Outlier points are marked separately.

Step 6 – Export your results. Use the Copy, Print, CSV, JSON, or TXT buttons to save or share your five number summary report.

Source: Verzani, J. (2014). Using R for Introductory Statistics, 2nd ed. CRC Press, Boca Raton, FL.

Five Number Summary vs Other Descriptive Statistics

The five number summary is one approach to describing data. Understanding how it compares to other methods helps you choose the right tool for your analysis.

Five Number Summary vs Mean & Standard Deviation

Mean and standard deviation assume data is roughly normally distributed. The five number summary makes no such assumption, making it better for skewed data, data with outliers, or small samples.

Five Number Summary vs Range

The simple range (Max − Min) captures full spread but is highly sensitive to a single extreme value. The IQR captures the middle 50% and is far more robust.

Source: Agresti, A. & Franklin, C. (2012). Statistics: The Art and Science of Learning from Data, 3rd ed. Pearson Education, Upper Saddle River, NJ.

Real-World Five Number Summary Examples

Example 1 — Student Test Scores (Personal/Academic)

Dataset: 45, 52, 55, 60, 67, 72, 76, 83, 88, 91, 95
Sorted (n=11). Min=45 · Q1=57.5 · Median=72 · Q3=85.5 · Max=95
IQR=28. Lower fence=57.5−42=15.5 · Upper fence=85.5+42=127.5 · No outliers.
The median of 72 and a wider upper half (Q3–Median=13.5 vs Median–Q1=14.5) suggest a roughly symmetric distribution.

Example 2 — Monthly Sales Revenue (Professional)

Dataset ($000s): 12, 18, 21, 25, 29, 33, 38, 44, 52, 61, 78, 142
Min=12 · Q1=23 · Median=35.5 · Q3=56.5 · Max=142
IQR=33.5. Upper fence=56.5+50.25=106.75. 142 is an outlier.
The median (35.5) is much closer to Q1 than Q3, confirming right skew driven by the outlier month.

Example 3 — Patient Blood Glucose (High-Stakes / Downstream)

Dataset (mg/dL): 82, 88, 91, 95, 98, 102, 108, 115, 121, 130, 145, 189
Min=82 · Q1=94.5 · Median=105 · Q3=127.5 · Max=189
IQR=33. Upper fence=127.5+49.5=177. 189 is a clinical outlier.
Downstream calculation: If the outlier (189 mg/dL) is excluded, the revised Median drops to 102 mg/dL — within normal fasting range. This illustrates how one extreme reading can distort clinical interpretation without outlier-aware statistics.

Source: Devore, J.L. (2015). Probability and Statistics for Engineering and the Sciences, 9th ed. Cengage Learning, Boston, MA.

Tips for Better Statistical Analysis

• Always visualise first. Compute the five number summary and draw a box plot before running any other statistical test. It reveals skew, outliers, and spread instantly.
• Investigate outliers — don't delete them automatically. An outlier may be a data entry error or a genuine extreme value. Each case requires judgment.
• Use IQR for outlier detection. The Tukey fence rule (±1.5×IQR) is robust. Applying it to the mean ± 3SD only works for near-normal distributions.
• Report both median and mean. If they differ substantially, your data is skewed. Report the median as the primary measure of centre.
• Use consistent quartile methods. When comparing two datasets, always use the same quartile method. Mixing inclusive and exclusive methods produces incomparable Q1 and Q3 values.
• Check sample size. For very small datasets (n < 5), the five number summary may produce identical quartiles. Interpret with caution.

Source: Hoaglin, D.C., Mosteller, F. & Tukey, J.W. (2000). Understanding Robust and Exploratory Data Analysis. Wiley-Interscience, New York.

Common Five Number Summary Mistakes to Avoid

• Forgetting to sort the data first. Q1, median, and Q3 are all position-based. Calculating them on unsorted data gives wrong answers every time.
• Mixing up inclusive and exclusive quartile methods. Textbooks, calculators, and software all differ. Always specify which method you used when reporting results.
• Treating the five number summary as a complete analysis. It describes distribution shape and spread but does not test hypotheses or establish causation.
• Ignoring the sample size. A five number summary from n=4 values is almost meaningless. Aim for at least n=10 for reliable quartile estimates.
• Confusing IQR with range. IQR = Q3 − Q1 (middle 50%). Range = Max − Min (full spread). They serve different purposes in statistical reporting.
• Assuming outliers are errors. Some outliers are real data points of high scientific interest. Always investigate before excluding them from analysis.

Source: Wild, C.J. & Pfannkuch, M. (1999). Statistical Thinking in Empirical Enquiry. International Statistical Review, 67(3), 223–248.

Frequently Asked Questions