Five Number Summary Visualizer
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Five Number Summary Examples
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What a Five Number Summary Actually Tells You
A five number summary is five values — minimum, first quartile (Q1), median, third quartile (Q3), and maximum — that describe where a dataset sits and how spread out it is. Unlike the mean and standard deviation, every one of these five numbers is a real data point (or sits exactly between two real data points), so a single huge value can't drag the whole summary off course the way it drags a mean.
That's the main reason statisticians reach for it on messy or skewed data: household income, hospital wait times, response times on a server, anything with a long tail. The five number summary is also what feeds directly into a box plot, so once you have the five values, the chart practically draws itself.
The Five Values, One at a Time
| Value | Percentile | What it marks | Where it sits on a box plot |
|---|---|---|---|
| Minimum | 0th | The smallest value in the dataset | End of the lower whisker (unless it's an outlier) |
| Q1 | 25th | Median of the lower half | Left/bottom edge of the box |
| Median | 50th | Middle value of the sorted dataset | Line drawn inside the box |
| Q3 | 75th | Median of the upper half | Right/top edge of the box |
| Maximum | 100th | The largest value in the dataset | End of the upper whisker (unless it's an outlier) |
How to Calculate It by Hand
Worked Example
Nine students' exam scores: 84, 67, 92, 75, 54, 88, 79, 98, 71.
With 9 values, the median is the 5th value: 79. Using the exclusive method, the lower half is {54, 67, 71, 75} and the upper half is {84, 88, 92, 98}. Q1 is the average of 67 and 71, which is 69. Q3 is the average of 88 and 92, which is 90. The minimum is 54 and the maximum is 98, so the five number summary reads 54, 69, 79, 90, 98.
From there, IQR = 90 − 69 = 21. The lower fence is 69 − 1.5(21) = 37.5 and the upper fence is 90 + 1.5(21) = 121.5. Both 54 and 98 fall inside those fences, so this dataset has no outliers.
The 1.5×IQR Outlier Rule
John Tukey proposed this rule in the 1970s as part of his work on exploratory data analysis, and it's still the default outlier check on almost every box plot drawn today. Take the interquartile range, multiply by 1.5, and use that as a buffer below Q1 and above Q3. Anything outside those two fences gets flagged and plotted as a separate point rather than absorbed into the whisker.
It's a useful default, not a law of nature. On a small dataset, one slightly unusual value can trip the fence even though nothing went wrong with the measurement. On a large dataset with heavy tails, the rule can flag dozens of points that are genuinely part of the distribution. Treat a flagged point as a prompt to look closer, not as automatic grounds to delete it.
A Second Worked Example, With an Outlier
Ten e-commerce order values: 12, 15, 22, 120, 18, 25, 31, 14, 19, 28.
With 10 values (even), the median is the average of the 5th and 6th values: (19 + 22) / 2 = 20.5. The lower half {12, 14, 15, 18, 19} has a median of 15 for Q1. The upper half {22, 25, 28, 31, 120} has a median of 28 for Q3. IQR = 28 − 15 = 13, so the upper fence sits at 28 + 1.5(13) = 47.5. The order of 120 is well past that fence, so it's plotted as an outlier and the upper whisker stops at 31, the largest value that isn't flagged.
Five Number Summary vs. Mean and Standard Deviation
Both approaches summarize a dataset, but they answer slightly different questions and behave differently when data misbehaves.
| Question | Five Number Summary | Mean & Standard Deviation |
|---|---|---|
| Sensitive to one extreme value? | No — it's based on rank, not magnitude | Yes — one huge value moves both |
| Assumes a particular shape? | No — works on skewed or lumpy data | Most interpretation assumes roughly symmetric data |
| Easy to read off a chart? | Yes — that's exactly what a box plot shows | Needs a separate calculation or error bar |
| Uses every data point's exact value? | No — only rank order matters | Yes — every value contributes to the calculation |
In practice, a lot of analysts report both. The five number summary (via a box plot) gives a fast visual read on skew and outliers; the mean and SD support the inferential tests — t-tests, confidence intervals, regression — that assume something closer to a normal distribution.
Common Mistakes
Pulling the "middle" value from an unsorted list gives a number that has nothing to do with the actual median.
If you're comparing your hand calculation to a calculator or to Excel's QUARTILE function, check which convention each one uses. Small odd-sized datasets are where the two methods diverge most.
The fences (Q1 − 1.5×IQR and Q3 + 1.5×IQR) are boundaries for the outlier test. The whisker itself should stop at the most extreme real value that falls inside those boundaries, not at the fence number itself.
IQR measures the width of the middle 50% of the data. It's a spread measure, but it isn't computed the same way as variance or standard deviation and the two won't generally match.
Quick Reference Code
If you're computing this in a script rather than by hand, here's the same calculation in Python and R.
# Python — pandas/numpy
import numpy as np
data = [54, 67, 71, 75, 79, 84, 88, 92, 98]
q1, median, q3 = np.percentile(data, [25, 50, 75])
iqr = q3 - q1
print(min(data), q1, median, q3, max(data), iqr)# R data <- c(54, 67, 71, 75, 79, 84, 88, 92, 98) fivenum(data) # returns min, Q1 (Tukey hinge), median, Q3 (Tukey hinge), max
Note that R's fivenum() uses Tukey's hinge method, while NumPy's default percentile() call uses linear interpolation — neither matches the exclusive method taught in many introductory statistics courses exactly. Always check which method a tool is using before comparing results across software.
Related Topics
This visualizer is part of Statistics Fundamentals, a free reference for descriptive statistics, hypothesis testing, and data visualization. Statistics Fundamentals is necessary reading if you want the underlying theory behind a tool like this one rather than just the chart it produces — the Statistics Fundamentals homepage links out to every topic area on the site.
Sources & further reading:
- Tukey, J.W. (1977). Exploratory Data Analysis. Addison-Wesley. [Origin of the box plot and the 1.5×IQR fence rule]
- NIST/SEMATECH Engineering Statistics Handbook — Box Plot
- Khan Academy — Box Plot Review
Frequently Asked Questions
It's the minimum, first quartile (Q1), median, third quartile (Q3), and maximum of a dataset. Together they describe the center and spread of the data using only rank-based values, which makes the summary resistant to outliers in a way the mean isn't. It's also exactly what you need to draw a box plot.
Sort the data and find the median, then split the dataset into a lower half and an upper half around that median. Q1 is the median of the lower half; Q3 is the median of the upper half. Whether the middle value is included in both halves (inclusive/Tukey method) or excluded from both (exclusive/Moore-McCabe method) depends on which convention you're using, and the two can give slightly different answers on small datasets with an odd number of values.
Calculate the interquartile range (IQR = Q3 − Q1), then build a lower fence at Q1 − 1.5×IQR and an upper fence at Q3 + 1.5×IQR. Any value outside those two fences is flagged as an outlier and plotted as an individual point rather than at the end of a whisker. This is the rule John Tukey introduced alongside the box plot itself.
A box plot shows skew, spread, and outliers at a glance, none of which a single mean value can convey on its own. It's especially useful for comparing several groups side by side, or for any dataset where a few extreme values would otherwise distort the average. For data that's roughly symmetric and outlier-free, mean and standard deviation work fine and are often easier to plug into further calculations.
Almost always it's a difference in quartile method, not an error. Software packages vary in whether they include the median in both halves when splitting the data (inclusive/Tukey hinges), exclude it from both (exclusive/Moore-McCabe), or use linear interpolation between ranked values (the method behind NumPy's default percentile() and many spreadsheet QUARTILE functions). The differences are usually small and disappear entirely once a dataset is large.
Yes — use the Compare Two tab above. It plots both box plots on a shared axis so you can see at a glance whether the medians differ, whether the spreads (IQRs) are similar, and how much the two interquartile boxes overlap. This is a common way to compare a before/after measurement or two treatment groups before running a formal hypothesis test.