The five number summary describes any dataset with five values: the minimum, the first quartile (Q1), the median (Q2), the third quartile (Q3) and the maximum.

Sort your numbers from smallest to largest. The median splits them in half, Q1 is the middle of the lower half, and Q3 is the middle of the upper half. Together these five numbers tell you the centre, the spread and the shape of the data without assuming it is bell-shaped. They are exactly the five values a box-and-whisker plot draws.

Rule of thumb: any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is a suspected outlier; beyond 3 × IQR it is an extreme outlier.

The five number summary is a compact description of a dataset built from five order statistics. It answers three questions at once: where is the middle, how spread out are the values, and is the distribution lopsided. Because every one of the five numbers is a position in the sorted data rather than an arithmetic average, the summary is robust: a single wild value can move the maximum but it cannot drag the median or the quartiles the way it drags the mean.

The five numbers, in order:

• Minimum, the smallest value in the dataset.
• First quartile (Q1), the 25th percentile. A quarter of the values fall below it.
• Median (Q2), the 50th percentile. Half the values fall below it.
• Third quartile (Q3), the 75th percentile. Three quarters of the values fall below it.
• Maximum, the largest value in the dataset.

What this calculator adds: the interquartile range (Q3 − Q1), the total range (max − min), the lower and upper Tukey fences, a list of any suspected and extreme outliers, the whisker ends actually used by a box plot, a skewness read from the quartile positions, and the sorted data with each quartile marked.

A worked one-liner: for the seven values 2, 4, 5, 7, 8, 9, 12 the median is 7, the lower half is 2, 4, 5 so Q1 = 4, the upper half is 8, 9, 12 so Q3 = 9, giving the summary 2, 4, 7, 9, 12 and an IQR of 5.

Who uses it: teachers introducing distributions before the normal curve, ecologists comparing body sizes across sites, clinicians summarising skewed measures such as length of stay, exam boards reporting score distributions, and analysts screening any new dataset for outliers before modelling.

4.1 Reading the five numbers in order

Read the summary from left to right and a picture of the distribution builds itself. The minimum and maximum tell you the total span of what was observed. Q1 and Q3 tell you where the bulk of the data sits: exactly half of every dataset falls between them, no matter how strange the shape. The median tells you the typical value in the only sense that is immune to extremes, since it is a position rather than an average.

The gap between the minimum and Q1 covers the lowest quarter of the data, and the gap between Q3 and the maximum covers the highest quarter. Each of those quarters contains exactly the same number of observations as the box does per half, so when one of them is much longer, the values in it are more thinly spread. That is the whole logic of reading a box plot.

4.2 What the interquartile range actually measures

The interquartile range is Q3 minus Q1, and it is the width of the middle 50% of your data. It is the natural partner to the median in the same way the standard deviation partners the mean. Its great virtue is resistance: you could take the largest value in your dataset and multiply it by a thousand, and the IQR would not move at all, because Q3 depends on position not magnitude. That is why skewed data, income, hospital stays, species counts, are almost always summarised with median and IQR rather than mean and standard deviation.

A small IQR relative to the total range means the data are concentrated with a few stragglers at the ends. An IQR that fills most of the range means the values are spread fairly evenly, with no dense core.

4.3 Judging skew from the quartiles alone

You do not need a skewness statistic to see asymmetry. Compare the two halves of the box. If the distance from the median to Q3 is clearly larger than the distance from Q1 to the median, the upper half of the middle data is more stretched, which indicates right skew, the direction typical of incomes, waiting times and counts. If the lower half is longer, the data are left skewed, which is common for scores near a ceiling such as easy exams or percentage completion.

Then check the whiskers, because they often disagree with the box. A short box with one very long whisker signals a tight core with a thin tail. When box and whiskers point in opposite directions, the skew is not consistent across the distribution and you should look at a histogram before summarising in words.

4.4 How the outlier fences are built and what they mean

The fences come from Tukey. The lower fence is Q1 minus 1.5 times the IQR and the upper fence is Q3 plus 1.5 times the IQR. Anything beyond a fence is flagged as a suspected outlier; anything beyond three times the IQR is called an extreme outlier. The 1.5 multiplier is a convention, not a law of nature. Tukey chose it because for roughly normal data it flags only about 0.7% of observations, which is rare enough to be interesting but common enough that a single flag is not alarming.

The critical point is what a flag means. It means look at this value. It does not mean delete it. An outlier can be a typing error, an instrument fault, a genuinely unusual case, or the single most scientifically interesting observation in the study. Deleting values simply because a fence flagged them is one of the most common forms of accidental data manipulation.

4.5 Why the whiskers may not reach the minimum and maximum

This trips up almost every student. In a standard box plot the whiskers do not extend to the extremes of the data. They extend to the furthest observation that still sits inside the fences, and anything beyond that is drawn as a separate dot. So when your data contain outliers, the whisker ends and the true minimum and maximum are different numbers. This calculator reports both, so you can label your figure correctly.

If your course or journal wants whiskers drawn all the way to the extremes, switch the whisker rule in section 2 to Always to the minimum and maximum. Just be sure the caption says which convention you used, because the two plots can look very different from the same data.

4.6 Why your quartiles may differ from another program

There is no single agreed definition of a quartile, and mainstream software genuinely disagrees. The Tukey or Moore and McCabe method finds the median, splits the data into two halves excluding that median when the count is odd, and takes the median of each half. This is what a TI-84 does and what most introductory textbooks teach. Excel's QUARTILE.INC, R's default type 7 and NumPy's percentile use linear interpolation across the whole sorted list, which usually gives a slightly narrower box. Excel's QUARTILE.EXC, R type 6 and Minitab use a different interpolation that usually gives a slightly wider box.

None is wrong. For large samples the three converge and the difference is irrelevant. For small samples the difference can be visible, which is why this calculator lets you pick the method and prints which one it used. If you are checking homework, choose Tukey. If you are reproducing a spreadsheet, choose the inclusive method.

4.7 What the four charts each show you

Chart 1 is the box plot itself, the direct picture of the five numbers, with any outliers as separate dots. This is the figure to put in your report.

Chart 2 shows the five values as bars so you can read the actual numbers off an axis, which is useful when the box plot is visually cramped because the data are tightly clustered.

Chart 3 plots every sorted observation with horizontal lines at Q1, the median and Q3. This makes gaps, ties and clumping visible in a way the box plot deliberately hides, and it is the fastest way to spot that your data are actually two groups mixed together.

Chart 4 is the cumulative distribution. Read across from 25%, 50% and 75% on the vertical axis and drop down to the horizontal axis to find each quartile. It shows why the quartiles sit where they do, and steep sections reveal where values pile up.

4.8 What the five number summary cannot tell you

The summary is deliberately coarse. It cannot reveal whether a distribution has two peaks, because two very different shapes can share the same five numbers. It says nothing about sample size, so a summary from six observations looks identical in form to one from six thousand and deserves far less trust. It ignores everything about the values inside each quarter, including ties and gaps. And, being purely descriptive, it offers no test, no p-value and no confidence interval, so two boxes that look different are not thereby significantly different.

4.9 Practical judgement

Use the five number summary as a first look, not a final answer. It is the right summary when the data are skewed, ordinal or outlier-prone, and it is the honest choice when you do not yet know the shape. Once you have seen the box plot, decide whether the mean is safe to report as well, whether the outliers are real, and whether the sample is large enough to say anything at all. Report the numbers alongside n, always, so the reader can calibrate how much weight the picture deserves.

▶ Run the analysis above to auto-fill all five examples with your results.

1. Enter your data. The default tab takes comma-separated numbers, exactly as the placeholder shows: 52, 48, 55, 61, 47, .... Newlines, tabs, semicolons and spaces work too, so a column pasted straight from Excel is fine. The data do not need to be sorted; the calculator sorts them for you.
2. Name each cluster. The group name field above every textarea is editable. Type something meaningful such as Site A, north slope and that name flows into the results table, all four charts and the exported report.
3. Add or remove clusters. Press Add cluster for a second, third or fourth group, so you can compare box plots side by side. Every column has its own Clear button to empty it and its own Remove button to delete it. Remove is disabled on the last remaining column.
4. Or upload a file. On the Upload tab, choose a CSV or Excel file and click the column names you want. Every column you click becomes its own cluster with its own five number summary. Click a highlighted column again to deselect it.
5. Try a sample dataset. Ten named datasets are built in, spanning ecology, clinical, education, business and two deliberately awkward sets, one with a strong outlier and one with heavy skew, so you can see what those look like in a box plot.
6. Choose your quartile method. This matters more than anything else on the page. Pick Tukey to match a TI-84 or a textbook, inclusive to match Excel's QUARTILE.INC or R's default, exclusive to match QUARTILE.EXC or Minitab.
7. Set the outlier and whisker rules. The standard 1.5 × IQR fence suits nearly everyone. Switch the whisker rule if your course requires whiskers drawn all the way to the minimum and maximum.
8. Press Calculate Five Number Summary. Nothing is computed until you do, and changing a dataset or a setting clears the results so you never read stale numbers.
9. Read the five coloured cards, then the working. With step-by-step working switched on, the calculator shows the sorted data, the median position, how each half was split and how each fence was computed, so you can check it by hand or copy it into homework.
10. Check the four charts and export. The box plot is the figure for your report. Section 5 auto-fills five reporting styles, and the two buttons under the charts export a plain-text report or a print-ready PDF.

Excel has no single "five number summary" command, so you build it from five functions: MIN, QUARTILE.INC, MEDIAN and MAX. This section shows the layout step by step with a picture of the spreadsheet each time, then covers the quartile method trap, box plots, and the equivalents in Google Sheets, R, Python, SPSS and the TI-84.

8.1 The functions you need

The second argument of QUARTILE.INC is the quart number: 0 = minimum, 1 = Q1, 2 = median, 3 = Q3, 4 = maximum. That is why one function can produce the whole summary.

8.2 Step by step, with the spreadsheet at every stage

8.3 The quartile trap: QUARTILE.INC vs QUARTILE.EXC

This is the single biggest reason an Excel answer disagrees with a textbook, a TI-84 or a classmate. Excel has two quartile functions and they give different numbers on the same data.

To reproduce the Tukey method in Excel you have to build it manually with MEDIAN on each half, using SMALL and LARGE to define the halves. It is fiddly and error-prone, which is exactly why this calculator offers all three methods with one click.

8.4 Making a box plot in Excel

1. Select your data column, including the header.
2. Go to Insert → Charts → Insert Statistic Chart → Box and Whisker. This exists in Excel 2016 and later.
3. Right-click the box and choose Format Data Series.
4. Tick Show inner points to plot every observation, and Show outlier points to draw values beyond the fences separately.
5. Under Quartile Calculation, choose Inclusive median or Exclusive median. This changes where the box edges sit, so pick the one matching the method you reported.

8.5 The same thing in Google Sheets

Sheets uses the same names but drops the .INC suffix on the default function.

Sheets' QUARTILE is the inclusive method, identical to Excel's QUARTILE.INC. Sheets also has QUARTILE.EXC. There is no native box plot, so people usually use a candlestick chart as a substitute.

8.6 R, Python, SPSS and the TI-84

On a TI-84 specifically: press STAT, choose 1:Edit, type your values into L1, press STAT again, arrow across to CALC, choose 1:1-Var Stats, press Enter twice, then scroll down past the mean and standard deviation. The last five lines are minX, Q1, Med, Q3 and maxX, which is your five number summary. For the box plot press 2nd then Y=, turn Plot1 on, choose the box plot icon with outliers, and press ZOOM then 9:ZoomStat.

8.7 Excel errors you will probably hit

9.1 The three quartile methods compared on one dataset

Using the nine values 2, 4, 5, 7, 8, 9, 12, 15, 20. Notice the median never changes; only Q1 and Q3 move.

Conclusion: the inclusive method usually gives the narrowest box. For n above about 50 the three methods differ by less than the rounding you would report anyway.

9.2 Where each of the five numbers sits

Conclusion: each of the four gaps between consecutive values contains exactly 25% of the observations, however wide or narrow it looks.

9.3 Outlier fences at a glance

Conclusion: with 1.5 × IQR you should expect roughly one flagged point in every 150 observations even when nothing is wrong, so a single flag in a large sample is not evidence of an error.

9.4 Reading skew from the quartile gaps

Conclusion: box and whiskers can disagree. When they do, plot a histogram before describing the shape in words.

9.5 Converting between the five numbers and other statistics

Conclusion: the IQR ÷ 1.349 shortcut is handy for meta-analysis when a paper reports only a box plot, but say clearly that it is an approximation.

Study design: the five number summary assumes a single set of observations measured on an ordered scale. Unlike the mean it does not require an interval scale, so ordinal data such as ratings and ranks are acceptable.

1. Define the population precisely. Write down exactly what could have been measured, because the quartiles describe that population and nothing else.
2. Build a sampling frame and randomise selection. Convenience sampling distorts the tails first, which is exactly where the minimum, maximum and outlier flags live.
3. Fix the measurement scale and resolution before you start. Record the units and the number of decimal places, and never change them mid-study. Coarse rounding creates artificial ties that can collapse the box.
4. Plan the sample size against what you want the summary to do. For a stable median aim for at least 15 to 20 observations; for stable quartiles aim for 30 or more; for a meaningful maximum and minimum you need considerably more, because extremes keep moving as the sample grows.
5. Measure each unit once. If you take replicates, average them into one value per unit first, otherwise the sample size is inflated and the quartiles are falsely precise.
6. Write your outlier policy before collection. Decide in advance what you will do with flagged values: investigate all of them, exclude only those with a documented cause, and report how many were excluded and why.
7. Decide the quartile method in advance and record it in the protocol, so results are reproducible by anyone using different software.
8. Record raw data in one column per group, exactly as the layout table below shows, ready to paste or upload here.

One column per group, one row per unit, numbers only below the header row. Blank cells are ignored, so unequal group sizes are fine.

Pre-registration and ethics: record the analysis plan, including the quartile method, the fence multiplier and the outlier policy, before you look at the data. Deciding an outlier rule after seeing the box plot is one of the easiest ways to bias a descriptive result without intending to.

Common collection mistakes:

• Rounding aggressively at collection, which creates ties and can give a zero-width box.
• Mixing two genuinely different populations into one column, which produces a wide box that describes neither.
• Recording a censored value such as "over 100" as 100, which corrupts the maximum and the upper quartile.
• Deleting flagged outliers before the summary is computed, then reporting the summary as if it were complete.
• Changing observer or instrument part way through, which widens the IQR for reasons that have nothing to do with the subject.

Use the five number summary whenever you want an honest first description of a dataset, especially before you know its shape.

Use it when:

• ✓ Your data are skewed, so the mean would be misleading.
• ✓ Your data contain outliers you want to identify rather than let them distort the summary.
• ✓ Your data are ordinal, such as Likert ratings, ranks or grades.
• ✓ You need to draw a box-and-whisker plot, which is built from exactly these five numbers.
• ✓ You are comparing several groups and want a compact visual comparison.
• ✓ You are screening a new dataset before modelling.

Do not rely on it alone when:

• ✗ The distribution may have two peaks, because two very different shapes can share the same five numbers. Plot a histogram as well.
• ✗ Your sample is very small, under about ten, where quartiles are unstable. Show the raw values instead.
• ✗ You need a formal comparison between groups. Use a Mann-Whitney or Kruskal-Wallis test; overlapping boxes are not a significance test.
• ✗ You need a confidence interval for the median. Use a bootstrap or the order-statistic method.
• ✗ Your data are categorical with no natural order. Quartiles have no meaning there.

Real-world examples:

1. Ecology, tree diameter or body mass per plot, compared as side-by-side box plots.
2. Medicine, hospital length of stay, which is always right skewed and always reported as median with IQR.
3. Education, exam score distributions per class, where the box plot shows spread and the outliers show who needs support.
4. Business, order values or delivery times per region, where a handful of very large orders would wreck a mean.
5. Quality control, machine output per shift, where the fences act as an automatic screening rule.

Decision tree: ordered numeric data → want centre and spread → skewed, ordinal or outlier-prone → use the five number summary. If the data are symmetric and clean, the mean and standard deviation carry more information. If you need to test a difference between groups, move to a rank-based test.

Assumptions

1. The data are at least ordinal. Values must be rankable from smallest to largest. Check the measurement scale. If the categories have no natural order, quartiles are meaningless.
2. Observations are independent. Check the sampling design. Repeated measures on the same unit make the sample look larger and the quartiles look more stable than they are.
3. The sample represents the population of interest. Check the sampling frame. Selection bias distorts the extremes first, so the minimum, maximum and outlier flags are the least trustworthy parts of a biased sample.
4. Enough observations for quartiles to be stable. Check n. Below about ten the quartiles move substantially when a single value changes, and below four the exclusive method is undefined.
5. The values are recorded at consistent precision. Check the raw data. Aggressive rounding manufactures ties, which can collapse the IQR to zero and make the fence rule unusable.
6. The data come from one population. Check chart 3 for two separate clumps. A box plot of two mixed groups gives a wide box that describes neither.

Limitations

• The five number summary cannot detect multiple peaks. Two very different distributions can produce identical summaries. Always pair it with a histogram or the sorted-data chart.
• It does not report sample size on its own, so always state n alongside it.
• It gives no measure of uncertainty. For a confidence interval around the median, use a bootstrap or the order-statistic method.
• It is purely descriptive. Non-overlapping boxes are suggestive but are not a hypothesis test; use Mann-Whitney or Kruskal-Wallis for that.
• The 1.5 × IQR fence is a convention, not a statistical test. It flags roughly 0.7% of perfectly normal data, so expect occasional false alarms in large samples.
• The mean cannot be recovered from the five numbers, and the standard deviation can only be approximated as IQR divided by 1.349, and only when the data are roughly normal.
• This tool assumes a simple unweighted sample. Weighted, stratified or clustered survey data need a survey package that computes design-consistent quantiles.

A five number summary calculator exists to do something a single average never can: describe a dataset honestly without assuming what shape it has. Minimum, Q1, median, Q3 and maximum give you the centre, the spread and the asymmetry in one line, and because every one of them is a position in the sorted data rather than an arithmetic average, no single freak value can quietly distort the picture. That robustness is the whole point.

The most important habit to take away is to read the five numbers as four gaps rather than five points. Each gap holds exactly a quarter of your observations, so a long gap means those values are thinly spread and a short gap means they are packed together. Once you read a box plot that way, skew becomes obvious at a glance and you no longer need a skewness statistic to describe the shape in words.

The second habit is to state your method. There is no universal definition of a quartile, and Excel, R, Minitab and a TI-84 will genuinely give you different values for Q1 on the same small dataset. None of them is broken. Say which method you used, and say what your whiskers represent and what fence multiplier flagged your outliers, and your result becomes reproducible by anyone. Those three sentences prevent almost every argument that starts with "but I got a different answer".

Treat flagged outliers as questions rather than verdicts. A value beyond a fence is asking you to go back to the raw record and check what happened. Sometimes it is a typing slip, sometimes an instrument fault, and sometimes it is the most interesting observation you collected. Deleting it because a formula flagged it is the one thing you should never do silently.

Paste your numbers into the calculator above, pick the quartile method your course or journal uses, read the box plot and the working, and copy the reporting sentence that fits. Then reach for this five number summary calculator again the next time you meet an unfamiliar dataset, because it is the fastest honest look you can take at data you do not yet understand.

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The following peer-reviewed references support the methods used in this five number summary calculator, covering quartile estimation, box plot construction, outlier detection and best practice in reporting descriptive statistics.

1. Tukey, J. W. (1977). Exploratory data analysis. Addison-Wesley. https://search.worldcat.org/title/3058187
2. Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365. https://doi.org/10.1080/00031305.1996.10473566
3. McGill, R., Tukey, J. W., & Larsen, W. A. (1978). Variations of box plots. The American Statistician, 32(1), 12–16. https://doi.org/10.1080/00031305.1978.10479236
4. Frigge, M., Hoaglin, D. C., & Iglewicz, B. (1989). Some implementations of the boxplot. The American Statistician, 43(1), 50–54. https://doi.org/10.1080/00031305.1989.10475612
5. Hoaglin, D. C., Iglewicz, B., & Tukey, J. W. (1986). Performance of some resistant rules for outlier labeling. Journal of the American Statistical Association, 81(396), 991–999. https://doi.org/10.1080/01621459.1986.10478363
6. Hoaglin, D. C., & Iglewicz, B. (1987). Fine-tuning some resistant rules for outlier labeling. Journal of the American Statistical Association, 82(400), 1147–1149. https://doi.org/10.1080/01621459.1987.10478551
7. Moore, D. S., & McCabe, G. P. (2005). Introduction to the practice of statistics (5th ed.). W. H. Freeman. https://search.worldcat.org/title/56617973
8. Wickham, H., & Stryjewski, L. (2011). 40 years of boxplots. Technical report, Rice University. https://vita.had.co.nz/papers/boxplots.pdf
9. Krzywinski, M., & Altman, N. (2014). Visualizing samples with box plots. Nature Methods, 11, 119–120. https://doi.org/10.1038/nmeth.2813
10. Streit, M., & Gehlenborg, N. (2014). Points of view: Bar charts and box plots. Nature Methods, 11, 117. https://doi.org/10.1038/nmeth.2807
11. Weissgerber, T. L., Milic, N. M., Winham, S. J., & Garovic, V. D. (2015). Beyond bar and line graphs: Time for a new data presentation paradigm. PLOS Biology, 13(4), e1002128. https://doi.org/10.1371/journal.pbio.1002128
12. Spitzer, M., Wildenhain, J., Rappsilber, J., & Tyers, M. (2014). BoxPlotR: A web tool for generation of box plots. Nature Methods, 11, 121–122. https://doi.org/10.1038/nmeth.2811
13. Wan, X., Wang, W., Liu, J., & Tong, T. (2014). Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC Medical Research Methodology, 14, 135. https://doi.org/10.1186/1471-2288-14-135
14. Leys, C., Ley, C., Klein, O., Bernard, P., & Licata, L. (2013). Detecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median. Journal of Experimental Social Psychology, 49(4), 764–766. https://doi.org/10.1016/j.jesp.2013.03.013
15. Aguinis, H., Gottfredson, R. K., & Joo, H. (2013). Best-practice recommendations for defining, identifying, and handling outliers. Organizational Research Methods, 16(2), 270–301. https://doi.org/10.1177/1094428112470848
16. Cox, N. J. (2009). Speaking Stata: Creating and varying box plots. The Stata Journal, 9(3), 478–496. https://doi.org/10.1177/1536867X0900900309
17. Marmolejo-Ramos, F., & Tian, T. S. (2010). The shifting boxplot: A boxplot based on essential summary statistics around the mean. International Journal of Psychological Research, 3(1), 37–45. https://doi.org/10.21500/20112084.823
18. American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000
19. NIST/SEMATECH. (2013). e-Handbook of statistical methods. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/
20. R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/