The quartile calculator above splits a sorted data set into four parts and reports the three cut points: Q1 at 25%, Q2 at the halfway mark, Q3 at 75%. What separates it from most free quartile tools is that it does not hide the awkward part. There is no single agreed definition of a quartile. Five conventions are in common use, they routinely return different numbers from identical data, and this page computes all five at once.
Arb Digital built it after seeing the same confusion repeatedly: a student gets one Q1 by hand, a spreadsheet returns another, a statistics package a third. Nobody made an arithmetic error. Each applied a different rule for a quartile position falling between two observations. Seeing all five together, the question stops being which is right and becomes which one your context expects.
What This Quartile Calculator Does
Quartiles are order statistics: they describe position within a ranked list, not distance from an average. That makes them robust — one absurd value at the top moves the mean substantially and usually leaves Q1 and Q3 untouched. It is why box plots, income bands and outlier rules are built on quartiles rather than standard deviations.
The tool parses your values, sorts them, reports the count n so a mis-paste is visible, and computes Q1, Q2 and Q3 five ways. The hero is Q1 under the method you selected, because Q1 is where the methods disagree most. The fourth tile is the gap between the highest and lowest Q1 across the five. When it reads zero, every convention agrees; when it does not, the method you cite matters as much as the number.
How to Use It
- Paste your values into the box. Commas, spaces, tabs, semicolons and line breaks all work, mixed freely. Trailing separators and blank lines are ignored, not counted as zeros.
- Check the parsed count. The panel states how many values it read. If that does not match expectations, something in the paste was not a number.
- Pick a method from the dropdown. If a spreadsheet is your reference, Type 7 matches PERCENTILE.INC and Type 6 matches PERCENTILE.EXC.
- Read the hero and grid. Q1, Q2 and Q3 come from your chosen method; the fourth tile measures how much the five disagree about Q1.
- Scan the five-method table before quoting anything. If the rows differ, state your method alongside your number.
The Five Methods and How Each One Is Calculated
All five sort the values ascending and let n be the count. They differ only in how a proportion p becomes a position, and what happens when that position is not a whole number.
Exclusive (Tukey hinges). Split the sorted list at the median. If n is odd, throw the middle value away and take the median of each remaining half. Q1 is the median of the lower half, Q3 of the upper. This is the hand method taught in most introductory courses and the one Tukey used for the original box plot.
Inclusive (Moore). Identical, except that when n is odd the middle value is kept in both halves rather than discarded. Each half is therefore one element longer, and Q1 and Q3 are pulled inward toward the median.
Nearest rank (Type 1). Take the value at rank ceil(p x n). No arithmetic between neighbours, so the answer is always an actual observation. For n = 11 and p = 0.25 the rank is ceil(2.75) = 3, so Q1 is the third smallest value.
Linear interpolation, Type 7. The position is h = (n - 1)p counted from zero. If h is not an integer, move the fractional distance from the value below it to the value above. This is the default in R and in NumPy, and it is what Excel PERCENTILE.INC and QUARTILE.INC return.
Type 6. The position is h = (n + 1)p counted from one, again with linear interpolation. Minitab and SPSS use it, and it matches Excel PERCENTILE.EXC and QUARTILE.EXC. Because (n + 1)p can fall below 1 or above n on small samples, this calculator clamps those cases to the minimum or maximum.
A Worked Example You Can Check Yourself
The calculator loads with eleven values: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49. Already sorted, so n = 11 and the median is the sixth value, 40. Every method agrees Q2 is 40, because with an odd n the median is an actual observation and there is nothing to interpolate. Q1 is where the trouble starts.
Exclusive by hand. Remove the median. The lower half is 6, 7, 15, 36, 39, whose middle value is 15. The upper half is 41, 42, 43, 47, 49, whose middle value is 43. So Q1 = 15, Q2 = 40, Q3 = 43.
Inclusive by hand. Keep the median in both halves. The lower half becomes 6, 7, 15, 36, 39, 40 — six values, so its median is the average of the third and fourth, (15 + 36) / 2 = 25.5. The upper half is 40, 41, 42, 43, 47, 49, whose median is (42 + 43) / 2 = 42.5. So Q1 = 25.5, Q2 = 40, Q3 = 42.5.
Type 7 by hand. The position is h = (n - 1)p = 10 x 0.25 = 2.5, counted from zero. Position 2 holds 15 and position 3 holds 36, so Q1 = 15 + 0.5 x (36 - 15) = 15 + 10.5 = 25.5. For Q3, h = 10 x 0.75 = 7.5; position 7 holds 42 and position 8 holds 43, giving 42 + 0.5 x 1 = 42.5.
Nearest rank. ceil(0.25 x 11) = ceil(2.75) = 3, the third value, 15. For Q3, ceil(8.25) = 9, the ninth value, 43. Type 6. h = (n + 1)p = 12 x 0.25 = 3 exactly, so Q1 is the third value, 15. For Q3, h = 9, the ninth value, 43.
Line them up. Q1 is 15 under exclusive, nearest rank and Type 6, and 25.5 under inclusive and Type 7. On the same eleven numbers the reported first quartile ranges from 15 to 25.5 — a gap of 10.5, making the larger answer 70% above the smaller. Q3 is steadier at 42.5 or 43, and Q2 is unanimous at 40. That pattern is typical: the median is stable, and the quartiles are where conventions fight. The NIST/SEMATECH e-Handbook chapter on exploratory data analysis sets out the order-statistic definitions formally.
The Even-n Case Where Two Methods Become One
Try a second example: 1, 2, 3, 4, 5, 6, 7, 8. Here n = 8 and there is no middle observation — the median is (4 + 5) / 2 = 4.5, a number absent from the data. That fact collapses part of the disagreement.
The exclusive method splits into 1, 2, 3, 4 and 5, 6, 7, 8, giving Q1 = 2.5 and Q3 = 6.5. The inclusive method returns exactly the same answers, and it has to: with an even n there is no median value to include or exclude, so the two rules describe the same split. Exclusive and inclusive can only ever differ when n is odd — which means half the arguments about Tukey versus Moore evaporate the moment your sample size is even.
The other three still diverge. Nearest rank gives 2, 4 and 6, from ceil(2), ceil(4) and ceil(6) — and note its Q2 is 4, not 4.5, since nearest rank never averages. Type 7 gives 2.75, 4.5 and 6.25, from positions 1.75, 3.5 and 5.25 counted from zero. Type 6 gives 2.25, 4.5 and 6.75, from positions 2.25, 4.5 and 6.75 counted from one. Type 6 always sits further out than Type 7, which is the point of the exclusive naming in Excel.
Which Method Should You Use
There is no statistically correct answer, only a contextually correct one. Three rules of thumb cover almost every case.
Match your reference. If a lecturer, marking scheme or journal specified a method, use it. If you are reproducing someone else's published figure, compare all five against their number to work out which they used.
Match your software. Reporting alongside R, Python or a modern spreadsheet means Type 7; alongside Minitab or SPSS, Type 6. Mixing conventions inside one document produces a box plot whose whiskers contradict the table beside it.
Prefer a real observation on ordinal data. Nearest rank never invents a number. For survey responses on a 1 to 5 scale, an interpolated Q1 of 2.75 is meaningless because nothing between 2 and 3 exists.
Sample size matters too. Small samples diverge most, because one position shift moves the answer a long way. By n = 100 the five answers usually cluster within a rounding error. The disagreement tile is effectively a small-sample warning light.
Where Quartiles Stop and the Neighbouring Tools Begin
This page owns one question: how the quartile methods differ and which number each one gives. Three closely related calculators pick up where it stops.
This calculator explains and compares the quartile conventions at 25, 50 and 75 per cent; if you want any other percentile answered under a single convention, use the percentile calculator instead. It reports the three quartiles and how the methods disagree about them; if you want the minimum and maximum added and the box plot drawn, use the five-number summary calculator instead. And it produces Q1 and Q3 as values; if you want those turned into a spread and run through the 1.5 x IQR outlier test, use the interquartile range calculator instead.
The mean, median and mode calculator gives the centre as single figures, and the standard deviation calculator measures spread around the mean — a figure pulled hard by the extremes quartiles shrug off.
Reading Quartiles Without Overreading Them
A quartile is a boundary, not a bucket. Q1 = 15 does not mean a quarter of your values equal 15; it means roughly a quarter fall at or below it. With small n that "roughly" does real work: eleven values cannot be split into four equal groups at all, since 11 divided by 4 is 2.75.
Ties compound this. If thirty of fifty survey responses are 4, Q1, Q2 and Q3 can all land on 4 and the box collapses to a line. Nothing is broken: the data has no spread across its middle half. Quartiles also say nothing about shape between the cut points, so two data sets can share identical quartiles and look nothing alike. The Penn State Eberly College of Science statistics courses derive the order statistics in full.
Arb Digital builds analytics setups with the definitions written down, so two dashboards on the same data never quietly disagree.
Browse All Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Quoting a quartile without naming the method — on small samples the same data can support two answers 70% apart.
- Assuming a spreadsheet and a textbook must agree — PERCENTILE.INC is Type 7 and most hand-taught courses use the exclusive Tukey split, so they disagree by design when n is odd.
- Sorting as text — a text sort places 100 before 20, wrecking every order statistic. This tool sorts numerically; a spreadsheet will not always do so.
- Interpolating on ordinal data — a Q1 of 2.75 on a five-point scale describes a response nobody gave.
- Reading Q1 as a group rather than a boundary — it marks the position below which about a quarter of the data lies, and with small n that quarter cannot come out exact.
Related Free Tools From Arb Digital
Answer any percentile with the percentile calculator, get the box-plot picture from the five-number summary calculator, or run the outlier test with the interquartile range calculator. For squared-unit spread there is the variance calculator, and the free online tools hub lists the whole statistics set.
Frequently Asked Questions
Because there is no single agreed definition. Five conventions are in common use and they differ in how a fractional position between two observations is resolved. On the eleven-value example loaded here, Q1 comes out as 15 under three methods and 25.5 under the other two.
QUARTILE.INC and PERCENTILE.INC use Type 7, the same default as R and NumPy. QUARTILE.EXC and PERCENTILE.EXC use Type 6, which matches Minitab and SPSS. The older QUARTILE function is the same as QUARTILE.INC.
Both split the sorted data at the median. Exclusive discards the middle value when n is odd, inclusive keeps it in both halves. When n is even there is no middle value to include or exclude, so the two methods always give identical answers.
Under four of the five methods, yes. The nearest rank method is the exception: it never averages two values, so on an even-sized data set it returns an actual observation rather than the midpoint. For the values 1 to 8 it gives Q2 as 4 rather than 4.5.
Four is the arithmetic minimum, but the methods disagree most sharply on small samples. Around n equals 100 the five answers usually differ by less than a rounding error, and beyond that the choice of convention rarely changes a conclusion.
Nearest rank, in most cases. It always returns a value that actually occurs in the data, so it will not report a first quartile of 2.75 on a five-point scale where only whole numbers exist.
No. The calculator sorts the values numerically before doing anything else, so the order you paste them in makes no difference. It also ignores blank lines and trailing separators rather than treating them as zeros.
This calculator applies published quartile definitions to the numbers you enter. Which definition suits your data, and what the resulting quartiles mean in your context, remain judgements the tool cannot make for you.
