Quartile Calculator

A quartile calculator splits a sorted data set into four equal parts and reports the three cut points: Q1, the median (Q2) and Q3. For the nine values 12, 15, 18, 22, 24, 27, 31, 35, 78 the median-excluded method taught in most textbooks gives Q1 = 16.5, Q2 = 24 and Q3 = 33, so the interquartile range is 16.5 and the value 78 is flagged as an outlier.

Two sister tools cover the neighboring questions. The Standard Deviation Calculator measures spread using every value and its distance from the mean, which suits roughly symmetric data. The Statistics Calculator returns the whole descriptive package at once. Use this page when you specifically need the quartiles, the interquartile range, or a defensible outlier test that a single extreme value cannot distort.

What the three quartiles actually mean

Sort any list of numbers from smallest to largest and you can cut it into four groups of equal size. The three cut points are the quartiles. Q1, the first or lower quartile, sits at the 25% mark: roughly a quarter of the values fall at or below it. Q2 is the median, the 50% mark, with half the data on either side. Q3, the third or upper quartile, sits at the 75% mark, with three quarters of the values at or below it and the top quarter above.

Because the quartiles are defined by position rather than by arithmetic, they behave very differently from the mean. Doubling the largest value in a data set moves the mean noticeably and the standard deviation a lot, but it usually leaves Q1, Q2 and Q3 exactly where they were. That resistance to extreme values is the whole reason quartiles exist, and it is why housing prices, salaries, hospital wait times and web response times are almost always reported with a median and quartiles rather than an average.

The quartile and IQR formulas

There is no single algebraic expression for a quartile the way there is for a mean. Quartiles are defined by a procedure on the sorted data, and the formulas that follow describe that procedure:

The median itself has a simple rule: with an odd count it is the single middle value, and with an even count it is the average of the two middle values. Everything else in the list above follows from applying that same median rule three times, once to the whole data set and once to each half.

Worked example: nine values with one outlier

Take the sorted data set 12, 15, 18, 22, 24, 27, 31, 35, 78. There are nine values, so the count is odd and the median is the fifth value: Q2 = 24. Under the median-excluded method the median is removed before splitting, which leaves a lower half of 12, 15, 18, 22 and an upper half of 27, 31, 35, 78. Each half has four values, so each median is the average of its two middle numbers:

Now build the fences. The lower fence is 16.5 − 1.5 × 16.5 = −8.25, and the upper fence is 33 + 1.5 × 16.5 = 57.75. Every value from 12 to 35 sits comfortably inside that band, but 78 is far above 57.75, so it is flagged as an outlier. The five-number summary is 12, 16.5, 24, 33, 78, and the full range is 78 − 12 = 66, four times the interquartile range, which is exactly the pattern you expect when one value sits well outside the pack.

The mean of the same nine numbers is 29.11 and the sample standard deviation is 19.76. Delete the single value 78 and the mean falls to 23.00 while the standard deviation collapses to 7.89, but the median only moves from 24 to 23. One point changed the mean by six units and cut the standard deviation by more than half; the median barely noticed. That contrast is the practical argument for quartile-based statistics.

Step by step through the sorted data

The table below shows the same nine values with the role each one plays under the median-excluded method, so you can see exactly where the halves begin and end.

Four quartile definitions: textbook halves, Tukey's hinges and Excel

If your textbook, your spreadsheet and your graphing calculator disagree about Q1, the cause is almost always the method. Two families are in common use. The half-based methods split the sorted list at the median and take the median of each half. When the count is even the list splits cleanly and both half-based methods agree; when the count is odd there is a leftover value in the middle, and they handle it differently. The interpolating methods used by spreadsheets compute a position in the sorted list and interpolate between the two neighboring values, so they can differ from the half-based answer even for an even count.

The median-excluded method, used by the TI-83 and TI-84 family as well as many introductory courses, removes the median before splitting. Each half then has fewer values, the halves sit further apart, and the resulting interquartile range is wider. The median-included method, Tukey's hinges from the original box plot, keeps the median in both halves. The halves overlap by one value, they sit closer together, and the interquartile range comes out narrower.

Excel's QUARTILE.EXC places Q1 at position 0.25 × (n + 1) and Q3 at 0.75 × (n + 1) in the sorted list, which is also the default percentile definition in the NIST/SEMATECH e-Handbook of Statistical Methods; it needs at least three values. QUARTILE.INC, which matches the older QUARTILE function and Google Sheets' QUARTILE, uses positions 1 + 0.25 × (n − 1) and 1 + 0.75 × (n − 1). A position such as 2.75 means “the 2nd value plus three quarters of the gap to the 3rd value.” For an odd count the half-based methods line up with Excel exactly: median excluded equals QUARTILE.EXC, and median included equals QUARTILE.INC.

None of these answers is wrong. They are reasonable conventions for the same idea, and the gap between them shrinks quickly as your data set grows. The calculator above always shows all four, so you can match whichever one your assignment or software expects. On the ten test scores used further down, the half-based methods give Q1 = 74 and Q3 = 91, QUARTILE.EXC gives 73.25 and 91.75, and QUARTILE.INC gives 75 and 90.25.

The two half-based methods compared on six data sets

Every figure in this table was computed with the same rules the calculator uses. Notice that the two even-count rows agree exactly, while every odd-count row shows a different Q1, Q3 and IQR. Because these counts are odd, the median-excluded columns also equal Excel’s QUARTILE.EXC and the median-included columns QUARTILE.INC; for the even rows Excel gives other values, as the calculator’s comparison table shows.

The last row shows how large the gap can get on a small, skewed sample: an IQR of 28 under one convention and 17 under the other, from identical numbers. On larger data sets the difference typically becomes negligible.

How many values land in each half

This table shows how the sorted list is divided by the two half-based methods for counts from 5 to 14. It is the quickest way to see why odd counts create two answers and even counts create one.

The 1.5 × IQR rule and the four fences

An outlier test needs a cutoff, and the interquartile range supplies one that scales with the data instead of assuming a fixed threshold. The standard rule multiplies the IQR by 1.5 and steps that distance out from each quartile. Anything past those inner fences is a candidate outlier, called a mild outlier in the NIST/SEMATECH e-Handbook. Stepping out three interquartile ranges instead gives the outer fences, and a value beyond them is an extreme outlier, unusual even among the unusual ones.

That last row is worth pausing on. The same value, 78, in the same data set, is a mild outlier under one convention and an extreme outlier under the other, purely because the narrower median-included IQR pulls the fences inward. When an outlier decision matters, state which quartile method you used.

Second worked example: ten test scores

Now take an even-count data set, ten exam scores: 62, 71, 74, 78, 82, 85, 88, 91, 94, 98. Ten is even, so the list splits cleanly into 62, 71, 74, 78, 82 and 85, 88, 91, 94, 98 with no leftover value, and both half-based methods return the same answer (Excel’s interpolating functions do not: 73.25 and 91.75 with QUARTILE.EXC, 75 and 90.25 with QUARTILE.INC). The median is the average of the fifth and sixth values, (82 + 85) ÷ 2 = 83.5. Each half has five values, so each quartile is simply the middle one of its half: Q1 = 74 and Q3 = 91.

The interquartile range is 91 − 74 = 17, meaning the middle half of the class scored within a 17-point band. The fences are 74 − 25.5 = 48.5 and 91 + 25.5 = 116.5, and since the lowest score is 62 and the highest is 98, no score is an outlier. The five-number summary is 62, 74, 83.5, 91, 98; the mean is 82.30 and the sample standard deviation is 11.19. Here the mean and the median sit close together, which is the signature of a fairly symmetric data set.

How to use this calculator

Paste or type your numbers into the box. Commas, spaces, semicolons and line breaks all work as separators, so a column copied straight out of a spreadsheet is fine. Dollar and percent signs are stripped, and any token that is not a number is skipped with a short note so a stray column heading does not break the calculation.

Pick the quartile method with the four buttons: median excluded, median included, QUARTILE.EXC or QUARTILE.INC. If you are checking homework, choose the one your course uses; if you are checking a spreadsheet, choose the Excel function it uses; if you are not sure, look at the side-by-side table further down, which always shows all four. Leave the decimal places on Exact, or round to 0, 1 or 2 places to match how you report the answer. The results update as you type: the headline is the interquartile range, the three tiles beneath it are Q1, Q2 and Q3, and the cards below give the five-number summary, the fences, the mild and extreme outliers, the method comparison, the steps and a sorted table with each value labeled. Commas separate values, so leave out thousands separators.

Who this calculator is for

Students working through a statistics unit use it to check quartile and box plot homework, and the step-by-step card shows the intermediate values a grader will want to see. Teachers use it to generate examples with predictable answers. Analysts and engineers use the IQR and the fences to screen a data set before modeling it, since a single mistyped value can wreck an average while leaving a median untouched. Quality and operations teams use quartiles to describe cycle times and defect counts, where the distribution is usually skewed and the mean is misleading. Anyone comparing two groups can also use the midhinge and the interquartile range as a compact, outlier-resistant summary of center and spread.

Key terms

• Quartile: one of the three cut points that divide sorted data into four equal-sized groups.
• Q1, first or lower quartile: the 25th percentile; the median of the lower half.
• Q2, median: the 50th percentile; the middle value of the whole data set.
• Q3, third or upper quartile: the 75th percentile; the median of the upper half.
• Interquartile range (IQR): Q3 − Q1; the width of the middle 50% of the data.
• Fence: a cutoff beyond a quartile used to flag outliers: the inner fences sit 1.5 interquartile ranges out, the outer fences 3.
• Outlier: a value beyond an inner fence (a mild outlier) or beyond an outer fence (an extreme outlier).
• Five-number summary: minimum, Q1, median, Q3 and maximum, the five values a box plot draws.
• Midhinge: the average of Q1 and Q3, an outlier-resistant estimate of the center.
• Quartile deviation, or semi-interquartile range: half the IQR, sometimes reported as a spread measure.
• Tukey's hinges: the quartile definition behind the original box plot, matching the median-included method here.
• QUARTILE.INC and QUARTILE.EXC: Excel's two interpolating quartile functions, at positions 1 + p(n − 1) and p(n + 1).

What changes your quartiles

Four things move the numbers, and knowing which one is responsible saves a lot of confusion when two tools disagree.

• The method. Median excluded versus median included on an odd count, or a textbook method versus an Excel function on an even count, is the single most common reason for a mismatch.
• The count. Adding or removing one value can flip the count from even to odd and change which rule applies. It also shifts every position in the sorted list.
• Duplicates. Repeated values are counted individually, not collapsed. Ten copies of the same number pull the quartiles toward it, and a data set with enough duplicates can end up with Q1 = Q3 and an IQR of zero.
• Values near the quartile positions. A change to the smallest or largest value usually leaves the quartiles untouched, while a change to a value sitting right at a quartile position moves it directly. That is the opposite of how the mean behaves.

Tips for reporting quartiles

• Always say which method you used when the count is odd, and say it in the same sentence as the number.
• Report the five-number summary as a unit rather than a lone IQR. Readers need the center and the endpoints to interpret the spread.
• Round the quartiles to the same precision as your raw data. Quoting Q1 = 16.5000 for whole-number measurements suggests a precision you do not have.
• For skewed data, quote the median and IQR instead of the mean and standard deviation, and say so explicitly.
• Investigate flagged outliers before doing anything with them, and report how many you found and what you decided to do.
• When comparing two groups, compare the interquartile ranges rather than the full ranges. The full range grows simply because a larger sample gives extreme values more chances to appear.

Limitations of quartiles and the IQR

Quartiles throw information away on purpose, and that has costs. They ignore the shape of the data inside each quarter, so two very different distributions can share an identical five-number summary. On small samples the quartiles are unstable: with eight or nine values, adding one number can move Q1 or Q3 noticeably, and the fences move with them.

The 1.5 × IQR rule is a convention, not a statistical test. It does not produce a probability, it has no significance level, and on a large data set from a normal distribution it will flag about 0.7% of the values, roughly one in 140, as outliers even when nothing is wrong. Treat a flagged point as a prompt to investigate, never as proof of an error. Finally, the IQR is undefined in any useful sense for categorical data, and for heavily rounded or discrete data it can land on a value that no observation actually takes.

Related calculators

• For spread measured from the mean, including sample and population variance, use the Standard Deviation Calculator.
• For a full descriptive summary of one data set in a single pass, use the Statistics Calculator.
• To find how many standard deviations a single value sits from the mean, use the Z-Score Calculator.
• For the mean, median and mode of a list on their own, use the Average Calculator.
• To express a quartile as a share of a total, use the Percentage Calculator.
• For chance and distribution questions built on the same data, use the Probability Calculator.

Sources

Quartiles, the interquartile range and the fence rule are definitions applied to your own numbers, not published statistics. The definitions follow these references, and every figure on this page was computed directly from the data sets shown:

• Percentile definitions. NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.6.2, Percentiles: the default position p(N + 1) with linear interpolation, and the alternative 1 + p(N − 1) that the handbook notes Excel uses.
• Excel functions. Microsoft’s documentation of QUARTILE.EXC (its example data, 6 to 49, gives Q1 = 15 and Q3 = 43, the last row of the comparison table above) and QUARTILE.INC.
• The fence rule. Inner fences at Q1 − 1.5 × IQR and Q3 + 1.5 × IQR, outer fences at 3 × IQR, mild and extreme outliers: NIST/SEMATECH e-Handbook, section 7.1.6, What are outliers in the data? They are rules of thumb by construction, not derived thresholds.
• The half-based conventions. The median-excluded method (median removed from the halves) and the median-included method (Tukey's hinges) are the two textbook conventions for odd-count data. Both are computed here so results can be matched to either.
• Worked examples and tables. All quartiles, interquartile ranges, fences, means and standard deviations shown above were computed from the listed data sets using these formulas.