Descriptive Statistics
Interquartile Range Calculator
The interquartile range is Q3 minus Q1, and the page also shows that gap as a share of the full range from minimum to maximum.
Worksheet
Press Calculate to see the result.
The interquartile range is Q3 minus Q1, and the page also shows that gap as a share of the full range from minimum to maximum.
What this calculator answers
Seven values, 11, 14, 18, 19, 22, 25, and 27, have a full range of 16 from the smallest to the largest. The middle half is narrower. Hyndman-Fan type 7 puts Q1 at 16 and Q3 at 23.5, so the interquartile range is 7.5. That 7.5 is the headline. Dividing by the full range says the middle half covers 0.4688 of the distance from 11 to 27. The page stops at that subtraction and that share. It does not place fences at 1.5 times the result.
The headline is Q3 minus Q1. The second figure is that difference divided by maximum minus minimum, when the range is not zero. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Interquartile range uses this relationship.
IQR = Q3 - Q1. Share of the range = IQR / (max - min), when max is not equal to min.
- Q1. Lower quartile, Hyndman-Fan type 7.
- Q3. Upper quartile, Hyndman-Fan type 7.
- IQR. Q3 minus Q1, the headline.
- range. Maximum minus minimum, the denominator of the share.
The central half of the ordered sample is the interval from Q1 to Q3. Its length ignores the outer quarters, which is why a change that stays inside a tail can leave the IQR still until it moves a quartile. The share compares that length with the full extremes. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The ordered values are 11, 14, 18, 19, 22, 25, 27. Hyndman-Fan type 7 gives Q1 = 16 and Q3 = 23.5. The interquartile range is 23.5 - 16 = 7.5.
That difference is the headline. The minimum and the maximum do not enter it. The full range is 27 - 11 = 16. The IQR is 7.5 / 16 = 0.4688 of the range.
Fences at 1.5 times 7.5 are not computed on this page. The count is 7. Quartile positions would be reported on the quartile page.
| Interquartile range | 7.5 |
|---|---|
| IQR as a share of the range | 0.4688 |
| Q1 | 16 |
| Q3 | 23.5 |
| Range | 16 |
A second situation
25 becomes 35, which lifts the upper quartile
The ordered values are 11, 14, 18, 19, 22, 27, 35. Hyndman-Fan type 7 gives Q1 = 16 and Q3 = 24.5. The interquartile range is 24.5 - 16 = 8.5.
That difference is the headline. The minimum and the maximum do not enter it. The full range is 35 - 11 = 24. The IQR is 8.5 / 24 = 0.3542 of the range.
Fences at 1.5 times 8.5 are not computed on this page. The count is 7. Quartile positions would be reported on the quartile page.
| Interquartile range | 8.5 |
|---|---|
| IQR as a share of the range | 0.3542 |
| Q1 | 16 |
| Q3 | 24.5 |
| Range | 24 |
Replacing 25 with 35 moves Q3 from 23.5 to 24.5, so the interquartile range rises from 7.5 to 8.5. The full range becomes 24, and the share falls from 0.4688 to 0.3542 because the extremes moved more than the quartiles. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
25 was replaced by 35. The headline result went from 7.5 to 8.5.
The nudged inputs
The ordered values are 11, 14, 18, 19, 22, 27, 35. Hyndman-Fan type 7 gives Q1 = 16 and Q3 = 24.5. The interquartile range is 24.5 - 16 = 8.5.
That difference is the headline. The minimum and the maximum do not enter it. The full range is 35 - 11 = 24. The IQR is 8.5 / 24 = 0.3542 of the range.
Fences at 1.5 times 8.5 are not computed on this page. The count is 7. Quartile positions would be reported on the quartile page.
| Interquartile range | 8.5 |
|---|---|
| IQR as a share of the range | 0.3542 |
| Q1 | 16 |
| Q3 | 24.5 |
| Range | 24 |
How to read the result
An IQR of 7.5 is in the unit of the data. The share 0.4688 has no unit. It says the middle half is a bit under half of the distance from the minimum to the maximum. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- Type 7 quartiles are the ones you want subtracted.
- The values are on one scale.
- A share is reported only when the maximum and minimum differ.
- You want the length of the central half, not the fences around it.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Subtracting the median from the maximum and calling that an IQR.
- Using a range of zero as a denominator.
- Building outlier fences on this page and thinking they were part of the headline.
- Comparing an IQR in dollars with an IQR in percentages.
Where a different calculator is the honest choice
Use the quartile calculator when you need the positions of Q1 and Q3. Use the lower and upper fence calculator when 1.5 times this gap should become a cutoff. Use the range calculator when only the extremes matter. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Quartile Calculator, Lower and Upper Fence Calculator, Range Calculator.
Check yourself
Values are 11, 14, 18, 19, 22, 25, and 27. What is the interquartile range?
The interquartile range is 7.5, which is 0.4688 of the range.
Using the result in a sentence
Income spreads, test-score boxes, and any summary that should ignore the outer quarter on each side start from this subtraction. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The IQR does not say where the central half sits, only how wide it is. Two batches can share an IQR of 7.5 and have very different medians. Report Q1 and Q3 beside it when location matters. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Interquartile Range Calculator commit to?
This page uses Interquartile range: IQR = Q3 - Q1. Share of the range = IQR / (max - min), when max is not equal to min.. The central half of the ordered sample is the interval from Q1 to Q3. Its length ignores the outer quarters, which is why a change that stays inside a tail can leave the IQR still until it moves a quartile. The share compares that length with the full extremes. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Numbers, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the interquartile range is 7.5. An IQR of 7.5 is in the unit of the data. The share 0.4688 has no unit. It says the middle half is a bit under half of the distance from the minimum to the maximum. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Subtracting the median from the maximum and calling that an IQR. The IQR does not say where the central half sits, only how wide it is. Two batches can share an IQR of 7.5 and have very different medians. Report Q1 and Q3 beside it when location matters. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the quartile calculator when you need the positions of Q1 and Q3. Use the lower and upper fence calculator when 1.5 times this gap should become a cutoff. Use the range calculator when only the extremes matter. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The interquartile range is 7.5, which is 0.4688 of the range. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.