Descriptive Statistics
Lower and Upper Fence Calculator
The inner fences sit one and a half interquartile ranges past the quartiles, and the extreme fences sit three interquartile ranges past them.
Worksheet
Press Calculate to see the result.
The inner fences sit one and a half interquartile ranges past the quartiles, and the extreme fences sit three interquartile ranges past them.
What this calculator answers
The batch 4, 7, 8, 9, 10, 11, 12, and 40 needs a rule that says whether 40 is just the maximum or a point past the inner fence. Quartiles are Hyndman-Fan type 7. The interquartile range is Q3 minus Q1. The lower fence is Q1 minus 1.5 times that range, and the upper fence is Q3 plus 1.5 times that range. The extreme fences use 3 instead of 1.5. Every pasted value beyond the inner fences is listed by name. This page reports those cutoffs. It does not draw the box.
The headline pair is the lower and upper inner fence. Extreme fences use three interquartile ranges, and the outlier list contains every typed value outside the inner fences. 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
Tukey inner and extreme fences uses this relationship.
lower = Q1 - 1.5 * IQR. upper = Q3 + 1.5 * IQR. Extreme fences replace 1.5 with 3.
- Q1. Lower quartile, type 7.
- Q3. Upper quartile, type 7.
- IQR. Q3 minus Q1.
- fence. Quartile plus or minus 1.5 or 3 times the IQR.
The fences are anchored to the central half, not to the mean. A point is an outlier here when it is strictly below the lower inner fence or strictly above the upper inner fence. The extreme fences use the same quartiles with a wider multiple. 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 4, 7, 8, 9, 10, 11, 12, 40. Hyndman-Fan type 7 gives Q1 = 7.75 and Q3 = 11.25. The interquartile range is 11.25 - 7.75 = 3.5.
The lower fence is 7.75 - 1.5 * 3.5 = 2.5. The upper fence is 11.25 + 1.5 * 3.5 = 16.5. The extreme fences use 3 instead of 1.5: −2.75 and 21.75.
Points strictly outside 2.5 to 16.5 are outliers: 40. Points strictly outside the extreme fences are: 40.
| Lower fence | 2.5 |
|---|---|
| Upper fence | 16.5 |
| Lower extreme fence | −2.75 |
| Upper extreme fence | 21.75 |
| Outliers beyond the inner fences | 40 |
| Extreme outliers | 40 |
A second situation
The point 40 is removed
The ordered values are 4, 7, 8, 9, 10, 11, 12. Hyndman-Fan type 7 gives Q1 = 7.5 and Q3 = 10.5. The interquartile range is 10.5 - 7.5 = 3.
The lower fence is 7.5 - 1.5 * 3 = 3. The upper fence is 10.5 + 1.5 * 3 = 15. The extreme fences use 3 instead of 1.5: −1.5 and 19.5.
Points strictly outside 3 to 15 are outliers: None. Points strictly outside the extreme fences are: None.
| Lower fence | 3 |
|---|---|
| Upper fence | 15 |
| Lower extreme fence | −1.5 |
| Upper extreme fence | 19.5 |
| Outliers beyond the inner fences | None |
| Extreme outliers | None |
Removing 40 recalculates the quartiles on the remaining seven values. The inner fences become 3 and 15, and no remaining point sits outside them, so the outlier list is empty. 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
40 was deleted, leaving the seven tighter values. The headline result went from 2.5 to 3.
The nudged inputs
The ordered values are 4, 7, 8, 9, 10, 11, 12. Hyndman-Fan type 7 gives Q1 = 7.5 and Q3 = 10.5. The interquartile range is 10.5 - 7.5 = 3.
The lower fence is 7.5 - 1.5 * 3 = 3. The upper fence is 10.5 + 1.5 * 3 = 15. The extreme fences use 3 instead of 1.5: −1.5 and 19.5.
Points strictly outside 3 to 15 are outliers: None. Points strictly outside the extreme fences are: None.
| Lower fence | 3 |
|---|---|
| Upper fence | 15 |
| Lower extreme fence | −1.5 |
| Upper extreme fence | 19.5 |
| Outliers beyond the inner fences | None |
| Extreme outliers | None |
How to read the result
A lower fence of 2.5 and an upper fence of 16.5 are cutoffs, not data values. The value 40 is past 16.5, and it is also past the extreme upper fence, so it is an extreme outlier on this rule. 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
- Hyndman-Fan type 7 is the quartile definition.
- The 1.5 and 3 multiples are the Tukey conventions you want.
- Points that land exactly on a fence are not outliers.
- The list you paste is the list that should be judged.
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
- Using the mean and the standard deviation to build these fences.
- Calling Q1 and Q3 the fences.
- Listing the maximum as an outlier when it still sits inside the upper fence.
- Drawing a box and forgetting to say which points were excluded from the whisker.
Where a different calculator is the honest choice
Use the box plot calculator when the fences should be drawn. Use the quartile calculator when you need positions rather than cutoffs. Use the interquartile range calculator when Q3 minus Q1 is the only figure. 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: Box Plot Calculator, Interquartile Range Calculator, Quartile Calculator.
Check yourself
Values are 4, 7, 8, 9, 10, 11, 12, and 40. What are the inner fences, and which point is an outlier?
The lower fence is 2.5 and the upper fence is 16.5, and 40 is the outlier.
Using the result in a sentence
Screening a small batch before a box plot, and checking a published upper whisker against the raw values, are the jobs of the fence arithmetic. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The 1.5 rule is a convention, not a law of which points are wrong. A heavy-tailed but genuine distribution will mark many real observations as outliers. Do not delete them only because they failed this screen. 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 Lower and Upper Fence Calculator commit to?
This page uses Tukey inner and extreme fences: lower = Q1 - 1.5 * IQR. upper = Q3 + 1.5 * IQR. Extreme fences replace 1.5 with 3.. The fences are anchored to the central half, not to the mean. A point is an outlier here when it is strictly below the lower inner fence or strictly above the upper inner fence. The extreme fences use the same quartiles with a wider multiple. 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 lower fence is 2.5. A lower fence of 2.5 and an upper fence of 16.5 are cutoffs, not data values. The value 40 is past 16.5, and it is also past the extreme upper fence, so it is an extreme outlier on this rule. 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?
Using the mean and the standard deviation to build these fences. The 1.5 rule is a convention, not a law of which points are wrong. A heavy-tailed but genuine distribution will mark many real observations as outliers. Do not delete them only because they failed this screen. 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 box plot calculator when the fences should be drawn. Use the quartile calculator when you need positions rather than cutoffs. Use the interquartile range calculator when Q3 minus Q1 is the only figure. 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 lower fence is 2.5 and the upper fence is 16.5, and 40 is the outlier. 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.