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Home›Probability and statistics›Averages

Outliers and the 1.5 × IQR rule

In OpenStax's evening class, the 1.5 × IQR rule marks the scores 45 and 25.5 as outliers because both are below 61.5.

Is a score of 45 unusual in this class?

Start with the lab below. It holds the 22 evening class scores from OpenStax, with the outlier fences switched on. Before you touch anything, guess. Is the 45 an outlier? What about the 25.5? Look for the red rings, then drag the 45 upward slowly and stop the moment its ring disappears. Note where that happens. The rest of this page explains the number you just found.

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The evening class with Spread and Outlier fences on. The red dashed lines are the fences; outliers get a red ring. Drag the 45 up past 61.5 and its ring disappears. Then drag a score near 98 to the far right and see whether the upper fence catches it.

An outlier is a value unusually far from the rest of the data. It might be a typing mistake or a broken sensor, but it might also be a real and important case: the one house that is a mansion, the one patient who reacts differently to a drug. OpenStax advises looking at outliers closely rather than deleting them automatically. To look, you first need a fair way to decide what counts as "unusually far".

Two fences, each 1.5 IQRs beyond a quartile

The usual rule builds two fences from the quartiles. The lower fence is Q1 − 1.5 × IQR and the upper fence is Q3 + 1.5 × IQR. A value below the lower fence or above the upper one is a potential outlier. The rule uses quartiles on purpose. Outliers cannot drag the fences outward the way they drag a mean or a standard deviation.

Back to the lab. The evening class has Q1 = 78, Q3 = 89 and IQR = 11. The fences are 78 − 1.5 × 11 = 61.5 and 89 + 1.5 × 11 = 105.5, and 61.5 is where the 45 lost its ring. The scores 45 and 25.5 are below 61.5, so both are outliers; no score is above 105.5. The day class, with IQR 26.5, has fences at 16.25 and 122.25, so even its 32s are inside. For the 11 OpenStax salaries, Q1 = 40.5, Q3 = 69 and IQR = 28.5. That gives an upper fence of 69 + 42.75 = 111.75 thousand dollars, and the $120,000 salary is a potential outlier. For the 13 home prices the upper fence is $1,159,375. The $5,500,000 house is far outside. The $1,095,000 house is just inside.

Box plot of the evening class scores. The box runs from Q1 = 78 to Q3 = 89 with the median 81. Red dashed lines are the fences at 61.5 and 105.5; the circled scores 25.5 and 45 lie outside, and the whiskers stop at the most extreme values inside the fences, 65 and 98.

Two low scores cost the mean more than 4 points

Compare the evening class with and without its two outliers. With all 22 scores the mean is about 79.05, the median 81 and the standard deviation about 16.8. Without 45 and 25.5, the 20 remaining scores have a mean of about 83.43, a median of 82 and a standard deviation of about 9.0. The mean rose by more than 4 points and the standard deviation almost halved, while the median moved by just 1. Salaries tell the same story: removing the $120,000 salary drops the mean from about $58,682 to $52,550, and the median stays at $54,000.

So what should you do with an outlier? First check whether it is a mistake, such as 25.5 typed instead of 85.5. If it is an error and you can fix it, fix it. If it is real, keep it and say so, and consider reporting the resistant measures, the median and IQR, alongside the mean and standard deviation. Deleting real values because they are inconvenient changes the story the data tell. The rule is a flag, not a verdict. "Potential outlier" means "look here".

OutlierA value unusually far from the rest of the data
Lower fenceQ1 − 1.5 × IQR
Upper fenceQ3 + 1.5 × IQR
First step with an outlierCheck whether it is an error; do not delete it automatically
  1. Evening class: Q1 = 78, IQR = 11. Lower fence?
  2. Evening class: Q3 = 89, IQR = 11. Upper fence?
  3. How many evening class scores are outliers?
  4. Salaries: Q3 = 69, IQR = 28.5 (thousand). Upper fence?
  5. Home prices: Q3 = 649,000, IQR = 340,250. Upper fence in dollars?
  6. Is the $1,095,000 house an outlier by the 1.5 × IQR rule?
    • Yes
    • No, it is below the upper fence
  7. Data 12, 15, 11, 14, 13, 40. Upper fence (Q1 = 12, Q3 = 15)?
  8. Evening class without its two outliers: mean to 2 decimal places?
  9. Removing the two low outliers from the evening class changes which value least?
    • Mean
    • Standard deviation
    • Median
    • Range
  10. You find a score of 25.5 that might be a typing error for 85.5. What first?
    • Delete it
    • Check the original record
    • Replace it with the mean
    • Ignore it
Common mistakes
  • Building the fences from the median: 81 − 16.5 = 64.5 and 81 + 16.5 = 97.5, which flags the three 98s. The fences start from the quartiles, not the median. From Q1 = 78 and Q3 = 89 they are 61.5 and 105.5, and the 98s are well inside.
  • Leaving out the 1.5: 78 − 11 = 67 and 89 + 11 = 100, which makes the 65 an outlier too. The rule multiplies the IQR by 1.5 before subtracting or adding. 65 is above the real lower fence, 61.5, so only 45 and 25.5 are outside.
  • Reporting only the mean salary, about $58,682, as what a typical employee earns. One $120,000 salary lifts that mean; without it the mean is $52,550. When the data hold a potential outlier, give the median, $54,000, beside the mean.
Teacher's noteRun the lab before showing the formula, and let students find 61.5 by dragging. The number then has somewhere to come from. Two errors are worth checking on paper: fences measured from the median, and fences built without the 1.5. Spend a moment on the word "potential". The rule only flags values. Whether to keep one is a judgement about the data, and the 25.5 that might be a typo for 85.5 is a good example to argue about.
Key points
  • Fences: Q1 − 1.5 × IQR and Q3 + 1.5 × IQR; values outside are potential outliers.
  • Outliers move the mean and standard deviation a lot, the median and IQR little.
  • Check outliers for errors; do not delete real values to tidy the data.
Sources for this course (7)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 5~6학년군 [6수04-01] 평균의 의미, 고려 사항 「평균을 구하는 방법뿐만 아니라 그 의미를 직관적으로 파악하게 한다」; 중학교 1~3학년군 [9수04-01] 중앙값·최빈값과 대푯값 고르기, [9수04-07] 분산과 표준편차, [9수04-08] 상자그림으로 분포 비교, 용어 변량·대푯값·중앙값·최빈값·도수분포표·산포도·편차·분산·표준편차·사분위수·상자그림; 고등학교 확률과 통계 용어 표본평균·모평균·표본분산·표본표준편차와 기호 X̄, S², S, N(m, σ²); 실용 통계 [12실통02-04] 대푯값과 산포도 고르기)
  • Middle-school variance as the mean of the squared deviations: EBS Math 중3 「분산과 표준편차」 (www.ebsmath.co.kr/resource/rscView?cate=123500&cate2=123510&cate3=123511&rscTpDscd=RTP10&grdCd=MGRD03&sno=24525&type=S, read 2026-10-19)
  • Textbook: OpenStax, Introductory Statistics 2e by Barbara Illowsky and Susan Dean, 2.3 measures of location, quartiles as medians of the lower and upper halves, IQR, 1.5 × IQR outlier rule, 13 real estate prices, 11 salaries, day and evening class test scores (openstax.org/books/introductory-statistics-2e/pages/2-3-measures-of-the-location-of-the-data)
  • 2.5 measures of the center (.../pages/2-5-measures-of-the-center-of-the-data)
  • 2.6 skewness and the mean, median and mode, the three ten- and sixteen-value data sets (.../pages/2-6-skewness-and-the-mean-median-and-mode)
  • 2.7 measures of the spread, sample standard deviation with n − 1 (.../pages/2-7-measures-of-the-spread-of-the-data)
  • Trimmed mean only: Federal Reserve Bank of Dallas, Trimmed Mean PCE inflation rate (dallasfed.org/research/pce and /research/pce/descr)

Checked: 2026-10-17