Choosing the right average
In OpenStax's 13 home prices, one $5,500,000 house lifts the mean to about $865,015, while the median stays at $488,800.
Nine bus waits: how many were longer than the mean?
Try this before reading on. A student times nine waits at the same bus stop, in minutes: 3, 4, 4, 5, 5, 6, 6, 7 and 40. The numbers are made up, and the 40 was the day a bus broke down. What is the mean wait, and on how many days was the wait longer than the mean? The total is 80 minutes, so the mean is 80 ÷ 9, about 8.9 minutes. Only one wait, the 40, is longer than that. Eight of the nine were 7 minutes or less. The median, the 5th value, is 5 minutes. A friend asks how long that bus usually takes. Which number do you give?
News reports about house prices and incomes nearly always quote the median. School reports quote mean marks. That is not a matter of taste. Each average reacts differently to the shape of the data, and the right one depends on what you want to describe.
One $5,500,000 house moves the mean by almost $400,000
Homes show the same thing with bigger numbers. OpenStax lists 13 real estate prices from $114,950 to $5,500,000. Sorted, the 7th price is $488,800, so that is the median. The mean is the total, $11,245,200, divided by 13: about $865,015. Only two of the 13 houses cost more than that mean. Take out the $5,500,000 house and the mean of the other 12 drops to about $478,767. Their median, $483,900, barely changes. One value at the far end moved the mean by almost $400,000 and the median by about $5,000.
Statisticians say the median is resistant and the mean is not. The mean uses the size of every value, so it feels the pull of extreme ones. The median uses only the order. How far out the extremes sit makes no difference to it. When data are skewed, with a long tail on one side, the mean is dragged towards the tail: above the median for a right tail, below it for a left tail. The bus waits have a right tail, which is why their mean of about 8.9 minutes sits above the 5-minute median.
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Totals call for the mean; long tails call for the median
Use the mean when the data are roughly symmetric with no extreme values, and when the total matters. A teacher checking whether a class did better overall needs the mean. So does a shop planning total sales from average spending, because mean × n gives back the total. Use the median for skewed data such as incomes, house prices, waiting times or the age at which people buy their first home, where a few very large values would make the mean describe almost nobody. Use the mode for categories, or when the single most common value is what you need. A shoe shop orders the most common size, not the mean size.
Reporting both is often the honest choice. If a company's mean salary is far above its median, that gap is news in itself. It says a few people earn much more than everyone else. A trimmed mean is a middle path. It drops a fixed share of the highest and lowest values before averaging, and it is used in some sports judging and in economic statistics.
Be careful when an advertisement or a report just says "average". The word can mean the mean, the median or the mode, and a seller may pick whichever looks best. "The average customer saves $300" could be a mean pushed up by a few big savers, while the median customer saves much less. If the kind of average is not stated, it is fair to ask which one it is and how many values it came from.
| Data | Best average | Why |
|---|---|---|
| Test scores, heights | Mean | Roughly symmetric; totals matter |
| Incomes, house prices | Median | Long right tail |
| Favourite colour, shoe size to stock | Mode | Categories or the most common value |
| Resistant | Hardly affected by extreme values (the median is) |
| Skewed right | Long tail to the right; mean > median |
| Skewed left | Long tail to the left; mean < median |
| Trimmed mean | Mean after dropping a share of the highest and lowest values |
- Median of the 13 OpenStax home prices, in dollars?
- Mean of the 13 home prices, to the nearest dollar?
- How many of the 13 houses cost more than the mean?
- Without the $5,500,000 house, what is the mean of the other 12, to the nearest dollar?
- For data with a long right tail, usually...
- mean < median
- mean = median
- mean > median
- there is no median
- A billionaire walks into a café of 30 people. Which changes a lot?
- The median wealth
- The mean wealth
- Neither
- Both equally
- A shoe shop wants to know which size to order most of. Which average?
- Mean
- Median
- Mode
- Range
- Why does a teacher comparing total class performance use the mean?
- It ignores weak students
- Mean × n gives the total, and scores are roughly symmetric
- It is always larger
- It is resistant
- Mean salary $90,000, median $50,000. What does the gap suggest?
- Salaries are symmetric
- A few very high salaries
- A few very low salaries
- A mistake in the data
- Median of the 12 home prices without the mansion, in dollars?
- Coding categories as numbers and averaging the codes: walk = 1, bus = 2, car = 3, answers 1, 1, 3, 3, so the "average" student takes the bus. The codes are labels, not amounts. Nobody in these data takes the bus. Report the counts or the mode; here there are two modes, walk and car.
- Believing the median never changes when the data change. It ignores how far the end values go, not every change. In the bus waits 3, 4, 4, 5, 5, 6, 6, 7, 40, turning the 40 into 400 leaves the median at 5. Turning one 4 into 9 moves a value across the middle, and the median becomes 6.
- Using the median to rebuild a total: pocket money of $20, $20, $30, $30 and $100 has median $30, so five friends must have 5 × 30 = $150 between them. Only the mean turns back into the total. The real total is $200, which is the mean, $40, times 5.
- The median is resistant; the mean follows extreme values.
- Skewed right: mean > median. Skewed left: mean < median.
- Mean for symmetric data and totals, median for skewed data, mode for categories.
- When mean and median differ a lot, report both.
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