Mean Median Mode Guide 2026: Averages Explained Plus Examples
Table of Contents
The Three Averages
Dataset: 4, 8, 15, 16, 23, 42
- Mean: 108 / 6 = 18
- Median: (15 + 16) / 2 = 15.5
- Mode: none (all unique)
Bimodal data has two modes, and that is information, not a defect. Shoe sizes 7, 7, 8, 9, 9 have modes 7 and 9 - two customer groups with different needs. The mean, 8.2, describes a size nobody bought. Whenever the mode splits in two, look for two underlying populations: the average is hiding a segmentation the business should be serving separately.
The trimmed mean borrows the median's outlier resistance without abandoning the mean entirely. For 4, 8, 15, 16, 23, 42, drop the extremes (4 and 42) and average the rest: (8 + 15 + 16 + 23)/4 = 15.5. Olympic scoring trims the highest and lowest judges for exactly this reason. A 10% trim each side is the standard compromise between robustness and using all the data.
When Each Misleads
Salaries: Nine people earn $50K, the boss earns $500K. Mean = $95K (misleading); median = $50K (truthful).
| Use | When |
|---|---|
| Mean | Symmetric data, no outliers (test scores) |
| Median | Skewed data (income, home prices) |
| Mode | Categories (most popular size, top vote-getter) |
Real estate: Agents quote median home prices because a few mansions skew the mean upward.
Never take the simple average of two averages. Class A: 20 students averaging 80; Class B: 10 students averaging 90. The combined average is (1600 + 900)/30 = 83.3, not 85. Unequal group sizes demand weighting by headcount - the naive average of 80 and 90 silently pretends the classes are the same size. This error appears in combined department metrics, merged survey results, and "average of averages" dashboards everywhere.
The "average customer" can be a fiction. A mean household size of 1.9 children describes zero actual families, yet product teams design for it. For discrete choices - how many seats, what size pack - the mode (2 children) is the actionable number. Match the average to the decision: means for totals, medians for typicals, modes for categories.
Weighted Mean
Course grade: Homework 90 (20%), midterm 80 (30%), final 85 (50%): (90x0.2 + 80x0.3 + 85x0.5) = 18 + 24 + 42.5 = 84.5.
Portfolio return: 60% stocks (+10%), 40% bonds (+3%): 0.6x10 + 0.4x3 = 7.2%.
GPA is the weighted mean everyone has computed. Grades of A (4.0) in 3 credits, B (3.0) in 4 credits, C (2.0) in 3 credits: (4x3 + 3x4 + 2x3)/10 = (12 + 12 + 6)/10 = 3.0. Credit hours are the weights, which is why a 4-credit B moves the needle more than a 3-credit A. Any time some observations count more than others, the weights belong in the denominator too.
Expected value is a weighted mean with probabilities as weights. A fair die averages (1+2+3+4+5+6)/6 = 3.5. A lottery ticket with a 50% chance at $100 and 50% at $0 has expected value $50. Casinos, insurers, and warranty companies all price from expected value - the house edge is just a weighted mean tilted in their favor, computed over millions of trials.
Range and Outliers
Dataset: 4, 8, 15, 16, 23, 42. Range = 38. Q1 = 8, Q3 = 23, IQR = 15.
Outlier rule: Values beyond Q1 - 1.5xIQR or Q3 + 1.5xIQR. Here: below -14.5 or above 45.5 - 42 is unusual but not a formal outlier.
The 1.5 x IQR rule turns "that one looks wrong" into arithmetic. For 12, 15, 18, 22, 25, 30, 31, 35, 100: Q1 = 16.5, Q3 = 33, IQR = 16.5. The upper fence is 33 + 1.5 x 16.5 = 57.75, so 100 is a formal outlier. Box plots draw exactly these fences - points beyond them get plotted as lone dots, which is why the rule matters for reading any published chart.
Do not delete outliers just because they are inconvenient. A $2M mansion among $300K homes is real data, not a measurement error - removing it falsifies the market. Report the median and IQR instead, which resist the distortion honestly. Delete only clear errors, like a 900-inch human height; for genuine extremes, change the statistic, not the dataset.
Frequently Asked Questions
Mean is the arithmetic average (sum/count); median is the middle value when sorted. Median resists outliers; mean uses all data.
For skewed data like income or home prices, where outliers distort the mean. Use mean for symmetric data like test scores.
Yes - bimodal (two modes) or multimodal datasets. Some datasets have no mode if all values are unique.
An average where values have different importance: sum(value x weight)/sum(weights). Used for grades, portfolio returns, GPA.
Strongly - one extreme value can drag the mean far from typical values, which is why median is preferred for skewed data.