Mean, Weighted & Running Average
Last reviewed: May 2026
Calculate arithmetic mean, weighted average, median, and mode for any data set. The average is the most commonly used summary statistic, but choosing the right type of average matters: mean for symmetric data, median for skewed data, and weighted average when items have different importance.1
| Type | Formula | Best For |
|---|---|---|
| Arithmetic mean | Sum / count | Symmetric data, general use |
| Weighted mean | Σ(value×weight) / Σweights | Grades, portfolios, surveys |
| Median | Middle value (sorted) | Skewed data, incomes, prices |
| Mode | Most frequent value | Categorical data |
| Geometric mean | (product)^(1/n) | Growth rates, returns |
| Average Type | Formula | Best For | Example |
|---|---|---|---|
| Arithmetic mean | Sum ÷ count | Symmetric data without outliers | Test scores: (85+90+78+92+88)/5 = 86.6 |
| Median | Middle value when sorted | Skewed data, income, home prices | Incomes: 30K, 35K, 40K, 45K, 200K → median = 40K |
| Mode | Most frequent value | Categorical data, sizes, ratings | Shoe sizes sold: 8, 9, 9, 10, 9 → mode = 9 |
| Weighted mean | Σ(value × weight) ÷ Σweights | Different importance levels | GPA: (A×4 + B×3 + A×3) / (4+3+3) = 3.4 |
| Geometric mean | ⁿ√(x₁ × x₂ × ... × xₙ) | Growth rates, ratios, percentages | Returns: 10%, 20%, −5% → geom = 7.9% |
| Harmonic mean | n ÷ Σ(1/xᵢ) | Rates, speeds, P/E ratios | Speeds: 30, 60 mph → harm = 40 mph |
The arithmetic mean is the default "average" but is frequently the wrong choice. Income data, home prices, and any dataset with extreme outliers require the median because a few very high values pull the mean far above the typical experience. A neighborhood where 9 homes sell for $300,000 and one sells for $3,000,000 has a mean price of $570,000 but a median of $300,000 — the median reflects the typical home far more accurately.
The arithmetic mean is sensitive to outliers — extreme values pull it away from the center of the data. Consider annual salaries at a small company: $40,000; $45,000; $50,000; $52,000; $55,000; $60,000; $500,000 (the CEO). The mean salary is $114,571 — but no non-CEO employee earns anywhere near that. The median ($52,000) represents the typical employee experience. This limitation makes the mean particularly misleading for income distributions, housing markets, medical costs, and any data with a long right tail. News headlines reporting "average" figures for these domains almost always use the mean, which overstates what most people actually experience.
Weighted averages appear more often than most people realize. Your GPA is a weighted average: courses worth more credit hours count more heavily. Investment portfolio returns are weighted by the amount invested in each asset — a 20% return on $1,000 and a 5% return on $9,000 produces a weighted average return of 6.5%, not the simple average of 12.5%. Teacher grade calculations weight categories differently: exams 40%, homework 30%, participation 15%, final project 15%. Baseball batting averages weight at-bats equally but contrast with on-base percentage, which weights walks alongside hits. Whenever components contribute unequally to a total, the weighted average provides the correct computation.
A moving average smooths noisy data to reveal underlying trends by averaging a fixed window of recent data points. Stock analysts use 50-day and 200-day moving averages to identify market trends — when the 50-day crosses above the 200-day (a "golden cross"), it signals bullish momentum. Weight trackers use 7-day or 14-day moving averages to see past daily fluctuations of 1–3 pounds caused by water retention, meal timing, and other temporary factors. Businesses use moving averages for sales forecasting, inventory planning, and quality control. The window size involves a tradeoff: shorter windows (5–10 days) respond quickly to changes but remain noisy; longer windows (50–200 days) are smoother but slower to reflect real shifts. Exponential moving averages weight recent data more heavily, providing a compromise between responsiveness and smoothness.
When averaging growth rates, percentages, or ratios, the geometric mean produces the correct result while the arithmetic mean can be significantly misleading. If an investment returns +50% in year 1 and −50% in year 2, the arithmetic mean return is 0% — suggesting you broke even. But starting with $100: year 1 gives $150, year 2 gives $75. You lost 25%. The geometric mean correctly identifies the average annual return as −13.4%. This matters enormously in financial planning: using arithmetic mean returns overstates expected outcomes for volatile investments. The geometric mean formula for returns: ((1+r₁) × (1+r₂) × ... × (1+rₙ))^(1/n) − 1. It always produces a value equal to or less than the arithmetic mean, with the gap increasing as volatility rises.
Averaging averages produces incorrect results when the groups have different sizes. If Class A (20 students) has an average score of 85 and Class B (30 students) has an average score of 90, the combined average is not (85+90)/2 = 87.5. The correct calculation weights by class size: (20×85 + 30×90) / (20+30) = (1,700+2,700)/50 = 88.0. This error, called Simpson's Paradox in its extreme form, can actually reverse conclusions when subgroup sizes differ dramatically. A hospital with a higher survival rate for both mild and severe cases can still have a lower overall survival rate than a competing hospital if it treats a much higher proportion of severe cases. Always verify that averages of averages account for group sizes before drawing conclusions.
When you want the simplicity of the mean but need protection from outliers, trimmed and Winsorized means offer robust alternatives. A 10% trimmed mean removes the top and bottom 10% of values before calculating the mean — Olympic figure skating uses this approach by dropping the highest and lowest judges' scores. A Winsorized mean replaces extreme values with the nearest non-extreme values rather than removing them. Both methods reduce outlier influence while preserving more data than simply switching to the median. The trimmed mean is especially useful for survey data where a few extreme responses (joke answers, misunderstandings) can distort results. Use our Standard Deviation Calculator to assess data variability alongside average calculations.
The word "average" in everyday language usually means "typical" — but statistically, the mean, median, and mode can all claim to represent the average, and they often disagree. For symmetric distributions (test scores, heights, many measurement errors), all three are nearly identical, so the mean works well. For right-skewed distributions (income, wealth, home prices, hospital stays), the median better represents the typical case. For categorical or discrete data (most popular color, most common family size), the mode is most informative. Reporting multiple measures is the most honest approach: "median household income is $75,000; the mean is $105,000" tells you both the typical experience and the influence of high earners. When someone reports just one number as "average," ask which type — the answer often reveals whether the statistic supports the narrative it is being used to tell.
Running averages update continuously as new data arrives, useful for real-time monitoring. A student who scored 78, 85, 92, and 88 on four exams has a running average of 78 → 81.5 → 85 → 85.75. To achieve a target average of 90 for the course, they need the fifth exam score to satisfy: (78+85+92+88+x)/5 = 90, giving x = 107 — impossible, which means the target is out of reach and recalibrating expectations is necessary. This reverse-calculation technique helps students determine exactly what scores they need on remaining assignments to achieve grade targets, or helps salespeople determine what performance is needed in remaining months to hit annual quotas. The formula to find the needed value: required = (target × total count) − current sum.
Averaging values that use different scales or units produces meaningless results without normalization. You cannot average a temperature (72°F), a humidity (65%), and a wind speed (12 mph) to get a "weather score" because the units are incomparable. Z-score standardization converts each measurement to its number of standard deviations from the mean, putting all values on a common scale. Alternatively, percentile ranks convert each value to its position within a reference population. Once standardized, averaging becomes meaningful — a composite fitness score might average standardized VO2 max, strength, flexibility, and body composition measurements. This principle applies broadly: college admissions combine standardized test scores, GPAs (already a weighted average), and subjective ratings through normalization frameworks.
Scientists report experimental results as mean ± standard error, indicating both the central tendency and the uncertainty of the measurement. Manufacturers use process averages to set quality control limits — the average weight of cereal boxes determines the center line on a control chart, with ±3 standard deviations marking acceptable variation boundaries. Meteorologists forecast temperatures as averages of multiple model outputs, weighting each model by historical accuracy. Actuaries calculate average life expectancy using weighted mortality tables that account for age, sex, and health factors — these averages directly determine insurance premiums and pension calculations. In every field, the choice of which average to compute and how to weight it shapes the conclusions drawn from data. Our Percentage Calculator can help with the underlying percentage computations used in many averaging scenarios.
→ Use median for skewed data. Income, home prices, wait times.
→ Weight matters for grades. A 40% final counts more than a 10% quiz.
→ Don't average percentages blindly. Different group sizes need weighted average.
→ Check for outliers. One extreme value can distort the mean significantly.
See also: Statistics · Grade · Percentage · Standard Deviation