Average Calculator

Mean = sum divided by count. The arithmetic average. Sensitive to outliers. Median = the middle value when numbers are sorted. If even count, the average of the two middle values. Robust to outliers. Mode = the most frequently occurring value. A dataset can have no mode, one mode, or multiple modes. Example: {1, 2, 2, 3, 100}. Mean = 21.6. Median = 2. Mode = 2. Median and mode are far more representative of this dataset because the outlier (100) pulls the mean far from the typical values.

Use median when data has outliers or is skewed. Median household income is preferred over mean income because a small number of very high earners dramatically inflate the mean. Home prices, hospital stay lengths, salary data, and web page load times are all typically reported as medians. Use mean when data is roughly symmetrical with no extreme outliers, such as heights, temperatures, or test scores in a normal distribution.

Standard deviation (σ) measures how spread out numbers are around the mean. Steps: (1) Find the mean. (2) Subtract the mean from each number and square the result. (3) Average those squared differences (this gives variance σ²). (4) Take the square root of the variance. This calculator uses population standard deviation (dividing by n). Sample standard deviation (used when the data is a sample from a larger population) divides by n−1 instead. A low standard deviation means values cluster close to the mean; a high one means they are spread out.

The geometric mean is the nth root of the product of n numbers. For {2, 8}: geometric mean = √(2×8) = √16 = 4. Unlike the arithmetic mean, the geometric mean is ideal for data that grows multiplicatively: investment returns, population growth rates, and ratios. Example: a stock grows 20% one year and 80% the next. Arithmetic mean = 50% (misleading). Geometric mean = √(1.20×1.80) − 1 ≈ 46.97% (actual compound growth rate). For investment data, always use geometric mean to find the true average rate of return.

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals: H = n / (1/x₁ + 1/x₂ + ... + 1/xₙ). It is used when averaging rates. Example: you drive 60 km/h for 100 km, then 120 km/h for 100 km. What is the average speed? Arithmetic mean = (60+120)/2 = 90 km/h (wrong). Harmonic mean = 2/(1/60+1/120) = 2/(3/120) = 80 km/h (correct). The harmonic mean is always the smallest of the three Pythagorean means (harmonic ≤ geometric ≤ arithmetic) when all values are positive.

Range = maximum − minimum. It is the simplest measure of spread (dispersion). A large range means the data spans a wide set of values; a small range means values are clustered close together. Example: test scores {70, 72, 73, 74, 99}. Range = 99−70 = 29. But standard deviation = 10.6, showing the data is less spread than the range suggests due to the outlier at 99. Range is easy to calculate but sensitive to outliers. For a more robust measure of spread, use standard deviation or interquartile range (IQR = Q3−Q1).

Variance (σ²) is the average of the squared differences from the mean. Standard deviation (σ) is simply the square root of the variance. Variance is in squared units (e.g., cm² if the data is in cm), which makes it hard to interpret directly. Standard deviation brings it back to the original units, making it more intuitive. Both measure the same thing: how spread out the data is. Variance is preferred in statistical formulas and probability theory because it adds up cleanly (variance of a sum of independent variables = sum of variances).

Yes. A dataset with two modes is bimodal; three or more modes is multimodal; no repeated values means no mode. Example: {1, 2, 2, 3, 3, 4} has two modes: 2 and 3. This is bimodal. In practice, bimodal distributions often indicate that two different groups are mixed together. Example: heights of adult humans are bimodal because men and women have different average heights. If all values occur equally often, technically every value is a mode, but this is usually reported as "no mode."

The mean appears in almost every field: Academic grading: GPA is a weighted mean of course grades. Weather: average temperature, rainfall. Finance: average daily return, moving averages in stock charts. Sports: batting average, points per game. Science: taking multiple measurements and averaging to reduce error. Quality control: monitoring average defect rates. Polling: average approval ratings. The key limitation is sensitivity to outliers — a single extreme value shifts the mean significantly, which is why median is often reported alongside mean for skewed data.

A weighted mean gives different importance to each value. Weighted mean = Σ(wᵢ × xᵢ) / Σwᵢ, where wᵢ is the weight of each value xᵢ. Example: a course with three exams weighted 20%, 30%, and 50%. Scores: 70, 80, 90. Weighted mean = (0.2×70 + 0.3×80 + 0.5×90) / 1 = 14 + 24 + 45 = 83. Unweighted mean = (70+80+90)/3 ≈ 80. Weighted means are used for GPA, price indices (CPI), portfolio returns, and any situation where some values should count more than others.