Mathematics · Updated 2026

Average Calculator

Calculate mean, median, mode, range, standard deviation, variance, and geometric mean for any set of numbers. Paste comma-separated values or enter one per line.

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Mean, Median, Mode
Std Deviation & Variance
Geometric Mean
Sorted Number Chips
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Separate with commas, spaces, or new lines. Type 1000, not 1,000.

Enter numbers to see mean, median, mode, and full statistical breakdown.

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How to Calculate an Average

The average, or arithmetic mean, is the sum of the numbers divided by how many there are: for 4, 8, 15, 16, 23 and 42 it is 108 ÷ 6 = 18. Paste your numbers above to get the mean along with the median, mode, range, standard deviation and geometric mean, so you can see whether an outlier is pulling the average.

  1. Add all the numbers together. For 4, 8, 15, 16, 23 and 42 the sum is 108.
  2. Count how many numbers there are: 6.
  3. Divide the sum by the count: 108 ÷ 6 = 18.

The median of the same list is 15.5, the average of the two middle values (15 and 16), because the list has an even number of values. When the mean and the median are far apart, one or two large or small values are pulling the mean.

Mean, Median and Mode Side by Side

Six small data sets and what each measure says about them. SD is the population standard deviation, the main figure the calculator shows.

NumbersMeanMedianModeSD
4, 8, 15, 16, 23, 421815.5None12.32
1, 2, 2, 3, 10021.62239.21
10, 20, 30, 402525None11.18
−5, 0, 5, 102.52.5None5.59
3, 3, 3, 33330
12, 7, 3, 14, 6, 11, 5, 4, 12, 148.8912 and 144.02

The second row shows why the median matters: one value of 100 lifts the mean to 21.6 while four of the five numbers are 3 or less. The last row is the Sample button's data, which has two modes.

Weighted Average: Worked Examples

The calculator above treats every number equally. When some values count more than others, multiply each value by its weight, add the results and divide by the total weight.

A course grade

Homework counts 20% (you scored 92), quizzes 30% (85) and the final exam 50% (78). The weighted average is 0.2 × 92 + 0.3 × 85 + 0.5 × 78 = 18.4 + 25.5 + 39 = 82.9. The plain average of the three scores would be 85, which overstates the grade because the lowest score carries the most weight.

An average purchase price

You buy 100 shares at $20 and later 300 shares at $24. The average cost is (100 × 20 + 300 × 24) ÷ 400 = 9,200 ÷ 400 = $23 a share, not the $22 you get by averaging the two prices, because more shares were bought at the higher price.

A grade point average works the same way, with credit hours as the weights; the GPA calculator handles that for you.

Which Average Should You Use?

  • Mean: for data without extreme values, such as daily temperatures or repeated measurements.
  • Median: for skewed data with outliers, such as incomes, home prices or response times.
  • Mode: for the most common choice or size, including non-numeric categories like shoe sizes sold.
  • Geometric mean: for growth rates and ratios that multiply. It needs every value above zero, so the calculator shows N/A if any value is 0 or negative.
  • Harmonic mean: for averaging rates over equal distances, such as speeds on two legs of the same length. It also needs positive values.

Edge cases the calculator handles

Negative numbers and zero are fine for the mean, median, mode, range and standard deviation. With a single number, the standard deviation is 0 and the sample standard deviation is not defined, because it divides by n − 1. If every value appears the same number of times, there is no mode. Do not type commas inside numbers: 1,000 is read as two numbers, 1 and 0. For a deeper look at spread, use the standard deviation calculator.

Method and sources. Mean = sum ÷ n. Median = middle value of the sorted list (average of the two middle values for an even count). Population standard deviation divides the sum of squared deviations by n, sample standard deviation by n − 1. Geometric mean = exp(average of ln x); harmonic mean = n ÷ sum of 1/x. Definitions follow the NIST/SEMATECH e-Handbook of Statistical Methods. Every figure in the table and examples was computed with these formulas.

Mean, Median, Mode Explained

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. The calculator shows the population standard deviation (dividing by n) and, below it, the sample standard deviation. 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 the standard deviation is 10.8, 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: finishing times in a race that mixes casual walkers and trained runners often show two separate peaks. 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.

Only average them directly when each percent has the same base. Otherwise weight by the base. If you scored 80% on a 50-point test and 90% on a 150-point test, you earned 40 + 135 = 175 of 200 points, which is 87.5%, not the 85% you get by averaging 80 and 90.

Multiply the target average by the new number of tests and subtract the points you already have. With an average of 84 over 5 tests and a target of 86 after the sixth, you need 6 × 86 − 5 × 84 = 516 − 420 = 96. If the answer is above the maximum score, the target is out of reach with one test.

Yes. Add them with their signs and divide by the count: the average of −5, 0, 5 and 10 is 10 ÷ 4 = 2.5. Zeros count as values, so leave them in unless they mean "missing". Only the geometric and harmonic means need every value to be positive.
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