Average Calculator

About the Average Calculator

An average is a single number that summarizes the center of a dataset. This free average calculator finds the arithmetic, geometric, and harmonic mean of any list of numbers: paste in your values, pick a type, and the tool returns the result along with the count, the sum, and a step-by-step explanation of the calculation.

The word "average" most often means the arithmetic mean, but it is not the only way to describe a typical value. The median and the mode answer the same question in different ways, and each measure shines in different situations. The sections below define all three, walk through a worked example, and show what happens when an outlier enters the data.

Mean, Median, and Mode

Mean (Arithmetic Average)

The mean is the sum of every value divided by how many values there are:

mean = (x1 + x2 + … + xn) ÷ n

• x1, x2, …, xn — the individual values in the dataset
• n — the number of values (the count)

Median

The median is the middle value once the data is sorted from smallest to largest. If the dataset has an odd count, the median is the center number; if the count is even, the median is the mean of the two center numbers.

Mode

The mode is the value that appears most often. A dataset can have one mode, several modes, or no mode at all if every value is unique.

Worked Example

Take the dataset 3, 7, 7, 9, 12, 15, 15, 15, 21.

• Mean: the sum is 3 + 7 + 7 + 9 + 12 + 15 + 15 + 15 + 21 = 104, and there are 9 values, so 104 ÷ 9 ≈ 11.56 (the calculator displays 11.5556 to four decimal places).
• Median: the list is already sorted, and with 9 values the middle one is the 5th value: 12. Four values sit below it and four sit above it.
• Mode: 15 appears three times, more than any other value, so the mode is 15.

All three measures land close together here, a sign that the data is fairly symmetric. Enter the same list into the tool above and it will confirm the mean, count, and sum instantly.

How an Outlier Changes the Average

Now add one extreme value — 100 — to the same dataset, giving 3, 7, 7, 9, 12, 15, 15, 15, 21, 100.

• The mean jumps from about 11.56 to 204 ÷ 10 = 20.4 — higher than nine of the ten values, so it no longer describes a typical number at all.
• The median barely moves: with ten values it is the mean of the 5th and 6th, (12 + 15) ÷ 2 = 13.5, up from 12.
• The mode is unchanged at 15.

This is why the median often tells the truer story for skewed data. Reports on household income and house prices quote medians rather than means because a handful of very high earners or luxury sales would pull the mean far above what most people actually experience. When a headline quotes an "average" home price or salary, it is worth asking which average was used.

Mean, Median, or Mode: Which Should You Use?

For a quick read on all three measures at once, try the mean, median, and mode calculator; for deeper analysis such as standard deviation, variance, and quartiles, use the statistics calculator.

Beyond the Simple Average

Weighted Average

Sometimes values do not count equally. A weighted average multiplies each value by its importance before dividing by the total of the weights:

weighted average = (w1x1 + w2x2 + … + wnxn) ÷ (w1 + w2 + … + wn)

Example: a course weights homework at 20%, a midterm at 30%, and the final exam at 50%. Scoring 85, 78, and 92 gives (85 × 0.20 + 78 × 0.30 + 92 × 0.50) ÷ 1.00 = 86.4. The grade calculator automates exactly this kind of weighted math, including "what score do I need on the final?" scenarios.

Trimmed Mean

A trimmed mean discards a fixed percentage of the highest and lowest values before averaging, combining the stability of the median with the mean's use of more data. Dropping just the smallest and largest values from the outlier dataset above leaves 7, 7, 9, 12, 15, 15, 15, 21, whose mean is 101 ÷ 8 = 12.625 — far closer to the original 11.56 than the untrimmed 20.4. Olympic judging and official inflation statistics use trimmed means for this reason.

Geometric and Harmonic Means

The geometric mean is the nth root of the product of the values and suits multiplicative processes such as compound growth and investment returns. The harmonic mean — the count divided by the sum of the reciprocals — is the right choice for averaging rates and ratios, like average speed over equal distances. Both are available in the type dropdown of the calculator above.