How to Calculate an Average

You drive to a city 60 miles away at 60 mph, then drive back at 40 mph. What's your average speed for the whole trip?

Most people say 50 mph. The correct answer is 48 mph. The trip there takes 1 hour; the trip back takes 1.5 hours. You covered 120 miles in 2.5 hours: 120 ÷ 2.5 = 48.

"Average" sounds like one calculation. In practice, there are several kinds, and choosing the right one matters. Here's how to calculate each, and when to use it.

The Right Average Depends on What You're Averaging

Adding numbers and dividing by how many there are, the arithmetic mean, is correct for many situations but wrong for others. Averaging speeds needs the harmonic mean. Averaging growth rates needs the geometric mean. Summarizing incomes usually calls for the median.

The question to ask first is: what does the average need to preserve?

1. The Mean (Arithmetic Average)

Add all the values, then divide by how many there are.

Mean = (x₁ + x₂ + … + xₙ) ÷ n

Example: Test scores of 72, 85, 90, 64, 85, 78 and 95.

Sum = 72 + 85 + 90 + 64 + 85 + 78 + 95 = 569
Mean = 569 ÷ 7 ≈ 81.3

Use it when: values are roughly symmetric, without extreme outliers, and adding them makes sense. Check your sums with the basic calculator.

2. The Median (Middle Value)

Sort the values and take the middle one.

Sorted scores: 64, 72, 78, 85, 85, 90, 95. The median is 85.

With an even number of values, average the two middle ones. For 3, 7, 9, 12, the median is (7 + 9) ÷ 2 = 8.

Use it when: data is skewed or has outliers, like house prices or incomes. One billionaire moves the mean dramatically but barely changes the median.

3. The Mode (Most Common Value)

Find the value that appears most often. In our scores, 85 appears twice, so the mode is 85.

A dataset can have no mode (all values different) or several modes.

Use it when: data is categorical, like the most common shoe size or most popular color, where a mean makes no sense.

4. The Geometric Mean (for Growth Rates)

Multiply the growth factors, then take the nth root.

Geometric mean = (x₁ × x₂ × … × xₙ)^(1/n)

Example: An investment gains 10% one year and loses 10% the next. The arithmetic mean of the returns is 0%. But:

(1.10 × 0.90)^(1/2) = 0.99^(1/2) ≈ 0.99499
Average growth ≈ −0.5% per year

You actually lost money: $100 becomes $99. Use it when: averaging rates of growth, returns or ratios that compound. See The Math Behind Retirement.

5. The Harmonic Mean (for Rates Like Speed)

Divide the count by the sum of reciprocals.

Harmonic mean = n ÷ (1/x₁ + 1/x₂ + … + 1/xₙ)

Example: The round trip at 60 mph and 40 mph, over equal distances:

2 ÷ (1/60 + 1/40) = 2 ÷ (0.01667 + 0.025) = 2 ÷ 0.04167 = 48 mph

Use it when: averaging rates over equal amounts of the thing in the denominator, like speeds over equal distances, or prices per share when investing equal dollar amounts. Convert speeds with the speed unit converter.

Which Average Should You Use?

For any set of positive numbers, a neat inequality always holds:

Harmonic mean ≤ Geometric mean ≤ Arithmetic mean

They're equal only when all the values are the same.

An Insider Reference: Quetelet's "Average Man"

In 1835, Belgian astronomer and statistician Adolphe Quetelet published A Treatise on Man, introducing the idea of l'homme moyen, the "average man." He measured chest sizes of Scottish soldiers and heights of French conscripts, finding they clustered in bell-shaped patterns around a mean.

Quetelet believed the average represented a kind of ideal, with deviations as errors. That idea was hugely influential, shaping social statistics and even the design of the body mass index, which Quetelet developed. It also sparked lasting criticism: averages describe groups, not any individual. See Why Averages Can Lie.

Two Concepts Worth Knowing

Outlier

An outlier is a value far from the rest. The mean is sensitive to outliers; the median is robust. Some competitions, like Olympic diving and gymnastics, drop the highest and lowest judges' scores before averaging, a version called a trimmed mean.

Measure of Central Tendency

Mean, median and mode are all measures of central tendency: single numbers that describe the "center" of a dataset. See the statistics formulas.

Quick Answer: How Do You Calculate an Average?

To calculate the mean, add all values and divide by how many there are. The median is the middle value when data is sorted, and the mode is the most frequent value. For growth rates use the geometric mean (multiply and take the nth root), and for speeds over equal distances use the harmonic mean (n divided by the sum of reciprocals).

Try Them Yourself

• Basic Calculator: sums and divisions for the mean
• Statistics Formulas: mean, median, mode and more
• Scientific Calculator: nth roots for the geometric mean
• Speed Unit Converter: harmonic mean in travel problems
• How to Calculate a Weighted Average: when values count differently
• How to Calculate Standard Deviation: measuring spread around the mean

Track your commute time for a week. Calculate the mean and the median. If one day was unusually slow, you'll see the two numbers disagree, and you'll know which one tells the truer story.