How to Calculate Averages (Mean, Median, Mode & More)

“Average” is one of the most used and most misunderstood words in everyday math. News reports cite average salaries, average home prices, and average test scores, but the specific type of average changes the story entirely. A CEO earning $10 million can make the “average salary” at a 50-person company look impressive even if most employees earn $55,000.

This guide breaks down the five most common types of averages, explains when each one is appropriate, and walks through worked examples. For instant calculations, use our Average Calculator.

Arithmetic Mean

The arithmetic mean is what most people think of when they hear “average.” Add up all the values and divide by the count.

Formula:

$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}$

Example: Your exam scores for the semester are 88, 92, 75, 91, and 84. What is your average score?

Sum: 88 + 92 + 75 + 91 + 84 = 430
Count: 5 exams
Mean: 430 / 5 = 86.0

When to Use the Mean

The mean works best when your data is roughly symmetrical and free of extreme outliers. It is the go-to measure for:

Test scores in a typical class
Daily temperatures over a month
Manufacturing measurements (bolt lengths, fill volumes)
Sports statistics (batting average, points per game)

When the Mean Misleads

Consider five houses on a street priced at $250,000, $275,000, $260,000, $280,000, and $1,900,000. The mean price is $593,000. But four of the five houses cost under $300,000. The single mansion drags the average up by more than $300,000, creating a number that describes none of the homes accurately.

This is the core weakness of the arithmetic mean: it gives every data point equal influence, so one extreme value can distort the result.

Median

The median is the middle value when all data points are sorted from lowest to highest. It splits the dataset in half — 50% of values fall below it and 50% above.

How to Find the Median:

Sort the values in ascending order.
If the count is odd, the median is the middle number.
If the count is even, the median is the average of the two middle numbers.

Example (odd count): Dataset: 12, 7, 3, 15, 9.

Sorted: 3, 7, 9, 12, 15
Middle value (3rd of 5): 9

Example (even count): Dataset: 12, 7, 3, 15, 9, 20.

Sorted: 3, 7, 9, 12, 15, 20
Two middle values (3rd and 4th): 9 and 12
Median: (9 + 12) / 2 = 10.5

When to Use the Median

The median is resistant to outliers, making it the preferred measure for skewed distributions:

Home prices: Median home price is the standard real estate metric because a few mansions would inflate the mean.
Income data: Median household income is more representative than mean income for the same reason.
Response times: Server response times often have long-tail outliers. Median gives a better picture of typical user experience.

Back to the housing example: Sorting the prices ($250K, $260K, $275K, $280K, $1.9M), the median is $275,000 — far more representative of the neighborhood than the $593,000 mean.

Mode

The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), two modes (bimodal), multiple modes (multimodal), or no mode at all (if every value is unique).

Example: Shoe sizes sold today: 8, 9, 10, 9, 11, 9, 10, 8, 9.

8 appears 2 times
9 appears 4 times
10 appears 2 times
11 appears 1 time
Mode: 9

When to Use the Mode

The mode is the only average that applies to categorical (non-numeric) data. If you survey 200 people about their favorite ice cream flavor and “chocolate” gets the most votes, chocolate is the mode. You can’t calculate a mean or median of flavors.

The mode is also useful for:

Inventory planning: The most common shoe size tells you what to stock most of.
Quality control: If most defects fall into one category, that category is the mode and your priority.
Bimodal detection: If your data has two modes (like test scores clustering at 60 and 90), it may indicate two distinct groups in your population.

Weighted Average

A weighted average assigns different levels of importance to each value. Instead of treating every data point equally, you multiply each value by a weight that reflects its significance.

Formula:

$\bar{x}_w = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}$

Example: Calculating a Course Grade

Component	Score	Weight
Homework	92	20%
Midterm	78	30%
Final Exam	85	40%
Participation	95	10%

Weighted average = (92 x 0.20) + (78 x 0.30) + (85 x 0.40) + (95 x 0.10) / (0.20 + 0.30 + 0.40 + 0.10)

= 18.4 + 23.4 + 34.0 + 9.5 = 85.3

If you had used a simple mean: (92 + 78 + 85 + 95) / 4 = 87.5. The weighted average is lower because the midterm (your lowest score) carries more weight than homework.

Other Weighted Average Applications

Investment portfolio returns: Weight each asset’s return by its share of the portfolio.
GPA calculation: Credit hours serve as the weight. A 4.0 in a 4-credit course matters more than a 4.0 in a 1-credit course.
Composite ratings: Product reviews weighted by the reviewer’s purchase verification status.

Trimmed Mean

A trimmed mean removes a fixed percentage of the highest and lowest values before calculating the mean. This reduces the influence of outliers without switching entirely to the median.

Example: Olympic diving scores discard the highest and lowest judges’ scores, then average the rest. This is a trimmed mean in action.

Worked example: Dataset: 2, 45, 48, 50, 51, 52, 53, 55, 98.

Regular mean: (2 + 45 + 48 + 50 + 51 + 52 + 53 + 55 + 98) / 9 = 50.4
10% trimmed mean (remove top and bottom value): (45 + 48 + 50 + 51 + 52 + 53 + 55) / 7 = 50.6
The outliers (2 and 98) barely affected this dataset, but in other cases the difference can be dramatic.

A 20% trimmed mean on a dataset of 20 values removes the 4 smallest and 4 largest, then averages the remaining 12. This is common in economic statistics — the Federal Reserve Bank of Cleveland publishes a 16% trimmed-mean CPI inflation measure that strips out volatile price changes.

Choosing the Right Average

Scenario	Best Average	Why
Exam scores (no outliers)	Mean	Symmetric data, every score matters equally
Home prices in a city	Median	Skewed by luxury properties
Most popular product color	Mode	Categorical data
Course grade with weighted components	Weighted mean	Different components have different importance
Survey responses with spam	Trimmed mean	Removes extreme/fake entries

Quick test: Calculate both the mean and the median. If they are close, the data is fairly symmetric and the mean is fine. If they differ by more than 10-15%, the data is skewed and you should report the median (or both).

Real-World Applications

Finance: Portfolio managers calculate weighted average return, weighted average cost of capital (WACC), and volume-weighted average price (VWAP) for stock orders.

Education: GPA is a weighted average of grade points by credit hours. Class rank typically uses this weighted GPA.

Sports analytics: A basketball player’s scoring average (mean points per game) is a standard metric, but analysts also look at the median game score to filter out blowout performances that inflate the mean.

Healthcare: Median survival time is reported in clinical trials rather than mean survival time, because a few long-surviving patients would skew the mean upward and create false optimism.

Compute any of these quickly with our Average Calculator.

Frequently Asked Questions

What is the difference between mean and average?

In everyday language, “average” and “mean” are used interchangeably, and both refer to the arithmetic mean. Technically, “average” is a broader term that includes the median, mode, and other measures of central tendency. When someone says “the average is 42” without further context, they almost always mean the arithmetic mean.

Can a dataset have no mode?

Yes. If every value in the dataset appears exactly once, there is no mode. For example, the dataset {3, 7, 12, 18, 25} has no repeating values and therefore no mode. Some textbooks say every value is the mode in this case; others say the mode is undefined. In practice, a dataset with no mode simply means no single value dominates.

How do outliers affect the mean vs. the median?

Outliers pull the mean toward them but leave the median virtually unchanged. Adding a $5 million salary to a dataset of fifty $60,000 salaries increases the mean from $60,000 to $156,862 — a 161% jump. The median stays at $60,000 because the middle position doesn’t shift. This is why analysts report the median for income, home prices, and any data prone to extreme values.

When should I use geometric mean instead of arithmetic mean?

Use the geometric mean when combining rates of change or ratios. If an investment returns +10%, -5%, and +20% over three years, the geometric mean return is (1.10 x 0.95 x 1.20)^(1/3) - 1 = 7.71%, which reflects the actual compound growth. The arithmetic mean of 8.33% overstates the real return. The geometric mean is also used for averaging ratios, index numbers, and biological growth rates.

How many decimal places should I report for an average?

Report one more decimal place than the original data. If your measurements are whole numbers (88, 92, 75), report the average to one decimal (85.0). If your data has one decimal place (3.2, 4.7, 5.1), report to two decimals (4.33). Reporting excessive decimal places implies false precision that the original data doesn’t support.