Average: Definition, Formula, and How to Calculate It

In mathematics, the word average usually refers to the arithmetic mean: add the values together and divide by how many values there are. It is one of the simplest ideas in statistics, but it is also one of the most useful.

An average takes a collection of numbers and replaces it with one number that represents the group. This makes it easier to describe a dataset, compare different groups, or get a quick idea of where the values are centered.

01 · Average BasicsWhat Is an Average? Understanding the Arithmetic Mean

Suppose the numbers are

There are five values. First add them:

Then divide the sum by the number of values:

So the arithmetic mean is 15.4.

The formula can be written more generally as

Using mathematical notation, if the observations are x₁, x₂, ..., x_n, the mean is

The symbol x̄ is read as “x-bar” and is commonly used for the arithmetic mean of a set of observations.

02 · Core IdeaHow Does an Average Work? The Balance Point

There is a useful way to understand the mean without thinking only about the formula. Imagine placing each number on a number line and putting equal weights on them. The mean is the point where the values would balance.

This leads to an important property of the arithmetic mean. If you subtract the mean from every value and add all those differences together, the result is zero.

For the numbers 4, 6, and 8, the mean is 6:

The values on the two sides of the mean balance in this sense.

How to Calculate an Average Step by Step

For a small dataset, the procedure is straightforward.

Take the values 12, 18, 21, 25, and 34.

The average is therefore 22.

Notice that 22 does not have to be one of the original observations. An average is a summary value; it does not necessarily have to appear in the dataset itself.

What Does the Average Tell You About a Dataset?

The mean gives a measure of the center of a group of numerical observations. If the values are reasonably balanced around a central point, the mean can give a useful description of a typical observation.

For example, consider the daily number of customers visiting a small business over five days:

The average is

So 100 customers per day is a useful summary of the five-day period.

But this does not mean that exactly 100 customers arrived on every day. The mean compresses several observations into one number and therefore removes some information about the variation in the original data.

The average can be affected by extreme values

One of the most important facts about the arithmetic mean is that it is sensitive to unusually large or unusually small observations.

Suppose five people have monthly incomes of $1,000, $1,000, $1,000, $1,000, and $16,000.

The average income is $4,000, but four of the five people earn only $1,000. The single very large value pulls the mean upward.

This does not make the calculation wrong. The arithmetic is correct. The issue is interpretation: a mathematically correct average can still be a poor description of what is typical in a dataset.

Mean versus median

The median is the middle value after the observations have been arranged in order. Unlike the mean, it is much less affected by an unusually large or small observation.

Consider:

The mean is

The median is 4 because 4 sits in the middle of the ordered list.

Both 6.8 and 4 are valid summaries, but they tell different stories. The mean incorporates every value numerically, while the median identifies the middle position and is less influenced by the value 20.

What happens when there is an even number of values?

With an even number of observations, there is no single middle observation. In that case, the median is found by averaging the two middle values.

For example:

The two middle values are 10 and 14, so

This is different from the arithmetic mean of the entire dataset, which is calculated by adding all six observations and dividing by 6.

08 · Weighted MeanWhat Is a Weighted Average? Formula and Example

Not every situation gives every value the same importance. When observations have different weights, a simple average can give the wrong result.

The weighted average is calculated by multiplying each value by its weight, adding those products, and then dividing by the sum of the weights.

Suppose a student's final grade consists of a 30% assignment score of 80 and a 70% exam score of 90.

The final weighted average is 87.

A simple average of 80 and 90 would be 85, but that would incorrectly assume the assignment and exam have equal importance.

Why averaging averages can be misleading

It is tempting to calculate several averages and then simply average those averages. Sometimes that works, but not always.

Suppose one class has 10 students with an average score of 90, while another class has 30 students with an average score of 60. The simple average of the two class averages is

But if we want the average score of all 40 students, each student must have equal influence:

The correct overall average is 67.5, not 75.

The difference occurs because the two groups contain different numbers of observations. In effect, the group averages need to be weighted by their sample sizes.

10 · PercentagesHow to Calculate an Average of Percentages

Percentages require a little care because an average of percentages is not automatically the overall percentage.

Suppose one test is taken by 10 students and the average score is 80%, while another test is taken by 30 students and the average score is 60%. A simple average gives 70%, but the combined percentage depends on the number of students represented by each result.

The same principle applies to business reports, survey results, class grades, conversion rates, and many other situations. Before averaging percentages, ask whether each percentage represents the same number of observations.

Average, median, mode, and midrange

The word “average” is sometimes used informally for several different measures of central tendency. The arithmetic mean is only one of them.

The mean uses every numerical value and divides their sum by the number of observations.

The median is the middle observation after sorting the data, or the average of the two middle observations when there is an even number of values.

The mode is the value that occurs most frequently. A dataset can have one mode, more than one mode, or no repeated value at all.

The midrange is the average of the smallest and largest observations:

These measures should not be treated as interchangeable. Each summarizes the data in a different way.

A useful check using the minimum and maximum

For a non-empty set of ordinary numerical observations, the arithmetic mean cannot be smaller than the smallest value or larger than the largest value.

This gives a quick error check.

If the values are 12, 15, 18, and 25, an average of 30 is impossible. Likewise, an average below 12 would indicate a calculation or data-entry problem.

This simple check is especially useful when working with a calculator, spreadsheet, or a large dataset.

13 · Updating DataHow to Update an Average When Adding a New Value

There is also a useful way to update an average without starting from the beginning.

Suppose the current average of 5 numbers is 20. The sum of those five numbers must be

Now add a sixth value of 32. The new sum is 132, and the new average is

So the average rises from 20 to 22.

This way of thinking is useful in real datasets because an average is always connected to a total and a count. If you know two of those quantities, you can usually recover the third.

14 · Reverse CalculationHow to Find a Missing Value From an Average

Suppose four numbers have an average of 18, and three of the numbers are 12, 17, and 20. What is the fourth number?

First find the total that four numbers must have:

Then add the known values:

The missing value must make the total equal to 72:

Check:

This is often easier than trying to manipulate the average formula all at once. Think in terms of the total first.

15 · StatisticsAverage in Statistics: Population Mean vs. Sample Mean

In statistics, the mean is a measure of central tendency. It gives a single value intended to summarize the center of a numerical dataset.

For a population, the mean is often represented by the Greek letter μ:

Here, N is the number of observations in the population.

For a sample, the mean is commonly written as x̄:

The calculation itself has the same basic structure. The notation changes because population and sample statistics have different roles in statistical analysis.

16 · Key TakeawayAverage Formula: The Main Idea to Remember

The arithmetic mean is simple:

But using an average correctly requires more than performing the division. You should also ask what the numbers represent, whether every observation should have equal weight, and whether unusual values are influencing the result.

For a clean dataset with no major outliers, the mean can be an excellent summary. When the data is skewed or contains extreme observations, the median may give a better picture of what is typical. When observations have different importance or represent different group sizes, a weighted mean may be necessary.

The best habit is therefore not simply to calculate the average, but to understand what that average is actually saying about the data.