Arithmetic Mean — Definition, Formula & Examples

Arithmetic Mean

The most commonly used type of average. To find the arithmetic mean of a set of n numbers, add the numbers in the set and divide the sum by n.

Formula:

Example:

For the numbers 4 and 9,

See also

Mean

Key Formula

\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum_{i=1}^{n} x_i}{n}

Where:

$\bar{x}$ = The arithmetic mean (read as 'x-bar')
$x_1, x_2, \ldots, x_n$ = The individual values in the data set
$n$ = The total number of values in the set
$\sum$ = Summation symbol, meaning 'add up all the values'

Worked Example

Problem: Find the arithmetic mean of the test scores 72, 85, 90, 68, and 95.

Step 1: Add all the values together.

72 + 85 + 90 + 68 + 95 = 410

Step 2: Count the number of values. There are 5 test scores, so n = 5.

n = 5

Step 3: Divide the sum by n.

\bar{x} = \frac{410}{5} = 82

Answer: The arithmetic mean of the test scores is 82.

Another Example

Problem: A student earns

8,

12,

10, and

14 from four different chores. What is the arithmetic mean of these earnings?

Step 1: Add all the earnings.

8 + 12 + 10 + 14 = 44

Step 2: There are 4 values, so divide by 4.

\bar{x} = \frac{44}{4} = 11

Answer: The arithmetic mean earning is $11 per chore.

Frequently Asked Questions

What is the difference between arithmetic mean and average?

In everyday language, 'average' and 'arithmetic mean' refer to the same thing: add up all the values and divide by the count. However, in statistics the word 'average' can sometimes refer to other measures of central tendency like the median or mode. When precision matters, saying 'arithmetic mean' makes it clear which calculation you intend.

Can the arithmetic mean be a number not in the data set?

Yes, and it often is. For example, the arithmetic mean of 3 and 8 is 5.5, which does not appear in the original set. The mean represents a central balance point of the data, not necessarily a value that was observed.

Arithmetic Mean vs. Median

The arithmetic mean adds all values and divides by the count, giving a result influenced by every number—including extreme outliers. The median is the middle value when the data is sorted, so it is resistant to outliers. For the set {1, 2, 3, 4, 100}, the arithmetic mean is 22, while the median is 3. Use the mean when data is roughly symmetric; prefer the median when data is heavily skewed.

Why It Matters

The arithmetic mean is one of the most fundamental calculations in statistics and appears throughout science, economics, and daily life. Grade point averages, batting averages, and average temperatures all rely on it. Understanding the mean also lays the groundwork for more advanced concepts like standard deviation and variance, which measure how spread out data is around the mean.

Common Mistakes

Mistake: Dividing by the wrong number, such as dividing by the sum instead of the count.

Correction: Always divide the sum of the values by n, the number of values. Double-check your count before dividing.

Mistake: Assuming the mean is always a good representation of a data set, even when outliers are present.

Correction: A single extreme value can pull the mean far from where most data points lie. For example, in {2, 3, 4, 5, 1000}, the mean is 202.8, which does not represent the typical value. Consider the median or note the outlier.

Related Terms

Average — General term often synonymous with arithmetic mean
Mean — Broader category that includes arithmetic mean
Median — Middle value; alternative measure of center
Mode — Most frequent value in a data set
Weighted Mean — Mean where some values count more than others
Geometric Mean — Mean using products instead of sums
Set — Collection of values the mean is computed from
Standard Deviation — Measures spread of data around the mean