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Mean — Definition, Formula & Examples

Mean

Another word for average. Mean almost always refers to arithmetic mean. In certain contexts, however, it could refer to the geometric mean, harmonic mean, or root mean square.

 

Note: For any positive numbers, harmonic mean ≤ geometric mean ≤ arithmetic mean ≤ root mean square

Four mean formulas for 4 and 9: harmonic=72/13≈5.54, geometric=√36=6, arithmetic=6.5, root mean square=√48.5≈6.96

 

See also

Median, mode

Key Formula

xˉ=x1+x2+⋯+xnn=∑i=1nxin\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\displaystyle\sum_{i=1}^{n} x_i}{n}
Where:
  • xˉ\bar{x} = The mean (read "x-bar")
  • x1,x2,…,xnx_1, x_2, \ldots, x_n = The individual data values
  • nn = The total number of values in the data set
  • ∑\sum = Summation — add all values from the first to the last

Worked Example

Problem: Find the mean of the data set: 4, 8, 6, 10, 12.
Step 1: Count the number of values in the set.
n=5n = 5
Step 2: Add all the values together.
4+8+6+10+12=404 + 8 + 6 + 10 + 12 = 40
Step 3: Divide the sum by the number of values.
xˉ=405=8\bar{x} = \frac{40}{5} = 8
Answer: The mean of the data set is 8.

Another Example

This example works backward from a desired mean to find a missing value — a common exam question that tests deeper understanding of the formula.

Problem: A student scores 85, 90, 78, and 95 on four tests. What score does she need on a fifth test to have an overall mean of 88?
Step 1: Set up the mean formula with the unknown fifth score. Let the unknown score be xx.
85+90+78+95+x5=88\frac{85 + 90 + 78 + 95 + x}{5} = 88
Step 2: Add the known scores.
85+90+78+95=34885 + 90 + 78 + 95 = 348
Step 3: Multiply both sides by 5 to clear the denominator.
348+x=440348 + x = 440
Step 4: Solve for xx.
x=440−348=92x = 440 - 348 = 92
Answer: She needs a score of 92 on the fifth test.

Frequently Asked Questions

What is the difference between mean and average?
In everyday use, "mean" and "average" refer to the same thing: the arithmetic mean. Technically, "average" is a broader term that can describe any single value representing a data set (including median and mode), while "mean" specifically refers to the sum of values divided by the count. In most math classes, you can treat them as interchangeable.
What is the difference between mean, median, and mode?
The mean is the sum of all values divided by the count. The median is the middle value when data is arranged in order. The mode is the value that appears most frequently. For a symmetric data set these three are often close or equal, but for skewed data they can differ significantly. Outliers affect the mean far more than the median or mode.
When should you use the mean versus the median?
Use the mean when your data is roughly symmetric and has no extreme outliers. Use the median when data is skewed or contains outliers, such as household incomes or home prices. The median resists the pull of extreme values, giving a more representative "typical" value in those cases.

Mean vs. Median

MeanMedian
DefinitionSum of all values divided by the countMiddle value when data is sorted in order
FormulaSum ÷ nMiddle position: (n + 1)/2 th value
Sensitivity to outliersHighly sensitive — one extreme value shifts the meanResistant — outliers barely change it
Best used whenData is symmetric with no extreme outliersData is skewed or has outliers
Example: {1, 2, 3, 4, 100}223

Why It Matters

The mean is one of the first statistics you learn and one of the most widely used. You encounter it when calculating grade point averages, batting averages in sports, and scientific measurements. Understanding how the mean behaves — especially how outliers pull it — is essential for interpreting data in statistics, science, economics, and everyday decision-making.

Common Mistakes

Mistake: Forgetting to divide by the correct number of values, especially when a zero or a repeated value is in the data set.
Correction: Always count every entry, including zeros and duplicates. A zero still counts as a data point and lowers the mean; a repeated value increases nn.
Mistake: Using the mean to summarize heavily skewed data without recognizing it may be misleading.
Correction: Check for outliers first. If one or two extreme values dominate the sum, report the median alongside the mean (or instead of it) to give a more accurate picture of the typical value.

Related Terms