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Sample Mean — Definition, Formula & Examples

The sample mean is the average of a set of data values, found by adding all the values together and dividing by how many there are. It represents the "center" of your data and is one of the most common ways to summarize a dataset with a single number.

Given a sample of nn observations x1,x2,…,xnx_1, x_2, \ldots, x_n, the sample mean xˉ\bar{x} is the arithmetic average of those observations. It serves as an unbiased point estimator of the population mean μ\mu.

Key Formula

xˉ=1n∑i=1nxi=x1+x2+⋯+xnn\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \cdots + x_n}{n}
Where:
  • xˉ\bar{x} = Sample mean
  • xix_i = Each individual data value
  • nn = Number of data values in the sample

How It Works

To find the sample mean, sum every data value in your sample and divide by the total count. The result tells you where the data tends to cluster. Because it uses every value in the dataset, the sample mean is sensitive to extreme values — a single outlier can pull it significantly higher or lower. In inferential statistics, xˉ\bar{x} is used to estimate the true population mean when measuring every member of the population is impractical.

Worked Example

Problem: A student scores 78, 85, 90, 92, and 95 on five math tests. Find the sample mean of the scores.
Sum the values: Add all five test scores together.
78+85+90+92+95=44078 + 85 + 90 + 92 + 95 = 440
Divide by the count: There are 5 scores, so divide the sum by 5.
xˉ=4405=88\bar{x} = \frac{440}{5} = 88
Answer: The sample mean test score is 88.

Why It Matters

The sample mean is the starting point for nearly every statistical analysis — from calculating standard deviation to running hypothesis tests. In fields like medicine, economics, and engineering, decisions worth millions of dollars hinge on sample means computed from collected data.

Common Mistakes

Mistake: Confusing the sample mean (xˉ\bar{x}) with the population mean (μ\mu).
Correction: Use xˉ\bar{x} when you have data from a subset (sample) and μ\mu when referring to the entire population. In practice, you almost always compute xˉ\bar{x} and use it to estimate μ\mu.