x-bar — Definition, Formula & Examples
X-bar (written ) is the symbol used in statistics to represent the mean of a sample — the average value calculated from a subset of a larger population.
The statistic denotes the arithmetic mean of observed values drawn from a population. It serves as a point estimator of the population mean .
Key Formula
Where:
- = Sample mean
- = Each individual data value in the sample
- = Number of data values in the sample
How It Works
To find , add up every data value in your sample and divide by the number of values. Because is computed from a sample rather than the entire population, its value changes from sample to sample. This variability is described by the sampling distribution of , which has standard deviation . In AP Statistics, appears in confidence intervals, hypothesis tests, and the Central Limit Theorem.
Worked Example
Problem: A random sample of 5 test scores is: 72, 85, 90, 68, 95. Find x-bar.
Sum the values: Add all five scores together.
Divide by n: There are 5 values, so divide the sum by 5.
Answer:
Why It Matters
Every confidence interval and many hypothesis tests in AP Statistics are built around . Understanding what the symbol means — and that it estimates with some uncertainty — is essential for interpreting inference results on the AP exam and in real-world survey or experimental data.
Common Mistakes
Mistake: Confusing (sample mean) with (population mean).
Correction: is a statistic computed from sample data and varies between samples. is a fixed parameter of the entire population. Use when reporting results from a sample; use when referring to the true population value.