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Population Mean — Definition, Formula & Examples

Population mean is the average you get when you add up every single value in an entire population and divide by the total number of values. It is represented by the Greek letter μ\mu (mu).

The population mean μ\mu is a parameter defined as the sum of all NN values in a finite population divided by NN. Unlike a sample mean, it involves no estimation — it is the true central value of the entire data set.

Key Formula

μ=∑i=1NxiN\mu = \frac{\sum_{i=1}^{N} x_i}{N}
Where:
  • μ\mu = Population mean
  • xix_i = Each individual value in the population
  • NN = Total number of values in the population

How It Works

To find the population mean, you must have data for every member of the population, not just a subset. Add all the values together, then divide by the total count NN. Because it uses every value, μ\mu is a fixed number (a parameter), not a variable estimate. In practice, you can rarely measure an entire population, so you often use a sample mean xˉ\bar{x} to estimate μ\mu.

Worked Example

Problem: A small school has exactly 5 teachers. Their ages are 28, 34, 41, 45, and 52. Find the population mean age.
Sum all values: Add every age together.
28+34+41+45+52=20028 + 34 + 41 + 45 + 52 = 200
Divide by N: There are 5 teachers in the entire population, so divide by 5.
μ=2005=40\mu = \frac{200}{5} = 40
Answer: The population mean age of the teachers is μ=40\mu = 40 years.

Why It Matters

Population mean is the benchmark that sample-based statistics try to estimate. In AP Statistics and introductory college courses, understanding the distinction between μ\mu and xˉ\bar{x} is essential for hypothesis testing and confidence intervals. Fields like quality control and public health rely on knowing — or accurately estimating — population means to set standards and detect problems.

Common Mistakes

Mistake: Using the sample mean symbol xˉ\bar{x} when reporting a population mean.
Correction: Use μ\mu for the population mean and xˉ\bar{x} for a sample mean. Mixing them up signals confusion between a parameter (population) and a statistic (sample).