1. Definition

  • The true mean is the population mean (μ) — the actual arithmetic average of an entire population.
  • It is a fixed value, but usually unknown because we can rarely measure the entire population.
  • Instead, we estimate it using the sample mean ($\bar{x}$).

2. Formula (Population Mean)

For a population of size $N$:

$μ = \frac{\sum_{i=1}^{N} x_i}{N}$

Where:

  • $x_i$​ = each observation in the population
  • $N$ = total population size

3. Sample Mean as an Estimate of the True Mean

  • Since the true mean μ is usually unknown, we use the sample mean:

$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

  • $\bar{x}$ is a statistic (random, changes from sample to sample).
  • μ is a parameter (fixed, but unknown).
  • By the Law of Large Numbers, as n increases, $\bar{x}$ gets closer to μ.

4. Example

Population

Suppose the population = {2, 4, 6, 8, 10}

$μ = \frac{2 + 4 + 6 + 8 + 10}{5} = 6$

Sample

Take a sample {4, 10} $\bar{x} = \frac{4 + 10}{2} = 7$

Sample mean = 7, which is an estimate of the true mean (6).

5. Inference and the True Mean

  • In hypothesis testing, we test claims about the true mean (μ).
    • Example: H₀: μ = 100.
  • In confidence intervals, we say:
    • “We are 95% confident that the true mean μ lies between X and Y.”

6. Key Takeaways

  • The true mean (μ) is the population’s average, fixed but usually unknown.
  • The sample mean ($\bar{x}$) is the best unbiased estimator of μ.
  • As n → large, $\bar{x} \to μ$.
  • Hypothesis tests and confidence intervals are built to infer μ from sample data.

In short:
The true mean (μ) is the real average of the population. We rarely know it exactly, so we estimate it using the sample mean ($\bar{x}$), and then use confidence intervals and hypothesis testing to make statements about μ.