Sample Mean

1. Definition

• The sample mean ($\bar{x}$) is the average value of a sample, i.e., the sum of all sample observations divided by the number of observations.
• It is a statistic used to estimate the true population mean (μ).

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2. Formula

For a sample of size $n$: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

Where:

• $x_i$​ = individual data points
• $n$ = number of observations
• $\bar{x}$ = sample mean

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3. Example

Example 1 – Test Scores

• Sample = {80, 85, 90, 95, 100}

$\bar{x} = \frac{80 + 85 + 90 + 95 + 100}{5} = \frac{450}{5} = 90$

Sample mean = 90

Example 2 – Heights

• 10 people measured, total height = 1720 cm

$\bar{x} = \frac{1720}{10} = 172 \text{ cm}$

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4. Properties of the Sample Mean

• Unbiased estimator:
  • $E[\bar{x}] = μ$
  • The expected value of the sample mean equals the true population mean.
• Sampling distribution:
  • If sample size $n$ is large → by the Central Limit Theorem (CLT), $\bar{x}$ follows approximately a normal distribution:
    • $\bar{x} \sim N \left( μ, \frac{σ^2}{n} \right)$
  • Standard error of the mean (SEM):
    • $SE = \frac{σ}{\sqrt{n}}$
• Sensitivity: The mean is affected by outliers (very high/low values).

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5. Sample Mean vs Population Mean

Concept	Symbol	Description
Population mean	μ	Fixed, true average of the population (usually unknown)
Sample mean	$\bar{x}$	Estimate of μ from a sample

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6. When It’s Used

• Describing sample data (descriptive statistics).
• Estimating the population mean (inferential statistics).
• Hypothesis testing (e.g., one-sample t-test).
• Building confidence intervals.

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In short:
The sample mean ($\bar{x}$) is the arithmetic average of sample data. It is the best unbiased estimator of the true population mean (μ), and its variability decreases as the sample size increases.