Sampling and Descriptive Statistics

Descriptive Statistics

Probability assumes the model is known. Statistics assumes the model is unknown and must be inferred from data.

The Empirical Distribution

Assigns probability $1/n$ to each observed data point.

Estimates

We use sample statistics to estimate population parameters:

Sample Mean ( $\bar{X}$ ) estimates $\mu$ .
Sample Variance ( $S^2$ ) estimates $\sigma^2$ .

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Unbiased Estimators

Both $\bar{X}$ and $S^2$ are unbiased: their expected values equal the true parameters they estimate.

Visualization

Histograms: For continuous data.
Q-Q Plots: Compare sample quantiles to theoretical quantiles to checking distributional assumptions.