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  • STA256 Lecture 01
  • STA258 Lecture 01

Statistic

• Numerical characteristic of a sample.

• Function of observations which summarizes or reduces the data.

• Example:

  • $\overline{x}$: Sample mean
  • $s$: Standard deviation

• Ex:

  • Mice in a lab
  • $\begin{array}{c}\text{Observations}\\ X_1=1\\ X_2=2& \overline{x}=\frac{1+2+3}{3}=2\\ X_3=3& \text{range}=3-1=2\\ & s^2\\ & max\{x_1,x_2,x_3\}=3\\ & min\{x_1,x_2,x_3\}=1\end{array}$
  • How can we collapse this into one number? A Statistic

• A numerical characteristic of a sample.

• Formal Definition:

  • A statistic is a function of the sample observations $X_1,\dots ,X_n$ that does not depend on any unknown parameters.
  • Notation: $\mathrm{\Gamma }=g(X_1,\dots ,X_n)$.

• Purpose:

  • Summarizes or reduces data into a single value.
  • Serves as an Estimator for unknown population parameters.

• Examples:

  • $\overline{X}=\frac{1}{n}\sum X_i$ (Sample Mean).
  • $X_{(n)}=max\{X_1,\dots ,X_n\}$ (Sample Maximum).
  • $S^2$ (Sample Variance).

• Footnotes