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  • STA258 Lecture 03 Raw

Normal Q-Q Plot

• If it's closer to a straight line, it's normal.
• Otherwise the more dispersed they are, they're less normal.
• Important, because if we can show that our sample is similar to a Normal Distribution, we can then use normal distribution facts to draw conclusions.
• Ex:
  • We have 10 observations
  • $X_{(j)}$ is ordered
  • Compute empirical probabilities
  • Quantiles are:
    • $P(Z\le z_0)=0.65$ as an example in the bottom.
      • $⟹P(Z\le q)=\dots ⟹q=\dots$
    • You can do this manually or through R.
    • $z_0=0.385$
  • $\begin{array}{ccc}\text{Ordered Observation}& \text{Probability Level}& \text{Standard Normal Quantiles}\\ X_{(j)}& \frac{(j-\frac{1}{2})}{n}& q_{(j)}\\ -1& 0.05& -1.645\\ -0.1& 0.15& -1.036\\ 0.16& 0.25& -0.674\\ 0.41& 0.35& -0.385\\ 0.62& 0.45& -0.125\\ 0.80& 0.55& 0.125\\ 1.26& 0.65& 0.385\\ 1.54& 0.75& 0.674\\ 1.71& 0.85& 1.036\\ 2.3& 0.95& 1.645\end{array}$
  • Theoretical Quantiles vs Empirical (Sample) Quantiles

    \begin{document}
      \begin{tikzpicture}[scale=1.5]
        % Draw grid for better readability
        \draw[very thin, gray!30] (-2.5,-2) grid (2.5,3);
    
        % Draw Axes
        \draw[->, thick] (-2.5,0) -- (2.5,0) node[right] {Theoretical Quantiles ($q_{(j)}$)};
        \draw[->, thick] (0,-2) -- (0,3) node[above] {Sample Quantiles ($X_{(j)}$)};
    
        % Draw Tick Marks
        \foreach \x in {-2,-1,1,2}
          \draw (\x, 0.1) -- (\x, -0.1) node[below] {\small \x};
        \foreach \y in {-1,1,2}
          \draw (0.1, \y) -- (-0.1, \y) node[left] {\small \y};
    
        % Reference Line (y = x + 0.65 approx)
        \draw[dashed, blue, thick] (-2.2, -1.55) -- (2.2, 2.85);
    
        % Data Points
        \filldraw[black] (-1.645, -1.0) circle (1.5pt);
        \filldraw[black] (-1.036, -0.1) circle (1.5pt);
        \filldraw[black] (-0.674, 0.16) circle (1.5pt);
        \filldraw[black] (-0.385, 0.41) circle (1.5pt);
        \filldraw[black] (-0.125, 0.62) circle (1.5pt);
        \filldraw[black] (0.125, 0.80) circle (1.5pt);
        \filldraw[black] (0.385, 1.26) circle (1.5pt);
        \filldraw[black] (0.674, 1.54) circle (1.5pt);
        \filldraw[black] (1.036, 1.71) circle (1.5pt);
        \filldraw[black] (1.645, 2.3) circle (1.5pt);
    
        % Legend or Title
        \node[draw, fill=white, font=\small] at (1.2, -1.5) {Sample Size $n=10$};
      \end{tikzpicture}
    \end{document}
    

• To plot a Q-Q in R.

• obs = c(-1,-0.1,...)
  probs = (seq(1,10)-0.5)/length(obs)
  plot(obs, probs)
  norm.quantiles = qnorm(probs) 
  plot(norm.quantiles, obs)