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  • STA258 Pre-Lecture 02

Skewness

• This is a measure of how much a distribution leans towards a particular side.
  • Symmetric
    • Observations are concentrated around the mean and fairly equally on both sides of the mean.
    • When Mode $=$ Median $=$ Mean
  • Right skewed / Positive
    • We observe a tail to the right side of the mean.
    • There are more observations on the smaller values
    • Think of it as stretching to the positive or the right side
    • When Mode $<$ Median $<$ Mean
  • Left skewed / Negative
    • A tail on the left side of the mean
    • More observations on larger values
    • Think of it as stretching to the negative or the left side
    • When Mode $>$ Median $>$ Mean
  • None
    • Can't determine visually
• We can see skewness by looking at a Histogram of our data.
• We can find it out by evaluating the relation between Expected Value Median and Mode.
• We can find it out by checking Box Plots too.
• (2. Descriptive Statistics, p.48)
• (2. Descriptive Statistics, p.49)

\begin{document}
  \begin{tikzpicture}[font=\sffamily, scale=1.2]

    % Axis
    \draw[->, thick] (0,-1) -- (10,-1) node[right] {Value};
    \foreach \x in {0,2,4,6,8,10}
      \draw (\x,-1.1) -- (\x,-0.9) node[below=5pt] {\x};

    % --- Symmetric ---
    \begin{scope}[shift={(0,4)}]
      \node[anchor=west] at (0,1.2) {\textbf{Symmetric} (Mean $\approx$ Median)};
      % Whiskers
      \draw[thick] (2,0.5) -- (4,0.5);
      \draw[thick] (6,0.5) -- (8,0.5);
      \draw[thick] (2,0.3) -- (2,0.7);
      \draw[thick] (8,0.3) -- (8,0.7);
      % Box
      \draw[thick, fill=blue!10] (4,0.2) rectangle (6,0.8);
      % Median
      \draw[very thick, red] (5,0.2) -- (5,0.8) node[above] {M};
    \end{scope}

    % --- Right Skewed ---
    \begin{scope}[shift={(0,2)}]
      \node[anchor=west] at (0,1.2) {\textbf{Right-Skewed} (Mode $<$ Median $<$ Mean)};
      % Whiskers
      \draw[thick] (1,0.5) -- (2,0.5);
      \draw[thick] (5,0.5) -- (9,0.5);
      \draw[thick] (1,0.3) -- (1,0.7);
      \draw[thick] (9,0.3) -- (9,0.7);
      % Box
      \draw[thick, fill=green!10] (2,0.2) rectangle (5,0.8);
      % Median
      \draw[very thick, red] (3,0.2) -- (3,0.8) node[above] {M};
      % Mean Marker (approximate)
      \draw[blue] (4.5,0.5) circle (2pt) node[below=2pt] {$\bar{x}$};
    \end{scope}

    % --- Left Skewed ---
    \begin{scope}[shift={(0,0)}]
      \node[anchor=west] at (0,1.2) {\textbf{Left-Skewed} (Mean $<$ Median $<$ Mode)};
      % Whiskers
      \draw[thick] (1,0.5) -- (5,0.5);
      \draw[thick] (8,0.5) -- (9,0.5);
      \draw[thick] (1,0.3) -- (1,0.7);
      \draw[thick] (9,0.3) -- (9,0.7);
      % Box
      \draw[thick, fill=orange!10] (5,0.2) rectangle (8,0.8);
      % Median
      \draw[very thick, red] (7,0.2) -- (7,0.8) node[above] {M};
      % Mean Marker (approximate)
      \draw[blue] (5.5,0.5) circle (2pt) node[below=2pt] {$\bar{x}$};
    \end{scope}

  \end{tikzpicture}
\end{document}