Symmetry

  • This video discusses symmetry and skewness as they relate to the shape of a distribution. We use histograms, stem plots, and box plots to talk about these concepts.

  • A distribution is symmetrical if it can be divided into two halves of equal size and the same shape (mirror images).

  • In a symmetrical distribution, the plane of symmetry runs through the center, and the mean is the balance point that lies at the center as well.

  • In the example given, the symmetric case has both the mean and the median equal to the central value: μ=x~=12.\mu = \tilde{x} = 12.

  • The mean is described as the balance point of a distribution.

Skewness

  • Skewness refers to asymmetry in a distribution.

  • Distributions can be skewed to the left or to the right.

  • Skewness is interpreted by the direction in which the data cluster or by the tail of the distribution.

  • A distribution is skewed to the left if it has a long tail that trails toward the left.

  • A distribution is skewed to the right if it has a long tail that trails toward the right.

  • The direction of skewness can be read from the plot (histogram, stem plot, box plot) to determine where the data are more spread out.

Stem plots and skewness

  • The same left/right skew concepts apply to stem plots.

  • A stem plot can be judged for skewness by flipping it onto its side to resemble a number line, with the leftmost value being the smallest and values increasing to the right.

  • If the stem plot has a longer tail on the right, it is skewed to the right; if it has a longer tail on the left, it is skewed to the left.

Boxplots and skewness

  • When determining skewness from a boxplot, outliers can affect interpretation.

  • Example: a regular boxplot might suggest left skew, but a modified boxplot could reveal right skew for the same data.

  • A strategy for determining skew in boxplots:

    • If the two boxes (quartile boxes) are unequal in size, the side of the box that is larger indicates the direction of skew.

    • If the boxes are equal in size, then look at the whiskers to determine skewness. The longer whisker indicates the skew direction (left or right).

    • If the boxes are equal in size and the whiskers are also equal in length, the distribution is symmetrical.

Practical example with a histogram

  • Histograms show frequency by bars.

  • By counting bars to the right and left of the central value (12 in the example), we can infer skewness.

  • In the given description, to the right of 12 there are more data values than to the left of 12, implying a shift in the center.

  • The median in this skewed example is located between 16 and 18, i.e., 16 < \tilde{x} < 18.

  • The mean, being the balance point, will be influenced by skewness and move toward the tail.

Mean and median in relation to symmetry and skewness

  • In a symmetrical distribution:

    • The plane of symmetry is at the median (the middle data point).

    • The mean is the balance point, so the mean equals the median: μ=x~.\mu = \tilde{x}.

    • In the symmetric example, both are equal to a central value (e.g., 12): μ=x~=12.\mu = \tilde{x} = 12.

  • In a skewed distribution, the mean and median are no longer equal:

    • Skewed to the left (tail toward the left): the mean is less than the median, i.e., \mu < \tilde{x}.

    • The mean tends to be pulled toward the left tail; the median sits closer to the right side of the distribution.

    • Skewed to the right (tail toward the right): the mean is greater than the median, i.e., \mu > \tilde{x}.

    • The mean tends to be pulled toward the right tail; the median sits closer to the left side of the distribution.

  • Summary relationships:

    • Symmetric: μ=x~.\mu = \tilde{x}.

    • Left-skewed: \mu < \tilde{x}. (mean closer to left tail, median closer to right side)

    • Right-skewed: \mu > \tilde{x}. ean closer to right tail, median closer to left side)

Key takeaways

  • Use histograms, stem plots, and box plots to assess symmetry and skewness.

  • Skewness is about asymmetry and is read from the direction of the tail or the concentration of data.

  • Boxplots require careful interpretation: unequal boxes point to skewness via the larger side; equal boxes require examining whiskers.

  • The relationship between mean and median reveals skewness: symmetry implies μ=x~\mu = \tilde{x}, left-skew implies \mu < \tilde{x}, and right-skew implies \mu > \tilde{x}.

  • In an example with central value 12, a symmetric distribution has both mean and median at 12; a skewed distribution shifts these values with the median landing between 16 and 18 in the described histogram.