Symmetry and Skewness

Key Concepts

• Symmetry and Skewness relate to the shape of a distribution.
• Tools used to display distributions include histograms, stem plots, and box plots.

Symmetric Distributions

• A distribution is symmetric if it can be divided into two equal halves of the same shape.

Skewed Distributions

• Skewness refers to the asymmetry in a distribution.
• Left-Skewed: Longer tail on the left side.
• Right-Skewed: Longer tail on the right side.
• Tools to assess skewness:
  • Histograms: Visual representation of frequency.
  • Stem Plots: Can be flipped onto their side; align stems like a number line.
  • Box Plots: The size of boxes and lengths of whiskers indicate skewness.
    • Larger box side or longer whisker determines the skew direction.

Interpreting Skewness in Box Plots

• Modified Box Plot: Consider outliers to interpret skewness better.
• Unequal boxes:
  • Larger side determines skew.
• Equal boxes:
  • Longer whisker determines skew.

Effects on Mean and Median

• Symmetrical Distribution:
  • Mean equals the median.
• Left-Skewed Distribution:
  • Mean < Median (mean closer to left tail).
• Right-Skewed Distribution:
  • Mean > Median (mean closer to right tail).

Practical Examples

• Example with histograms and skewness discussion based on the position of values relative to 12.
• Illustrations using flipped stem plots and modified box plots to highlight the skew direction.