Boxplot

Boxplots display individual outliers and robust statistics for the data, useful for identifying outliers and for comparisons of distributions across subgroups.

• Boxplots follow the 5-data summary

  • min, first quartile/ lower hinge, median, third quartile/ upper hinge, max

    • Sometimes quartiles are different than hinges

      • Hinge
        • The lower hinge is the median of the lower half of the data up to the median, and including the median if and only if the median is a real data point
          • i.e., the median is not the average of two points; i.e., the size of the data is odd
        • The upper hinge is the median of the upper half of the data up to the median, and including the median if and only if the median is a real data point
      • Quartile
        • Quartiles are special percentiles
        • For $k$th percentile:
          1. Rank the data
          2. Find $k$% of the sample size
          3. If this is an integer, add 0.5; if it isn't an integer round up
          4. Find the datum in this position; If your depth ends in 0.5, then take the midpoint between the two data points
      • Therefore, the hinges are the same as the quartiles unless the remainder when dividing the sample size by 4 is 3

• Area != number of data points; each area contains the same number of points

  • Area represents the density

• Outliers: data outside the fences

  • fences: upper-hinge + 1.5 x H-spread, lower-hinge - 1.5 x H-spread
  • H-spread/ interquartile/ forth spread: upper-hinge - lower-hinge

• A multiple-class boxplot should be re-ordered by the median

Elements

• Box width
  • A disadvantage of boxplots is that they don't convey information on how big the different groups are
  • Set varwidth = TRUE in ggplot2, the width of boxes will be proportional to $\sqrt{n}$
• Extreme outliers
  • We can define outer fences to be over 3 times the box length away from the box; then outliers outside outer fences are extreme outliers

by zcysxy