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Boxplots and Outliers

Boxplots

• Here are the directions for drawing a boxplot:
  
  1. Compute Q1, Q2 and Q3. Also, compute the interquartile range IQR = Q3 - Q1.
     Example:  Suppose that the dataset consists of these hypothetical test scores:
     
     • >5  39  75  79  85  90  91  93  93  98

       Q1 = 75, Q2 = 87.5, Q3 = 93. IQR = 93 - 75 = 18.

  2. Draw three horizontal lines, all of the same length and all starting at the same x-value: one at height Q1, the second at Q2 and the third at Q3.
     Example:   Boxplot after Step 2.
  3. Draw two vertical lines, one at connecting the left endpoints of the lines and the other connecting their right endpoints.
     Example:   Updated boxplot after Step 3.
  4. Compute the inner fences IF1 = Q1 - 1.5 * IQR and IF2 = Q3 + 1.5 * IQR.
     Example: The inner fences are
     
     • IF1 = 75 - 1.5 * 18 = 48 and IF2 = 92 + 1.5 * 18 = 119.
  5. Draw a whisker downward from Q1 to IF1 or Q0, whichever comes first. Draw a whisker upward from Q3 to IF2 or Q4, whichever comes first.
     Example:   Boxplot after adding the whiskers in Step 4.
  6. Compute the outer fences OF1 = Q1 - 3 * IQR and OF2 = Q3 + 3 * IQR.
     Example:   The outer fences are
     
     • OF1 = 75 - 3 * 18 = 21 and OF2 = 92 + 3 * 18 = 146.
  7. Extreme outliers are observations that are beyond one of the outer fences OF1 or OF2. Mark any extreme outliers on the boxplot with an asterisk (*).
     Example:   The only observation less than OF1 = 21 is 5.
     Boxplot after marking 5 with a *.
  8. Mild outliers are observations that are between an inner and outer fence. Mild outliers are marked with a circle (O).
     Example:   The only observation that is between an inner fence and an outer fence is 39, which is between IF1 = 48 and OF1 = 21.
     Boxplot after marking 39 with a O.
• Compare your boxplot with one constructed by R from the same data.

Mild vs. Extreme Outliers

• Extreme outliers are data points that are more extreme than Q1 - 3 * IQR or Q3 + 3 * IQR.
• Extreme outliers are outside of either outer fence.
• Extreme outliers are marked with an asterisk (*) on the boxplot.
• Mild outliers are data points that are more extreme than than Q1 - 1.5 * IQR or Q3 + 1.5 * IQR, but are not extreme outliers.
• Mild outliers are outside of either inner fence, but not outside of any outer fence.
• Mild outliers are marked with a circle (O) on the boxplot.