Value of whisker in boxplot for 99.7 coverage

Question

I am trying to identify outliers from a boxplot using MATLAB. The function has a default whisker value of 1.5 that provides +- 2.7*sigma or 99.3 coverage. However, I want 99.7 or 3*sigma coverage. What could be the value of whisker in this case? I did not want to make a random guess, so need help from you guys. Thanks

Answer 1

In general, let:

Q1 = icdf('norm',0.25,0,1);
Q3 = icdf('norm',0.75,0,1);
IQR = Q3-Q1;

• Now if you have a constant k (BOXPLOT by default has k=1.5 for the whisker length), then the IQR outlier test identifies values outside the range: [Q1 - k*IQR, Q3 + k*IQR] as outliers, which corresponds to:

  >> k = 1.5;
  >> sdCov = [Q1 - k*IQR, Q3 + k*IQR]      %# +/-2.698*sigma coverage
  sdCov =
         -2.698        2.698
  

  or (in terms of area under the curve):

  >> area = 2*normcdf(sdCov(2), 0, 1)-1    %# 99.3% coverage
  area =
         0.99302
  

• In the opposite direction, if you want a sdCov*sigma coverage, then:

  >> sdCov = 3;
  >> k = (Q1+sdCov)/IQR
  k =
         1.7239
  

  or:

  >> area = 0.9973;
  >> sdCov = norminv(1-(1-area)/2);
  >> k = (Q1+sdCov)/IQR
  

  Therefore use the following in your case:

  boxplot(data, 'whisker',1.7239)
  

Here is an illustration borrowed from Wikipedia: