Originally posted on http://www.howtomeasureanything.com/forums/ on Thursday, April 30, 2009 6:20:57 AM.

“Hi Douglas,

I want to thank you for your work in this area .Using the information in your book I used Minitab 15 and created an attribute agreement analysis plot. The master has 10 correct and I then plotted 9,8,7,6,5,4,3,2,1,0. From that I can see the overconfidence limits you refer to in the book. Based on the graph there does not appear to be an ability to state if someone is under-confident. Do you agree?

Can you assist me in the origin of the second portion of the test where you use the figure of -2.5 as part of the calculation in under-confidence?
I want to use the questionnaire as part of Black Belt training for development. I anticipate that someone will ask how the limits are generated and would like to be prepared.

Thanks in advance – Hugh”

The figure of 2.5 is based on an average of how confidently people answer the questions. We use a binomial distribution to work out the probability of just being unlucky when you answer. For example, if you are well-calibrated, and you answer an average of 85% confidence (expecting to get 8.5 out of 10 correct), then there is about a 5% chance of getting 6 or less correct (cumulative). In other words, at that level is is more likely that you were not just unlikely, but actually overconfident.

I took a full distribution of how people answer these questions. Some say they are an average of 70% confident, some say 90%, and so on. Each one has a different level for which there is a 5% chance that the person was just unlucky as opposed to overconfident. But given the average of how most people answer these questions, having a difference of larger than 2.5 out of 10 between the expected and actual means that there is generally less than a 5% chance a calibrated person would just be unlucky.

It’s a rule of thumb. A larger number of questions and a specific set of answered probabilities would allow us to compute this more accurately for an individual.