Simpson's paradox

No, the Simpson 's paradox is not named for Homer , but to Edward Simpson , the statistician who described for the first time in 1951. It is one of those paradoxes that mathematics can we make knots to the head, but which unfortunately is more than just a curiosity to understand this paradox may be essential to make the right decisions! So if you do not know this statistical phenomenon very against -intuitive , read on , and the arms should you fall!
 Kidney stones: which treatment to choose?

No luck, we just discovered your calculations to the kidney. Fortunately treatments exist, and at the hospital the doctor presents in two. The first (call "Treatment A") is in an open surgery, while the second ("Treatment B") is a surgery that is done through small holes drilled through the skin. The doctor asks you what treatment you prefer. As desired above all to heal, you ask the successful practitioner statistics of these two treatments.
"Oh it's very simple, you answered the doctor, both treatments were tested every 350 patients, and here are the numbers: A treatment has worked in 273 cases and treatment B 289".
 The case seems understood, treatment B market with 83% success, against 79% only for treatment A. So you choose the treatment B. But just leaving the hospital, you cross another doctor who you ask his opinion about treatment. "Oh it's very simple, you he answers: both treatments were tested 350 times on each patient, the latter can be achieved either 'small' calculations, or 'big' calculations, and here are the figures" :

 

As you can see, if you have large stones, treatment A works best , and if you have small stones , treatment A is also the most effective. That is in complete contradiction to what you said the first doctor. And yet you have good counting and recounting , the "Total" line , it is indeed the same figures as those presented by the first doctor ...  
How is it possible that treatment B is better to global, but it is lower than both treatment A small than large stones? And it is not a joke , these numbers are from a real study [1 ] ! There are no statistics entourloupe or no manipulation , what you read there, that's the reality of the figures . You have a fine example of Simpson 's paradox.