Judges Gone Wild: Revised

I want to revise my previous post, Judges Gone Wild. First, the numbers are in error. I left out a term in my calculation. That term was near unity, so it won't make a huge difference. That term belonged in the denominator, however, so the corrected answer I will soon present below is actually more distressing.

I also decided that I tried to do too much with a single plot. I've therefore created two to replace the one. I now present them below, one for the judges ... 

 ... and one for the juries.

Before I started this, I would have been appalled by the thought of an innocent person having more than a 25% chance of wrongful conviction. I'm still appalled, but I'm now admittedly distracted by calculations that suggest an innocent person has substantially more than a 50% chance of of being convicted during a bench trial. The odds range from a "low" of 50% for forcible rape to nearly 84% for muder/manslaughter. Since judges convict 86% of all defendants in the murder/manslaughter category, the number indicates that judges are incapable of distinguishing cases of innocence from cases of guilt.

WARNING: Mathematical Addendum

At least with respect to the plot for the juries, Spencer and I agree reasonably well on the overall chance of being convicted by a jury despite being innocent. He calculates 25% for all trials combined. I calculate 28%. 

We disagree substantially, however, with respect to the overall chance of being convicted by a judge despite being innocent. He calculates 37% for all trials. I calculate 62%. The discrepancy can be understood by looking at the terms that go into the calculation.

The first term is the convictions per trial. We extract that number directly from the judge / jury agreement table. For Spencer and the NCSC data, that number was 80%.  For my work and the Kalven-Zeisel data, the number was 83%.  The two sets of data were close on this number, despite being separated in time by half a decade. 

The second term was the percentage of innocent defendants. Spencer calculated 28%. I calculated 26%. Again we are close. 

The third term was the wrongful conviction rate. Recall that the wrongful conviction rate is the number of wrongful convictions divided by the number of convictions. It is one of the primary results from our analysis of the judge / jury agreement data.  Spencer calculated 13% for judges from the NCSC data, using his methods. I calculated 20% for judges from the Kalven-Zeisel  using my methods. That's our largest discrepancy. The comparison is not great, but not terrible.

When we perform the final calculation, however, the differences combine in such a fashion as to amplify the difference of the results.

Using Spencer's numbers:  0.80 x 0.13 / 0.28  =  .37  =  37%

Using my numbers:  0.83 x 0.196 / 0.26  =  0.64  =  64%

As it turned out, when my numbers were larger, they were in the numerator. When my number was smaller, it was in the denominator. The result was a substantial difference between our calculated values for the chance of being convicted by a judge despite being innocent.