Is there a more tragic story than an innocent person going to prison? Tragic, and powerful. That’s why The Shawshank Redemption is one of the most beloved movies of our time. And why we’ve all heard of this quote from an esoteric 18th-century English guy, William Blackstone: “It is better that ten guilty persons escape than that one innocent suffer.” And why real-life stories of the exonerated always make headlines. Here’s the first line of a Washington Post story about Glenn Ford, who was exonerated last month:

“My sons, when I left, was babies,” Louisiana’s longest-serving death row inmate told reporters after his release late Tuesday. “Now they’re grown men with babies.”

It hits you in the gut. You first think about this particular person, this man who lost his family, who spent decades in some awful cell believing he was going to be electrocuted. And then you think that other frightening thought, the bugaboo lurking behind all exoneration stories: How many other Glenn Fords are still behind bars? How many will die there? Just how often does our venerated justice system fail?

Rarely, at least according to U.S. Supreme Court Justice Antonin Scalia. In a 2006 opinion he cited an approximate error rate of 0.027 percent, based on back-of-the-envelope calculations by an Oregon district attorney in a fiery op-ed for the New York Times. The op-ed was in response to a report by Samuel Gross, a law professor at the University of Michigan, cataloguing 340 exonerations between 1989 and 2003. “Let’s give the professor the benefit of the doubt,” the op-ed read. “Let’s assume that he understated the number of innocents by roughly a factor of 10, that instead of 340 there were 4,000 people in prison who weren’t involved in the crime in any way. During that same 15 years, there were more than 15 million felony convictions across the country. That would make the error rate .027 percent — or, to put it another way, a success rate of 99.973 percent.”

But that claim, Gross writes in today’s issue of the Proceedings of the National Academy of Sciences, “is silly.” Here’s the problem with its logic. The known exonerations were almost all murder and rape cases, which get much more post-conviction attention, whereas the total number of felonies also includes burglary, car theft, tax fraud, and drug possession. Some 95 percent of felony convictions are the result of plea bargains, with no formal evidence ever presented, and most never bother with an appeal.

There’s a more rigorous way to crunch the numbers, according to Gross’s new study. And that approach leads to a false conviction rate that was high enough to make me gasp — 4.1 percent.

To be more precise: Gross and his colleagues calculated a 4.1 percent error rate among people who are sentenced to death. This is a small subset (less than 0.1 percent) of the total number of prison sentences but, because of the stakes, these cases are scrutinized far more than most. For capital cases, Gross writes, “everyone from defense lawyers to innocence projects to governors and state and federal judges is likely to be particularly careful to avoid the execution of innocent defendants.”

Because of all of the resources spent on capital cases, the researchers reason, it’s likely that many (and perhaps a majority) of innocent defendants on death row will ultimately be exonerated. Calculating this rate would then give an approximate, albeit low estimate of the real false conviction rate.

Even with this small subset, though, getting a reasonable estimate is tricky, for two reasons. One is that the exoneration process takes time. To date 143 people on death row have been exonerated, and their time spent on death row ranged from 1 to 33 years, with an average of 10. That means there are some number of innocent people on death row today who have not yet been exonerated but will be in the future.

The other issue is that some number of innocent people on death row will have their sentence reduced to life in prison but will never be freed. (Why? Because once they’re off death row, their case is no longer under such intense scrutiny and exoneration is unlikely.)

Gross and his colleagues collected data on the 7,482 people who were sentenced to death between 1973 — the first year of modern death-penalty laws — and 2004. Of these, 117 were exonerated, or 1.6 percent. But among these, 107 were exonerated while they were still on death row, whereas only 10 were exonerated after their sentence had been reduced to life in prison.

This leads to a bizarre situation. If you’re on death row and your sentence is reduced to life in prison, you’re suddenly much less likely to be exonerated than someone who stays on death row.

To account for the X number of innocent defendants who were lost to life in prison, Gross’s team borrowed a statistical approach from medical research called “survival analysis.” Researchers typically track how long it takes a group of people with some disease, say diabetes, to achieve a certain outcome, such as death. Survival curves for diabetics on some new drug treatment, say, can then be compared with those on no treatment.

In this study, the “disease” is being on death row, the “outcome” is exoneration, and the “treatment” is being moved to life in prison. (Just as a medical treatment is likely to lower a diabetic’s risk of death, being moved to life in prison lowers the risk of exoneration.) Using this mathematical approach*, the researchers were able to calculate the cumulative risk of exoneration for the “untreated” population — that is, people who were never removed from death row. Because these cases get the most intense scrutiny, the researchers say, this is the closest we can get to the true rate of death-row false convictions. And that rate is 4.1 percent.

All of this assumes that the rate of innocent people being sentenced to death hasn’t changed over the past four decades. The authors say there’s no specific evidence to support a change, though it’s true that we’re sending far fewer people to death row than we used to. I thought that improvements in DNA technology might have lowered our error rate, but the researchers don’t think it makes much of a dent. Just 18 of the 142 exonerations since 1973 were thanks to DNA testing.

So then the next terrifying question is, geeze, how many innocent people have actually been executed?

Fortunately it’s probably not many. Innocent defendants are far more likley to have their sentenced changed to life in prison than to be executed. Still, with an error rate of 4 percent, the researchers write, “it is all but certain that several of the 1,320 defendants executed since 1977 were innocent.”

It’s impossible to say whether this 4.1 percent false conviction rate applies to defendants who never went to death row. But I’ll leave you with one last depressing thought. Of all of the people found guilty of capital murder, less than half actually get a death-penalty sentence. And when juries are determining whether to send a defendant to death row or to life in prison, surveys show that they tend to choose life sentences when they have “residual doubt” about the defendant’s guilt.

That means, then, that the rate of innocent defendants serving life in prison is higher than those on death row. “They are sentenced,” the authors write, “and then forgotten.”

*If you want to know about the particular equations involved in the survival analysis, check out Gross’s paper or the Wikipedia article on the method.