UPDATE (June 27, 11 a.m.): Monday morning the Supreme Court struck down Texas’s stringent abortion regulations in a 5-3 decision. The court found that requiring doctors who perform abortions to have admitting privileges at a nearby hospital and requiring abortion facilities to meet the standards of surgical centers constituted an undue burden. Earlier this year, when the case was debated in oral argument, I looked at what qualified as “undue burden,” a hazy term that dated to a 1992 Supreme Court decision in Planned Parenthood v. Casey. At its core, the “undue burden” definition was a math problem — one that was poorly defined.

But Monday, in the majority decision in Whole Woman’s Health v. Hellerstedt, Justice Stephen Breyer articulated how it should be calculated: “Here, as in Casey, the relevant denominator is ‘those [women] for whom [the provision] is an actual rather than an irrelevant restriction.'” In other words, whether the burden of the law is “undue” should be judged by the effect it has on women in Texas seeking abortions who would be deterred by the law, not all women in Texas or even all women seeking abortions in Texas. That’s a definition that makes it easier to overturn any future restrictions on abortion access.

You can read the original article, from March, about what defines an undue burden below.

The Supreme Court heard arguments Wednesday in Whole Woman’s Health v. Hellerstedt, a case on Texas’s abortion restrictions that might result in a deadlock in the absence of Justice Antonin Scalia, who died Feb. 13. Because of court rules, a tie decision wouldn’t become precedent, which would mute its significance nationally. But even a 4-4 ruling could resonate for years: It could give all courts a clearer sense of how to apply something called the “undue burden” standard that’s at the heart of the case.

In 1992, the Supreme Court decided Planned Parenthood v. Casey and changed the way that the legality of abortion restrictions is judged. According to that decision, which upheld Pennsylvania’s parental notification laws for minors but struck down spousal notification requirements, laws cannot place an “undue burden” on women seeking an abortion. What’s an undue burden? It’s something that prevents a “large fraction” of women from having an abortion. And what’s a “large fraction”? Well, there’s the problem.

I reached out to advocates on each side of the abortion debate and went through cases to try to find out how many people a law had to affect before its impact qualified as an “undue burden.” I found that not only are there very few cases that cite specific numbers, but even a circuit court that had unusually good data to work with got snarled when it tried to do the math.

In 2006, the 6th U.S. Circuit Court of Appeals tried to figure out how much of a problem it was for women in Ohio to have to meet with a doctor at least 24 hours before having an abortion. Some clinics had already adopted that rule voluntarily, so the court had real data about how many women it had deterred.

For every 1,000 women who sought an abortion at one of those clinics, 50 to 100 received an exemption from the meeting because coming in twice would be a hardship. About a quarter of these exemptions went to women in abusive situations. Half of the women experiencing abuse said that if they hadn’t gotten an exemption, they couldn’t have had the abortion (six to 13 women out of 1,000).

So, is that enough women to count as a “large fraction”? It depends on the universe of women you look at.

In Casey, the Supreme Court said what it thinks the denominator should be. The majority opinion, written by Justices David Souter, Sandra Day O’Connor and Anthony Kennedy, said that burden should be measured for women “for whom the law is a restriction, not the group for whom the law is irrelevant.” When the Supreme Court set this standard in Casey, it said the right way to judge the law’s spousal notice requirement was to consider how it affected women who didn’t want to tell their husbands about an abortion, rather than how it affected all women.

Which brings us back to the 6th Circuit. In that case, the state argued that only six to 13 women out of 1,000 seeking abortions would wind up not having them as a result of the additional required doctor visit; less than 1 percent wasn’t a “large fraction.”

Those opposing the law said this was the wrong comparison to make. The lawyer opposing Ohio’s law argued that those six to 13 women in 1,000 who would lose the chance to have an abortion should be compared to the 50 to 100 women who had been exempted from the extra meeting (six out of 50 or 13 out of 100 is about 12.5 percent).

The court rejected both these approaches. Instead — and stick with me here — the judges wanted to compare the number of women deterred by the extra meeting to all the women who asked for an exemption (rather than all women who received an exemption). That latter number wasn’t available, so the judges couldn’t determine the fraction of women affected by their own standard. But they knew that because they were considering a larger universe of women, the answer was lower than 12.5 percent. And 12.5 percent, the court decided, didn’t count as a “large fraction” anyway, so they upheld the part of the law requiring the meetings.

The final opinion from the 6th Circuit sounds a little apologetic about the calculations, conceding that the standard was “more conceptual than mathematical.”

In the new case being argued in front of the Supreme Court on Wednesday, both sides were getting stuck on how to choose a universe of women again. Texas has required abortion clinics to meet stringent health and safety requirements, and many clinics have closed in the wake of the law. Justice Ruth Bader Ginsburg and Texas Solicitor General Scott Keller argued about what the appropriate denominator is to test whether Texas’s law burdens a “large fraction” of women. Ginsburg argued that women who live near still-open clinics shouldn’t be counted as affected (excluding these women would make the fraction larger), and Keller argued that, because the Texas law places restrictions on clinics, not individual women, the correct denominator is the one that the 5th Circuit used in this case: all Texas women of reproductive age (such a large universe of women would make it unlikely that a “large fraction” of women would be burdened).

Even if the case doesn’t establish a precedent on abortion laws, here’s hoping it offers a clearer way to do the math to meet the standards that the court has already set.

CORRECTION (March 3, 12:14 p.m.): An earlier version of this article misstated the author of the Supreme Court’s opinion in Planned Parenthood v. Casey. It was a joint opinion from Justices O’Connor, Kennedy and Souter, not Justice Souter alone.

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