Shameful Innumeracy in the New York Times

I've been writing this blog for a long time - nearly four years. You'd think that
after all of the bad math I've written about, I must have reached the point where
I wouldn't be surprised at the sheer innumeracy of most people - even most supposedly
educated people. But alas for me, I'm a hopeless idealist. I just never quite
manage to absorb how clueless the average person is.

Today in the New York Times, there's an editorial which talks about
the difficulties faced by the children of immigrants. In the course of
their argument, they describe what they claim is the difference between
the academic performance of native-born versus immigrant children:

Whereas native-born children's language skills follow a bell
curve, immigrants' children were crowded in the lower ranks: More than
three-quarters of the sample scored below the 85th percentile in English
proficiency.

Scoring in the 85th percentile on a test means that you did better on that
test than 85 percent of the people who took it. So for the population as a
whole
, 85% of the people who took it scored below the 85th percentile -
by definition. So, if the immigrant population were perfectly matched
with the population as a whole, then you'd expect more than 3/4s the
score below the 85th percentile.

As they reported it, the most reasonable conclusion would be that on the
whole, immigrant children do better than native-born children! The
population of test takers consists of native-born children and immigrant
children. (There's no third option - if you're going to school here, either
you were born here, or you weren't.) If 3/4s of immigrant children are scoring
85th percentile or below, then that means that more than 85% of
the non-immigrant children are scoring below 85th percentile.

I have no idea where they're getting their data. Nor do I have any idea of
what they thought they were saying. But what they actually said is a
mind-boggling stupid thing, and I can't imagine how anyone who had the most
cursory understanding of what it actually meant would miss the fact that
the statistic doesn't in any way, shape, or form support the statement it's
attached to.

The people who write the editorials for the New York Times don't even
know what percentiles mean. It's appalling. It's worse that appalling - it's
an absolute disgrace.

• Brian says:

While you're of course right, to be pedantic about it, it does depend on how much greater than 3/4 scored below the 85th percentile.

• GDC says:

I predict that next, they will reveal the appalling statistic that fully half of them scored below average!

• Mike says:

GDC, it's more appalling if they reveal half of them scored below the median. FAIL.

• Shawn Smith says:

I guess the New York Times editorial staff lives in Lake Wobegone, where all the children are above average. It's just typical journalism school innumeracy. Getting worked up about it is probably more pointless than trying to convince Vorlath that infinitely sized sets can have the same cardinality even if one is a proper subset of another.

• Baldeagle says:

Maybe it was a typo... it could be the journalist scribbled '35th percentile' and mis-read it later...

• Rich says:

Some well established federal organizations (ie, NIH) do percentiles backwards. That is, 85 percentile score better than 15%. Maybe that's what's going on....

• justawriter says:

Whereas native-born children's language skills follow a bell curve, immigrants' children were crowded in the lower ranks: More than three-quarters of the sample scored below the 85th percentile in English proficiency.

I think it is likely, given the lead in phrase, that the author meant 15th percentile.

• Alex R says:

You wrote: "I have no idea where they're getting their data. Nor do I have any idea of what they thought they were saying."
The cool thing about the modern web is that you can often figure out what the source of a piece of data is. While I don't think I have the exact source of this one, I think the this document (PDF) is pretty close. In it, the authors look at a sample of 274 immigrant students -- probably a subsample of the 400 students referred to in the editorial. This article says

...19 students (or 7.4% of the sample) scored at or above the normed mean for English speakers of the same age on the English Language Proficiency subtest (a standard score of 100 or greater...). ... On average, the sample demonstrated academic English proficiency scores equivalent to the second percentile of native English-speaking peers. Three fourths of participants fell more than one standard deviation (15 points) under the mean.

So, we find that three-fourths of the immigrant students scored more than one S.D. below the mean -- which in a normal distribution, would be about the 16th percentile. But, amusingly, on this test which has a mean score of 100 and a standard deviation of 15 points, this also corresponds to a score of 85 points on the test. While I don't know what the authors of the editorial thought they were saying either, it appears that they thought that "percentile" was just a fancy way of saying "score" -- not realizing that it didn't sound so smart in the given context.

• B-Con says:

Maybe the journalist simply got it backwards. Ie, maybe they had the statistic "over 75% immigrant children score in the bottom 25%" and they translated it into different words to sound better, but they translated it wrong.
@#6:
Wouldn't that mean the other way around? The standard way of doing it is 85% mean that 85% of the population is below you. So the "backwards" way of doing it would be to say that 15% is better than 85%, ie, 15 percentile means that only 15% are better than you.

• Phil says:

Why that's almost half the population.......

• I can't help but think that, for the average person, courses in statistics are more valuable than courses in geometry, calculus and the like.
My mathematical education ended with calculus, which was interesting... but useless to me since the day I left the classroom.
Statistics, on the other hand, comes up over and over and over again, and what little I've absorbed via osmosis has been absolutely essential to me as a reporter.
And even someone as mathematically illiterate as me recognizes the fallacy in the results stated in this article. ðŸ™‚

• anonymous says:

@Alex R: I found that article too, and while that's also my best guess as to where the numbers 3/4 and 85 came from, the NYT editorial also said "more than three quarters" whereas the article says (in the sentence right after your quote) "Only 25.2% of the total sample fell within one standard deviation of the average native English speaker of their age," implying that 74.8% scored below 85 points [1].
So if that is the source, apparently the editors don't understand the meaning of "more than" either.
[1] this assumes 100 is the average of native speakers, although if 100 is the score of everybody that would imply that natives had a higher average, so being one SD away from the native average means a score above 85, which makes the "more than" statement even more wrong.

• staticvars says:

Equally ridiculous was the statement that the scores for the population don't "follow a bell curve", based on where they fall in the distribution for another population. It must be uncomfortable for all of those people "crowded into the lower ranks". We should find some more spacious ranks for them.
If I were an editor, I would have a special flag that wouldn't let any article (or editorial) using the word "percentile" escape without a having a person with at least a high school understanding statistics reviewing it.

• anonymous says:

Incidentally, 1 standard deviation below the mean in a Normal distribution is not too far from the 15th percentile, for those who think the NYT editors just got the percentile definition backwards. Coincidence? I am being way too entertained by this.

• z k says:

I hate nitpicking assholes like you.
''Scoring in the 85th percentile on a test means that you did better on that test than 85 percent of the people who took it."
Every kid in high school knows that this is NOT the case. In many high school and college class rooms it is common for the AVERAGE score to be approxiamtely 85. But, alas, you the "google genius guy" has to use semantics and very strict definition of words to try and make other people sound like idiots, when in actuality you are just a judgmental nerd who never gets chicks. Tool.

• Barry Kelly says:

Good lord, "z k". Have a look at a definition of percentile before you get on your high moral ground.

• e says:

I'm assuming it's a typo

• Dave Munger says:

Z K:
Wow. That was rude, arrogant, *and* ignorant.
If you think the difference between "percentile" and "percentage" is nitpicking, you probably shouldn't be hanging out on a site called "good math, bad math."
I'd suggest sticking to sites with lots of pictures. And no commenting.

• MeBigFatGuy says:

I think the reason the immigrants did so poorly was all those questionable 'sick' days they were taking. It's been reported, that nearly 40% of all sick days -- 40% -- were on monday's and fridays.... 40%!!!!

• Rodrigo says:

Z K did give me a chuckle, saying "you never get chicks" to a married person ðŸ˜€

• You Suck says:

Your personality consistently scores above the 85th percentile on the Insufferable-Douchebag index.

• B-Con says:

Typo in my previous post (#9), I meant:
> "over 75% immigrant children score in the bottom 15%"
not 25%.

• Robert says:

The thing that disturbed me most was not that percentile thing (I'm not really used to that way of representing statistics, and naturally assumed they meant they were below 85% of normal students) but the implication that normal students follow a bell curve and immigrants don't.
That just shouts 'small sample size' to me.
Besides that, I think 'following' a bell curve implies some kind of time-series, wouldn't it be more clear to say 'are distributed along a bell curve'?

• Brian says:

ZK: It appears that the reporter got the numbers exactly backwards and you call that "nitpicking"? That was a Poe, right? Right? I mean ... c'mon.

• Brett Cox says:

Immigrants don't follow a bell curve (Gaussian distribution), most tragically their intelligences instead follow a hyperbolic distribution with a vast concentration of morons at the 0th percentile.

• Janne says:

It's the same everywhere. We had an opinion piece in Sweden's largest newspaper last week arguing against detailed safety regulations since they don't work. As proof the writer brought up smoke alarms, showing that even though 90% of homes had them, still almost half the lethal fires were in homes with smoke alarms. Which, according to the writer, would show that smoke alarms were useless...
I wrote a bit about exactly how that is wrong here:
http://janneinosaka.blogspot.com/2009/11/smoke-detector-stats.html

"Just think how stupid the AVERAGE American is, and then realize that HALF of them are STUPIDER than THAT!"
- George Carlin

• Niall says:

The author may have been referring to the 85th percentile of the immigrant-native population combined.
Lets say that 15% of all 14 year old children get a D or lower in English and 75% of all immigrant 14 year olds get a D or lower.
In this case, 3/4 of immigrant children (of the sample they measured) scored at or less than the 85 percentile of the set of all Yankee children.
Ironically this may be more a case of poor language skills than a genuine mathematical slip.

• christopher says:

If you just reverse the quote to look at the positive side, it clearly sounds like a compliment for immigrant children.
"Nearly a quarter of immigrant children were in the top 15% in English proficiency." That sounds like they're crowded in the upper ranks.

• ASFalcon13 says:

Sorry to dig up the corpse of issues past here, but this reminds me of Richard Cohen's "What Is the Value of Algebra?" op-ed in the Washington Post a few years ago. To review, Richard claims that he's never used algebra and never regretted that he didn't know how. Although it's dealing with statistics instead of algebra, the article Mark's discussing here serves as an interesting counterpoint: if you don't have a fundamental knowledge of math, sometimes you don't even realize that you're attempting to apply it, and you can't identify that you're getting it horribly wrong.
Oh, and z k: I hate nitpicking assholes like you. You came to a blog that has the goal of "Squashing bad math and the fools who promote it". What did you expect? Also, consider that, when you accuse Mark of being, "a judgmental nerd who never gets chicks", you come off looking just as judgemental, nerdy, and lonely on a Saturday night as you think Mark is. Tool.

• Paul M says:

Z K
Like most people who read this blog I actually enjoy it, but ignoranuses like you really grate. What you did was a bit like walking into someones house and insulting them for no reason. Something I'm willing to bet you would be too gutless to actually do for fear of loosing a few more of your teeth. If you actually have something constructive to say then out with it, otherwise please go away.

• Escuerd says:

Niall @ 28,
It doesn't matter whether they were referring to the 85th percentile of the American-born population or of the immigrant population. You'd still expect more than 3/4 to be below the 85th percentile unless the immigrants' English skills were better, which seems both unlikely and like the opposite of what they were trying to get across.

Lets say that 15% of all 14 year old children get a D or lower in English and 75% of all immigrant 14 year olds get a D or lower.
In this case, 3/4 of immigrant children (of the sample they measured) scored at or less than the 85 percentile of the set of all Yankee children.

That would be a lower score than 85% of the population, but that would make it the 15th percentile (very much less than the 85th). If one scores at the nth percentile, that implies that n percent scored lower.

• Escuerd says:

''Scoring in the 85th percentile on a test means that you did better on that test than 85 percent of the people who took it."
Every kid in high school knows that this is NOT the case. In many high school and college class rooms it is common for the AVERAGE score to be approxiamtely 85.

85 percent is not the same as 85th percentile.
Maybe you'd have done a bit better in school if you understood that. But where were we? Oh yes, I'll just have a small order of fries, thanks.

• Mu says:

It looks to me that the author confuses percentile with an IQ type score. Percentile has 100 max, 50 average; IQ has 100 average. So my reading would be that 75% of immigrant children fall in the 42.5 percentile group.

• Al says:

Two thirds of all people don't understand percentiles, .....That's almost half!

• Doug Little says:

In many high school and college class rooms it is common for the AVERAGE score to be approxiamtely 85.

If you mean 85/100 then the teachers need to do a better job of making the exam harder, there is not enough overhead here to really differentiate the exceptional students from the good/average students.

• cm says:

The language used here, both by the Times and by Mark, are unclear. For example, Mark wrote:

Scoring in the 85th percentile on a test means that you did better on that test than 85 percent of the people who took it. So for the population as a whole, 85% of the people who took it scored below the 85th percentile, by definition. So, if the immigrant population were perfectly matched with the population as a whole, then you'd expect more than 3/4s the score below the 85th percentile.

I had to read that a few times to get it. Maybe this revision is a bit clearer?

Scoring in the 85th percentile means that you did better on that test than 85 percent of the people who took it. That is, 85% of the people who took it scored below the 85th percentileâ€”by definition. So, if the immigrant population's distribution of scores were the same as the population as a whole, then, by definition, "more than 3/4s" (that is, 75%) would score below the 85th percentile, because 85% is more than 75%.

• cm says:

Darn, even better:

Scoring in the 85th percentile means that you did better on that test than 85 percent of the people who took it. That is, 85% of the people who took it scored below the 85th percentileâ€”by definition. So, if the immigrant population's distribution of scores were the same as the population as a whole, then, by definition, "more than 3/4s" (that is, more than 75%) would score below the 85th percentile, because 75% is less than 85%.

• ms physics says:

Speaking as a "chick," I think it's good that my husband knows the difference between percent and percentile. If he didn't, well, I'd probably would have thought he was an idiot and not worth my time. Kind of like folks who call themselves z k.

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