Struggle With Statistics? Your 'Fixed Mindset' Might Be To Blame (arstechnica.com) 151
A new study in Frontiers in Psychology examined why people struggle so much to solve statistical problems, particularly why we show a marked preference for complicated solutions over simpler, more intuitive ones. Chalk it up to our resistance to change. From a report: The study concluded that fixed mindsets are to blame: we tend to stick with the familiar methods we learned in school, blinding us to the existence of a simpler solution. Roughly 96 percent of the general population struggles with solving problems relating to statistics and probability. Yet being a well-informed citizen in the 21st century requires us to be able to engage competently with these kinds of tasks, even if we don't encounter them in a professional setting. "As soon as you pick up a newspaper, you're confronted with so many numbers and statistics that you need to interpret correctly," says co-author Patrick Weber, a graduate student in math education at the University of Regensburg in Germany. Most of us fall far short of the mark.
Part of the problem is the counterintuitive way in which such problems are typically presented. Meadows presented his evidence in the so-called "natural frequency format" (for example, 1 in 10 people), rather than in terms of a percentage (10 percent of the population). That was a smart decision, since 1-in-10 a more intuitive, jury-friendly approach. Recent studies have shown that performance rates on many statistical tasks increased from four percent to 24 percent when the problems were presented using the natural frequency format.
Part of the problem is the counterintuitive way in which such problems are typically presented. Meadows presented his evidence in the so-called "natural frequency format" (for example, 1 in 10 people), rather than in terms of a percentage (10 percent of the population). That was a smart decision, since 1-in-10 a more intuitive, jury-friendly approach. Recent studies have shown that performance rates on many statistical tasks increased from four percent to 24 percent when the problems were presented using the natural frequency format.
Huh? (Score:5, Insightful)
Part of the problem is the counterintuitive way in which such problems are typically presented. Meadows presented his evidence in the so-called "natural frequency format" (for example, 1 in 10 people), rather than in terms of a percentage (10 percent of the population). That was a smart decision, since 1-in-10 a more intuitive, jury-friendly approach. Recent studies have shown that performance rates on many statistical tasks increased from four percent to 24 percent when the problems were presented using the natural frequency format.
I've heard this argument before, and I just don't get it. "Percent" means per hundred, as the word is derived from the Latin "per centum," literally, "per hundred." It's a natural frequency format, just as much as saying "1 in 10 people." It's saying "10 per 100" people. What's so confusing?!?
Re:Huh? (Score:5, Insightful)
I've heard this argument before, and I just don't get it. "Percent" means per hundred, as the word is derived from the Latin "per centum," literally, "per hundred." It's a natural frequency format, just as much as saying "1 in 10 people." It's saying "10 per 100" people. What's so confusing?!?
It's not confusing, it's just that many people don't do the conversion in their heads. Further, presenting the natural frequency is more useful for small percentages: e.g. 1 in 4,000 is definitely easier to digest than 0.025%
Re: (Score:2)
It's not confusing, it's just that many people don't do the conversion in their heads. Further, presenting the natural frequency is more useful for small percentages: e.g. 1 in 4,000 is definitely easier to digest than 0.025%
1. What "conversion"?
2. What makes "1 in 4,000" easier to digest than "0.025%"?
Re: (Score:3)
2. What makes "1 in 4,000" easier to digest than "0.025%"?
Probably people being under the illusion they have an accurate image how many 4000 people are (for example). They do not.
Re: (Score:2)
0.025% is just meaningless.
That's the kind of statement I don't understand. It's 25 out of 100,000 people. I mean, if moving the decimal point is such an advanced concept, how is it that the metric system is so successful?
Re: (Score:3, Insightful)
Because most people, especially now, have little physical grasp of numbers, beyond what they can count on their hands. Expressing numbers in ratios, rather than percentages, becomes much more meaningful when the percentage gets smaller and the numbers become indivisible by 2. 1 in 700 is much easier to grasp than 0.142%. To paraphrase George Carlin," think of how dumb the average person is, and remember that half of them are dumber than that."
Re: (Score:3)
Yeah, so?
Mean, median, and mode are all types of averages.
Perhaps the "even dumber than that" segment of the population includes you.
Re: (Score:2)
Re: (Score:2)
Percentages outside the 0.1-1000 range can be very confusing, at least for westerners. We tend to group numbers by thousands, not hundreds, and percentages with a large number of digits are a mix of both. For example 10000% is x100, even though we spell it TEN thousand percent.
And 0.00025 is less confusing than 0.025%, at least for me, because I can almost see the 100000 in the 0.00025 as they are both around the same length.
Re:Huh? (Score:5, Informative)
That's not it.
It's easier to grasp what to do to those numbers. Presented with percentages it's often hard to see what mathematical operations are necessary to arrive at the desired answer in bayesian statistics problems.
E.g.
A medicinal test for disease X gives false positives in 0.1% of cases. It gives a false negative in 1% of the cases (i.e. correct positive in 99% of the cases). The disease afflicts 0.01%.
Of those tested positive, how many have disease X.
Of course one now could employ the statistics toolbox to solve that problem. OTOH one could compare the 10 in 10,000 false positives (with a slight error since only 9,999 are without disease), to the 1 in 10,000 diseased (noticing that the false negatives have negligible impact for the question at hand and we can work with 100% correct positives as well as 99% if we want an estimate).
So now we need to compare only small numbers, 10 false positives to 1 diseased positive or 1 in 11 which is about 9%.
(the correct result without the approximations is 10 in 111 or 9,009...%).
Also note the easy expansion of 1 in 1,000 to 10 in 10,000 to get to comparable numbers. It's not important to have an accurate image of those 10,000, what's of interest is to compare the 10 false positives to the 1 diseased.
Such medicinal tests help a lot to find candidates that should undergo more sophisticated (and much more expensive) tests, to see if they really have X (it'll reduce the expensive tests by a factor of 1,000), but patients need to be informed even with a "positive" result it's still unlikely that they have X, but advisable to do the more sophisticated test. One might think that the test is pretty useless if it delivers 91% false positives when in fact it is pretty accurate, only the occurence of disease X is so rare.
So such "frequencies" do not only help to get a (pretty) correct result without knowing any bayesian statistics tools, but also to understand how the information affects the result, and how the unintuitive (to someone not used to such statistics) result comes about.
Re:Huh? (Score:4, Insightful)
It's not confusing, it's just that many people don't do the conversion in their heads. Further, presenting the natural frequency is more useful for small percentages: e.g. 1 in 4,000 is definitely easier to digest than 0.025%
1. What "conversion"?
2. What makes "1 in 4,000" easier to digest than "0.025%"?
I suppose it's because .025% is a poor choice of a way to express a value. Percent means parts of a hundred, and they make more sense when the values is between 1 and 100. When you're using percents that are far less than 1%, it is hard to get an intuitive feel for the relative size of whatever is being measured. Sure it's easy enough to do the conversion, but why not express it as a number that is scaled to the measurement in the first place.
It's sort of like when someone asked for the height of my son. I could say he is 0.0011 mile tall, and although you may have a good feeling for how long a mile is, you have no idea whether he is average, short, or tall until you've done the conversion.
Re: (Score:2)
And after you've done the conversion, you still have no idea because there are not enough significant digits.
Re: (Score:3, Insightful)
Re: (Score:3)
Duh
Re: (Score:1)
Obviously 0.352 rods is about average.
Re: (Score:3)
0.025% is not a "people friendly" number. It's a tiny fraction that most people will never see in their lives.
As an example, show me 0.025% of something. Anything. Perhaps 0.025% of a TV. Or a cup of water. Or your phone's storage. It's hard to visualize simply because it seems to imply "a portion of".
But 1 in 4000 is more "people friendly" because you're not asking a tiny part of something, but now one unit in a bunch of units. You don't show me 0
Re: (Score:2)
1. What "conversion"?
Multiplication.
2. What makes "1 in 4,000" easier to digest than "0.025%"?
If you're asking this question then you don't understand the human mind in terms of it's ability to visualise and process numbers. That isn't a bad thing, it's probably that you are smart and surround yourself with smart people, but it always pays to understand how to talk with people who aren't so clever.
Simple tip: No decimals, base number of 1 as the lowest common denominator. Everything else should be stacked in terms of a 1. It is intuitively easier to understand for everyone than presen
meatheads of planet Ostrich (Score:1)
Okay, so you missed the first five minutes of the class where the concept of 'percent' is introduced in your grade six or grade seven math class.
Unintended consequence: dunderhead for life.
Bad five minutes to take two extra puffs after the recess bell rings.
Thereafter, it takes a SPECTACULAR level of blockheaded arrogance (I'm not going to learn, and you can't make me, but I sure can beat the living crap out of this giant
Re: (Score:2)
Thereafter, it takes a SPECTACULAR level of blockheaded arrogance (I'm not going to learn, and you can't make me ...
And this differs from the state of affairs in America . . . how?
Re: (Score:2)
I would rathter people just say "zero point 1" for 1 in 10, or "zero point zero zero zero two five" for 0.00025 (25 in 100,000).
When folks start spewing fractions like "1 / 4000" they cannot be compared mentally to other fractions like that have a different base, such as "276 in 4500".
Re: (Score:2)
First, understanding how to move a decimal place does not mean doing so in tens with such frequency that it is automatic. Especially with large and small decimal numbers. It becomes very easy to be off by an order of magnitude as you get lost in a sea of zeros.
Second, percentages are used to express probability. They are not blind numbers on a page, to grasp them you must associate them with
Re: (Score:2)
Especially with large and small decimal numbers. It becomes very easy to be off by an order of magnitude
You should be able to deal with zeros, but if you hate them, the solution is not to write things like 0.0025% for 1 fourty-thousandth --
you can write: that the fraction was 2.5E-4 and that's perfectly fine.
Second, percentages are used to express probability. They are not blind numbers on a page,
Probabilities Are by definition simple blind numbers on a page --- rational numbers no less than zero a
Re: (Score:2)
25 to 1/4 is one of the decimal to fraction conversions the schools drilled into students in the 3rd grade
to instantly recognize - (How else would you remember what 25% means or that a quarter equals 25 cents?)."
Okay, yet again someone explaining the math as if anyone didn't understand it. Nobody is confused about the math. If nothing else at least acknowledge that it makes little to no sense to arbitr
Re:Huh? (Score:5, Insightful)
Nobody is saying that 1-in-10 is mathematically different than 10%. It is heuristically more helpful to people with less mathematical competence.
When you're good at math, you naturally line up all the "givens" in a problem. You go over each one an interpret what it means, "So that means if I had 100 people, ten of them would prefer vanilla to chocolate..." It's like a wood carver examining a block of wood to find a good place to start cutting. You do this so automatically it seems intuitive to you, but it's actually the result of long training and practice.
To people who aren't as well trained in math, the "givens" look like an impenetrable wall of text, because the individual bricks in the wall don't instantly convey useful information to them. Well, of course they don't; you have to *think* about them, and the less accustomed you are to numbers, the more work it is for you for less certainty of reward.
But if you put a picture into peoples' heads, you give them an immediate handhold on the problem. It's not difficult for a mathematically fluent person to make his own handhold, but it is a stumbling block for a lot of people.
Re: (Score:2)
But if you put a picture into peoples' heads, you give them an immediate handhold on the problem.
So you're saying that the point is, if one replaces the term "10 percent" with "10 per hundred," people would understand the question better?
Re: (Score:3)
So you're saying that the point is, if one replaces the term "10 percent" with "10 per hundred," people would understand the question better?
Yes. I know it sounds weird, but it's really what you automatically do without planning when you approach a problem. But even if you're pretty good at math, having the problem stated in a slightly different way can help you when you're tackling problems that are hard for you. It's what a professor automatically does when a lot of students get stuck on a problem set question: he restates the problem in a way that enables the students to relate it to things they've already learned.
Re: (Score:2)
So you're saying that the point is, if one replaces the term "10 percent" with "10 per hundred," people would understand the question better?
Yes.
That's just sad.
What happens if you have a preamble to the question that defines "percent," e.g., "'Percent' means 'per hundred'"? Does that help?
Re: (Score:3, Insightful)
You seem to be having trouble understanding that different people think in different ways.
Would it help with a preamble saying that other people are different from you?
Re: (Score:3)
Well, that's just another kind of "literacy" too. Everyone I think struggles with the fact that other people are different than they are; but I think quite a few people don't have any idea how different other people can be in their education and abilities.
Re: (Score:2, Insightful)
That's a huge problem. Some people never actually grok'ed percent, or division, or power, or exp, or e, or phi, or differential equations, or derivates of such.. Someone could actually be a good mathematician, but still have holes in their elementary understanding. Someone could be an excellent people person and make better estimates, but nobody ever knew why everything they touched turned to gold.
When you measure everything, you lose the value of everything.
Captcha: circus
Re: (Score:2)
In mathematics as an educational subject, learning each topic is to an unusual degree dependent on having a high degree of mastery of the prior topics. This means that in a mass educational system where students are put into a big room with a lot other students, missing a couple of topics can start an avalanche effect, the end result is the disaster of a person who "just can't do mathematics".
Any normally intelligent human being should be able to do math at a level which is quite rare in our society. T
Re: (Score:3)
Any normally intelligent human being should be able to do math at a level which is quite rare in our society. The problem is that our educational system only produces mathematically literate adults as statistical outliers.
I think that it is just taught incorrectly. Back in Junior High, I seriously sucked at maths. Algebra was a struggle.
Then in High school, I had a teacher who insisted on us picking up and learning slide rules, even though they were rapidly becoming obsolete. The moment I finished my first lesson, it was like several locks in my mind opened - almost mentally painful. What was once a pain in the ass was now painfully but joyously obvious.
I don't know if it was just a fluke for how my particular brain o
Re: (Score:2)
This is true. A lot of people with "learning disorders" don't have learning disorders, they received poor quality of education. Sometimes when you have a student that's not doing their work, failing to remember what to do, writing it poorly or staring off into space through the classes, that isn't a learning disorder it's a bad teacher or series of bad teachers.
And my algebra teacher was just that - bad. In addition to struggling, I was so bored that I wanted to scream. And the assholes in school and my parents were just "You're such a smart boy Ol, but you're lazy and don't apply yourself."
Dare I say that they were abysmally stupid assholes? A guy gets A's in everything else, and struggles with maths and anything beyond, and it is because he is lazy?
Then that slide rule epiphany .
And the aftermath was not all roses and honey either. I then understood tha
Re: (Score:2)
Different views may sound trivial, but don't sell them short for that reason. There's the Linear Algebra concept of the Change of Basis, there's transforms such as the Laplace and Fourier, and more.
For a simple example, it is easy to multiply by ten in base ten. If we wanted to multiply by some other number, often, it could make sense to change the base. Easier to multiply by 2 if the numbers are represented in base 2.
Re: (Score:2)
And then you boil it down to lowest common denominator, which is 1 in 10, yes.
Re: (Score:2)
But if you put a picture into peoples' heads, you give them an immediate handhold on the problem.
So you're saying that the point is, if one replaces the term "10 percent" with "10 per hundred," people would understand the question better?
Even better if you say it as "1 out of 10".
There is a certain (small) number of objects that people can sort of intuitively grasp. More than that, and it becomes an exercise in abstract thought, not concrete thought.
Above that small number, the numbers become symbols, and symbol manipulation capability is almost a synonym for IQ.
Re: (Score:2)
But if you put a picture into peoples' heads, you give them an immediate handhold on the problem. It's not difficult for a mathematically fluent person to make his own handhold, but it is a stumbling block for a lot of people.
Or in other words the usual people are all different and what one see/finds easy is not going to be what the next does. Apply this to everything and anything you won't go wrong!
If fact I find the opposite happens to me in this situation. Present some numbers to me and ask a question and I'm like fine. Tart it up with extra nonsence to help people form a picture (I'm looking at you stupid maths questions at school!) and I just stare blankly into the distance. I can usualy work it out (though I do find someti
Re: (Score:3, Insightful)
"Percent" means per hundred
A lot of people don't even get that. The problem is innumeracy compounded by poor vocabulary.
Re: Huh? (Score:1)
Exactly. Some people have no problem that kind of equivalency. A lot of people do. I'm glad this is getting some study by psychologists.
Re: (Score:2)
I think you vastly overestimate how well average people understand percentages and fractions.
"Alice's class has 30 pupils, where 10% prefer ice cream and 60% chocolate. How many have a different favorite?"
"Bob's class has 30 pupils, where 1/5th prefer ice cream and 1/3rd chocolate. How many have a different favorite?"
I think you'll be amazed to know how many can't find 9 and 14 without pen and paper. And a surprising number not even with pen and paper...
Not people. *Robots*. (Score:1)
This is a problem for a certain kind of ... people.
Those who are ideologistic instead of curious. Those who call everyone who changed his views because he learned something new a "flip-flopper" too. (And not just those who say whatever pleases the listener of the day.)
Those who are dumb enough that they don't know the constant stream of doubt that comes from constantly coming up with all the things that something could be wrong.
Which is, sadly, true for a much bigger part of the population than we'd like to
I find it the other way round (Score:2)
Percent is and its cousins are fine, but "natural frequency" is anything but "natural" to me. I have to convert it to make sense of it.
Also "jury friendly"? Does "success" here mean to get a conviction?
Re: I find it the other way round (Score:1)
Success is a conviction or aquittal, depending on your interest in the trial. I know which one defendants is hoping for.
Re: (Score:2)
"You've got to remember that these are just simple farmers. These are people of the land. The common clay of the new West. You know... morons."
Re: (Score:2)
Nice quote!
Was it natural frequency format, (Score:2, Funny)
or did they also avoid using words with more that 5 letters? #MAGA!
Radical shift in curriculum (Score:2)
I've thought long and hard about this problem. After wrestling with it for a few years I'm ready to support a radical shift in public education.
Use the term "math" (or "maths", depending on your version of English) to refer to arithmatic, geometry, and algebra
"quantithinking classes".
Dump trigonometry and calculus from the curriculum. Replace with statistics, probability, design of experiments, and critical thinking. Call this new subject "quantitative thinking" (or whatever name you want) and give it equal
Intuitive And Jury Friendly (Score:5, Funny)
Recent studies have shown that performance rates on many statistical tasks increased from four percent to 24 percent when the problems were presented using the natural frequency format.
Hmm, let me make that more intuitive and jury friendly for everyone: recent studies have shown that performance rates on many statistical tasks increased from 1 in 25 to about 1 in 4 when the problems were presented using the natural frequency format.
Re: (Score:2)
Re: (Score:2)
Actually, I'm just fine using percentages. "4 out of 5" is for dentist endorsement of toothpaste.
Re: (Score:2)
FTFY
Do sports fans understand this any better? (Score:2)
Being exposed to so much of this statistics, do sports fans use these in their real life? When they buy a car and they read a review, "10% chance of major repair in five years" do they think as often as "that kicker misses extra point"?
Re: (Score:2)
You're more likely to learn something if the subject is interesting to you...
So for many people, "math" might be perceived as boring, while they like "sports"...
Re: (Score:2)
Even more jury-friendly ... (Score:4, Funny)
... since 1-in-10 a more intuitive, jury-friendly approach.
Which would be 1-in-12.
alternatives (Score:2)
1) People who can't understand statistics shouldn't be put in charge of important decisions (un-democratic, I know), or
2) Statistics education needs to be made mandatory to qualify as high school educated in this country.
Re: (Score:2)
A semester of statistics and a semester of geometry would be much more beneficial than the typical full semester of geometry to the majority of high school students.
Bound for failure (Score:5, Insightful)
If you can't figure out 10% is 1-in-10, you have no hope of wading through the standard level of obfuscation added to any publication when discussing statistics.
Re: (Score:2)
If you can't figure out 10% is 1-in-10, you have no hope of wading through the standard level of obfuscation added to any publication when discussing statistics.
Sure. If you can't figure out the most simple case you're quite right. That doesn't change the fact that numbers can be presented in a more simple and intuitive way to aid the understanding of a wider audience.
Let's flip that argument:
There are some people who can't figure out that 10% is 1 in 10. However those that can, likely have no problem figuring out that 1 in 10 is 10%, so by expressing the number in its easiest to understand format you capture the understanding of the largest possible audience. The
Re: Bound for failure (Score:2)
The problem is people don't know which is bigger, 7-in-10 or 13-in-25, so by using those ratios, you're guaranteeing those with numeric illiteracy have no chance of understanding the numbers. Everyone knows 52% is less than 70%, and has a good idea how big the difference is.
Re: (Score:2)
The problem is people don't know which is bigger, 7-in-10 or 13-in-25
Let me repeat myself: "The problem is always finding out what the easiest actually is.
Also 7 in 10 and 5 in 10 are also easy to understand. As are 70 out of 100 and 52 out of 100. Again, you target your lowest common denominator with the language they should understand. Your example effectively is the same as comparing numbers with non equal base units. If someone needs to convert something in their head then you've done something wrong in the way you've presented data.
Re: Bound for failure (Score:2)
Percentages are a way of applying consistent base units, but you argue against it?
If you can't move a decimal point, you are not worth explaining statistics to.
Re: (Score:2)
Percentages are a way of applying consistent base units, but you argue against it?
I'm struggling to understand what it is you don't get when I say: "The problem is always finding out what the easiest actually is." You persist in talking about absolutes showing a fundamental lack of understanding of how to communicate with people. I never dismissed percentages, I only argued that you should explain something in its simplest terms, quite often that isn't in percentages.
If you can't move a decimal point, you are not worth explaining statistics to.
How elitist of you. That's a fast track way to get yourself killed by anti-vaxxers, or by a superbug thanks to the thinkin
Re: Bound for failure (Score:2)
When someone is illiterate, you teach them to read. You don't just start writing everything at a kindergarten level.
The same applies to numeric literacy.
Re: (Score:2)
Indeed. So do you start by throwing the complete works of shakespere at them?
Also we're not teaching people, we're disseminating information. Or maybe you intend to start every post where you wish to use a percentage symbol with a 2 page primer of mathematics. In which case I tip my hat to you. That's more effort than most people will put in. The rest of us just find appropriate methods of communication.
Re: Bound for failure (Score:2)
Correct. We're not teaching people. Education is orthogonal to the dissemination of information. Shakespeare doesn't teach you how to read his works, he expects you to become educated before the attempt. Don't try to tell Shakespeare to write like Dr. Seuss just because you find One Fish, Two Fish... more accessible than Hamlet. Learn to read. Then your opinion on the topic might begin to matter. Before then, you are an illiterate not worth taking the time for.
Re: (Score:2)
So we've gone full circle. Let me repeat and we'll see if we come with a different more sane outcome:
How elitist of you. That's a fast track way to get yourself killed by anti-vaxxers, or by a superbug thanks to the thinking that the uneducated are not worth speaking to.
Since the authors of the study... (Score:2)
Re: (Score:2)
Yeah, I'll give the mindset problem some names (Score:2)
Marketing, lobbying, politics.
Let's actually use statistics (Score:1)
Stupid (Score:2)
"As soon as you pick up a newspaper, you're confronted with so many numbers and statistics that you need to interpret correctly," says co-author Patrick Weber, a graduate student in math education at the University of Regensburg in Germany. Most of us fall far short of the mark.
Or maybe just acknowledge that the vast majority of humanity is just plain stupid, and that you are part of it...
If you can't make the numbers dance... (Score:2)
If you can't make the numbers dance like the Bistromathics drive on the Heart of Gold, you don't understand statistics.
10%. 9 to 1 against. 1 in 10. They all mean exactly the same thing. If you haven't grasped this yet, it's not because you have a "fixed mindset". It's because you don't understand statistics.
Even the article gets it wrong... (Score:2)
Shouldn't that be from 1-in-25 to 1-in-4?
3 in 7 or 7 in 20... (Score:2)
... which is better?
I think that's enough to show that we need to normalize fractions into a common unit.
Lets use "ppm" for very low amount, and "percent" for higher quantities.
Statistics (Score:5, Interesting)
I'm a mathematician.
The second someone digs out statistics, I can always pick 20 holes with their methodology used, presentation of, choice of, or analysis of their numbers. Usually, I could make things come out "oppositely" with only minor tweaking and use of the other statistics from the same dataset that they discarded out of hand, and usually I could provide much better justification for the numbers I used than the ones they did.
People use statistics to back up their claims. That's it. And if you go looking hard enough and present statistics to do that, you can ignore all the stuff that doesn't match your claims. It's really easy to do.
And because nobody understand statistics (I would posit this category even includes statisticians!), you can get away with it.
I like to shout at shampoo adverts when they say "Women agree*" where the * leads you to a footnote saying they tested 19 women and 67% of those agreed... so you're telling me that, actually, worldwide, 12.7 women agreed... What the hell kind of selection criteria did you use to get that, and what use is that if you don't specify that they were random women from the street in a controlled trial rather than, say, the people who work in the office?
The old saying is right - there are lies, damn lies and statistics. If someone quotes statistics are you, assume it's a lie. It almost always is. Even when it's not, it's merely the portion of the truth that can be spun positively if you don't mind looking through an n-dimensional kaleidoscope at the data.
And what are they trying to do by telling you this? They're trying to tell you "Hey, look, you're stupid and have no idea what's going on and actually everyone else is really onboard". Statistics are used as the worst kind of "peer pressure" - if they are trying to convince you of something, rather than inform you.
Statistics can be incredibly useful, very revealing, and can lead to a better understanding. If you get them from a professional. Who'll then tell you what the statistics *mean* and whether or not they have caveats.
If you get them from shampoo ads and random junk on the Internet, they are no better than any other "fact" spewed at you in such a manner... wrong.
(Interesting fact: There's a TV program called QI, in which all kinds of "you'll never believe" stuff is presented in what's supposed to be a highly intellectual quiz show. QI facts are heavily researched, almost always counter-intuitive or contrary to what everyone has been told, and they spend years with some of the cleverest people from the top universities doing research for each series. And in one episode they reveal that the portion of facts that they themselves got wrong, or which have changed since, was over 50%).
If someone quotes statistics at you, don't just nod and go "Oh really?" because you're then likely to repeat that statistic without every checking it. The correct response is to think "What's he trying to convince me of?". Because you don't use statistics for anything else.
And unless you understand, truly understand, statistics, you know that any kind of amateur data-gathering or analysis by even the most well-intentioned people is a bottomless pit of potential failure.
Hell, to be honest, 99% of the time, I can't even work out the right answer for some statistics so I make it up and say 99%.
"Natural frequency" loses consistent scale (Score:2)