Ideas - We Give You Five: Odd in More Ways Than One

Episode Date: January 2, 2025

Five: a simple, easy number with a diabolical side. As we continue our series, The Greatest Numbers of All Time, meet the Janus-faced figure of five and find out how the number has acquired its personality for people in the arts and sciences. *This episode originally aired on Sept. 28, 2023.

My name is Graham Isidor. I have a progressive eye disease called keratoconus. Unmaying I'm losing my vision has been hard, but explaining it to other people has been harder. Lately, I've been trying to talk about it. Short Sighted is an attempt to explain what vision loss feels like by exploring how it sounds. By sharing my story, we get into all the things you don't see

about hidden disabilities. Short Sighted, from CBC's Personally, available now. This is a CBC Podcast. Welcome to Ideas. I'm Nala Ayed. With our special series, The Greatest Numbers of All Time. We humans, as the saying goes, are meaning-making machines. We can draw meaning and story and emotional qualities from virtually anything. The way stars are configured, the color of the sky at dusk and dawn, even cold, hard numbers.

Different numbers have different personalities. I feel like five is mean. It's as if each number has its own kind of force field radiating out from it that influences the things that it affects. Five is just nice. Like, I don't know. Like, I think there is something about it. Five is nice. Four and six. No way! Oh my god, no. I'm so on the uneven. It's like a lucky charm. If I pick five or the fifth ones, that I usually get it right.

the fifth ones that I usually get it right. In this episode, we investigate how the number five has acquired its personality for people in the arts and sciences. I think what's interesting about numbers is, you know, are they external to us or are they part of our internal makeup? Can you do it in French? Un, deux, trois, quatre, cinq. The number 5% is arbitrary, but that's the threshold for statistical significance.

Whether it's at a hinge in a musical phrase or the threshold point where scattered data cross over into scientific truth, the number five has a peculiar habit of showing up right in the middle of things. It's like anything could happen on a five chord. Ideas producer Tom Howell takes it from here. And now, the group you've all been waiting for, the Lovers of Fire!

No one's done more than Sesame Street to give personality to a number. Hello, I'm Bob. I like a number that's easy to count to. That's why I really appreciate the number five. You just go one, two, three, four, and there you are at five. Look at that five standing up there so straight and tall. I'm proud of you, number five. Ooh, give me five, give me five. Full-on anthropomorphizing of abstract symbols is something we're supposed to grow out of, I think.

But one doesn't need to be puristic about it. A residue of the impulse remains. If you want to see what I'm seeing, you can have a look at the screen in front of you. And if you don't, you have to close your eyes. I was thinking about where the numerical becomes human the other day while getting some earwax removed. Just get that out. Do you want to see what that looks like?

Yeah. All right. So we'll have a look back in there and observe the actual eardrum itself. Great. Now we can see. We should be able to see it. Clayton Fisher is an audiologist in Ottawa. So I'm just going to zoom in on that.

And that is a healthy looking eardrum. Great. The eardrum is translucent and round, appearing in the darkness like an alien planet or a deep-sea jellyfish. Its job is to separate the outer ear, or ear canal, from the middle ear.

The outer ear and the middle ear is doing what it's supposed to to sort of boost the vibrations into the inner ear. We are, believe it or not, approaching the number five. But what the audiologist wants to get to the bottom of right now is an imperfection in the way I hear certain sounds. It occurs at a mid to high frequency range. It is in the inner ear where you have thousands of these little microscopic sensory cells that are taking vibrations up and down to amplify the uh the movement of what's called the basilar membrane to allow us to hear like as soft as as we normally would

okay but it's these outer hair cells that are most vulnerable to yeah like you know noise exposure being being on a rock stage, right? In my 20s, I played keyboard and violin in some pop rock bands, and Clayton suspects this period is to blame for a dent in the contour of my hearing range. Unfortunately, it's right around the most important frequencies in the human voice. If you have a 10 dB difference, which you do from 4,000 to 8,000 Hz between your ears, smaller differences, 5 dB difference the other way at some of the most important speech frequencies, having differences between ears like that will also make it more difficult for you in the noisy environment because the brain is very much relying on a perfectly balanced signal

to figure out who to attend to and who to not attend to. And, you know, in your case, to a lesser extent, it would be, you know, localizing sound and so on. Your ears are EQing the sound. EQing the sound. Now, without going full Sesame Street, there are a couple of cute, slightly furry characters that we need to meet. They play an important role in how we hear music, and they live deep inside your ear, beyond the eardrum. So the cochlea is in the inner ear,

and it's called a cochlea because it's kind of shaped like a snail and it's essentially like a backwards piano that's kind of coiled up and it works very much like that where right at the base of the cochlea it's those cells that code for the very highest frequency sounds and then it starts getting lower and lower and lower in frequency as it goes towards the apex. As vibrations move through the inner ear, you get the basilar membrane vibrating and resonating to those specific frequencies. So it's actually tuned, this membrane itself. When you say backwards piano, you mean like how on the piano the biggest strings are at the bottom

and the smallest strings are at the top for the high notes? Yeah, so it's the other way in in the in the inner ear of course we don't have strings in there but but right at the base of the cochlea it is significantly stiffer than it is right at the apex of the cochlea that codes for those low frequency sounds not only do you get less sensitivity when you have damage to those cells for the softest sounds, but you also get less frequency specificity, okay? Because those outer hair cells are specifically going to move and fire right at the area of the cochlea where this frequency is going into.

So if you have damage to those cells, then there's a little bit more frequency smearing. Coincidentally, frequency smearing was the name of one of the bands I played in in my 20s. That's not true. But it is more or less true that inside our ears, we have a pair of pianos coiled up and upside down. And sitting there behind each piano is a tiny genius mathematician without whose impressive work calculating ratios, music would be meaningless noise. Thank you. Yes, do you have a question?

I do have a question, if it's okay. I am tasked by the CBC radio show Ideas to find out the role of five in concepts of beauty and truth. Are you using the number five in the work you're doing here this evening? Thank you, Tom, for asking about the number five. In music theory, the number five. In music theory, the number five really pertains to the chord, the five chord,

which is a very important chord. Oh, how so? Because it's like a question mark. This is Rebecca Hennessy with her band Makeshift Island. I've interrupted their gig at the Transact Club in Toronto for a dose of music theory. It's like anything can happen on a five chord. Oh. Really.

Can you demonstrate in some way? Well, here, check this out. I'm going to play the piano for you. Okay, okay. So if we're in this key, if this is the one key, the one chord. The five chord. That's a chord? Well, this is the one chord.

This would be the one scale, maybe. C major scale, for those of you who care. When we get to here... What does that make you want to do? Anybody? Go to the next place. Go to the... It really makes you want to go there.

Uh-huh. So this is the question mark. But we could go a lot of different places. You can go... How about this one? Yeah, I i mean i'm hearing all the sounds but where's the five and all this the five is how do we resolve what is the it's the the cadence well i think the thing rebecca didn't answer is that the chord she's talking about is built on the fifth note of the scale that's what the five chord is i might need to hear it can you play a one chord and then and then take it to the scam. That's what the five chord is. I might need to hear it. Can you play a one chord and then take it to the five?

Is that possible? Yeah, let's jam on the one chord, everyone. And B flat, C, C, where's C? Okay. One, one. Okay, one. One's one, one. And then we're gonna go to the five chord.

Ready? One, two, 3, 4. Back to the 1-4. There you go. Take it to the 5. There you go. Check this out. I'm playing in five.

Thanks for asking, Tom. Anything could happen on a fine chord. I'm now sitting at a standard upright acoustic piano. It has 88 keys. Here they all are. Try to count them as I go. Now, unless you're extremely quick, that was probably too fast for you to count. But in another part of your brain, you may have been doing much more complicated math effortlessly at breakneck speed without meaning to. All of these keys are named using only seven letters. That's A.

That's also called A. That's also called A. That's also called A. Y. We have 26 perfectly good letters of the alphabet, and if we must use letters for some reason to describe these keys, why not use them all? Even if you've had no musical training, it's likely your brain already knows the answer, or at least you can feel why. The frequency of each A is double the frequency of the last one. If my piano was in tune, this A would be at 440 Hz. Physically speaking, waves of pressure, jostling air particles, hitting your eardrum exactly 440 times per second, and your brain somehow keeps count. That's remarkable. Even more remarkable, here is 880 Hz.

It's not universal that people pick up on the connection between these two sounds, but most people do. It's your brain recognizing a ratio of 2 to 1. 880 is exactly twice 440. Impressive, but multiplying by 2 is still fairly easy. What about a ratio of 3 to 2? 440 times 1.5. 660 Hz. That's called a perfect fifth. The first note in a musical scale and the fifth note of that scale. A, B, C, D, E. It's a multiplication relationship. The same multiplication relationship that 2 has with 3. Just in case you're not really sure you're tuned into this,

that's the pair of frequencies with a more difficult relationship. It's square root of 2 to 1. Now square root of 2 is a famous irrational number. The first Greek mathematician to go on about it to his colleagues was reportedly murdered for causing trouble. And this sound, known as the devil's chord or the flat fifth, is generally considered to be jarring, assuming that your brain is doing the math. Now, that said, of course, you can acquire a taste for it. Okay, let's put the devil back in the hole. We're now getting to how the V chord means what it does in music. Both types of fifth note, the diabolical one and the perfect one, can claim to mark the halfway point between an A and the next A, or a B and the next B for that matter. The reason the perfect fifth is halfway is because that's where a ratio of three to two gets you. This E at 660 hertz is halfway, mathematically, between 440 hertz and 880.

This is the sense of suspension at a halfway point that gives the V chord its personality. It's like a question mark. Or for some, it's like the turnaround point in a journey that takes you out and then back. Each note in the V chord is 1.5 times the frequency of each note in the chord you started with. At that halfway point, we can pause, stop for lunch, maybe.

We're not seriously uncomfortable for a while. But the tiny mathematician at the piano inside your ear can tell you we're not home. We're dealing here still with a fractional relationship. And it feels good when we land safely back on that whole number, the ratio of 1 to 1, 2 to 1, 4 to 1, or 8 to 1. A whole number feels like home. Hi, I'm Kate Kiragawa, historian of mathematics, and I've been writing a book with Timothy Revell, The Secret Lives of Numbers. The habit of turning mathematical patterns into beauty and meaning goes right back to our points of origin. All my life has been associated with mathematics, and I love it that way.

Numbers in East Asian cultures have some sort of symbolism. So one and six are associated with the direction north, and season winter, and the color black black that's one and six and it goes like two and seven associated with south and summer and red and three and eight or east spring blue four and nine or west autumn and white and then what's five right what's five now used up all the directions yes and then it's the central position and associated with the color yellow. From those kind of symbolism embedded in cultures that I started to feel like five is just quite so special. Because, you know, it's just the only number that's coming into, you know, a center of all other directions. So five started to become my favorite since my young days at school in Japan.

Five is your favorite number? Five has been my favorite numbers. Yeah. And also it's like a lucky charm. So if I pick five or the fifth ones that I usually get it right. Numerology refers to giving mystical meaning and power to numbers. It provides answers to many of life's questions, but not many mathematics questions. For that, we want number theory, a practice that may be even more ancient. One of the oldest and most famous archives of numbers is on the bone. And usually it's called a shango bone. It was found in current-day Uganda, probably dated around 20,000

BCE. So that's really, really long time ago. I can just show you an example more closely. Kate shows me a picture of this Ischango bone. It once belonged to a baboon or a large cat, but has clearly been marked up by a human. There are notches all along the length of it. So this is like quite a mystery. And if you see those scratches on the bones, we examined and counted those numbers. First column added up to 48, and the second and the third columns as up to 60. So those are really interesting, distinct segments. And then out of those 60 notches, they're split into groups of 11, 13, 17, and 19. And what are those numbers? They're all prime numbers. Right.

They're all prime numbers. Right. It might not be anything, but still, like, you know, they could be a proof of, like, you know, they might have known something about numbers. Number theory is fundamental. This is, like, behind all the other mathematics. mathematics. Number theory concerns integers, whole numbers, and also like, you know, how things have been composed, not just the study of numbers, but the relations between them. Like prime numbers, why it's so important is we have the big numbers and we can break it down to the smaller numbers, right? And we can just break it down all the way. 100 is 2 times 50. 50 is 10 times 5. 10 is 2 times 5. 100 is 2 squared times 5 squared. And then what's left there is the prime numbers. You know, that's the easiest way to think about it. I mean, it's all about numbers,

but it's not all about numbers. It's fun for some, this finding of patterns amid numbers. For the rest of us, it's often a hard sell. I've personally always been drawn more to words, so if I hope in appreciating the other, I turned to The Joy of Mathematics, a series of video lectures published in 2007 by The Great Courses, a resource available for free through my local library.

Welcome to The Joy of Math. I'm Arthur Benjamin, professor of mathematics at Harvey Mudd College. Now, many people, when they hear the joy of math, boy, that sounds like a contradiction in terms. I mean, for many people, math is a four-letter word, something to be afraid of, not something to be in love with. And yet, in these lectures, I hope to show you why... I have to say, I was hooked.

But the course doesn't include a lecture specifically on the number five, so I went straight back to the source for some additional content. Hello, my name's Arthur Benjamin. I'm a professor of mathematics at Harvey Mudd College in Claremont, California. I'm also a mathemagician, and I tour the world combining my loves of math and magic, so people fall in love with mathematics like I do. Only in mathematics can you say things with absolute certainty, and there's even a certain amount of inevitability to it all. You might take this approach to solving the problem, you might try this problem geometrically,

or you might try it algebraically. But again, if you're careful, you'll always reach the same answer. And that is great in this chaotic world that we live in. All of the producers making this series are non-mathematicians, and it's now become a little bit competitive. You know, the zero person thinks zero is the coolest. And any suggestions of where one might start if you're looking to make five seem especially charming, important? I'll give you a great example. So one of the things I love to do is square numbers, take numbers and multiply them by themselves, right? So five squared would be 25. There's only two things you have to remember when squaring a number that ends in five. The first is that the answer will always, always end with 25. Okay, no matter what number you're squaring, if it ends in five, the answer will end in 25. Okay, that's fact number one. How does the

number begin? It begins by taking the first digit, multiplying it by the next higher digit, and that's how the answer begins. So, for example, let's say you were squaring 35. Okay, we know it ends in 25. How does it begin? Well, 35 begins with 3. Take the next higher number, that's 4. 3 times 4 is? 12.

12. And there's your answer. 1225. 1225. I know you wouldn't be telling me this if you didn't know it was true, but I still want to check. You gotta check. Of course. Be a critical thinker. 35 times 35

is 1225. Also known as 1,225. So now it's your turn. Let's say you were to square the number 65. Okay? It ends in 25. Good. And 6 times 7 is?

42. And so the answer is? 4,225. You got it. Yes. Right on. You're listening to a documentary called We Give You Five. It's part of our Ideas series, the greatest numbers of all time. You can also hear about rival claimants to greatness like Zero or the number three.

The complete series is available for download on our podcast feed. Ideas is a podcast and a broadcast heard on CBC Radio 1 in Canada, on US Public Radio, across North America on Sirius XM, in Australia on ABC Radio National and around the world at cbc.ca slash ideas. I'm Nala Ayed.

My name is Graham Isidore. I have a progressive eye disease called keratoconus. Unmaying I'm losing my vision has been hard, but explaining it to other people has been harder. Lately, I've been trying to talk about it. Short-sighted is an attempt to explain what vision loss feels like by exploring how it sounds. By sharing my story, we get into all the things you don't see

about hidden disabilities. Short Sighted, from CBC's Personally, available now. Our episode on zero characterizes that number as a beginning. Let's say you're zero years old. That's the beginning of your life. The number five is a midpoint. For most of us, it was once halfway to learning to count for the first time on our own fingers or toes.

Want to try the other foot? One, two, three, four, five. Five toes. Five is the dot in the center of the face of a die. It's the hinge of a musical phrase. More controversially, for the past century, five has guarded the border in science between statistical significance and a null result.

I think some person decided if the probability for the null to be true is 5%, that's rare enough. We would call that to be statistically significant. Ideas producer Tom Howell finds out what this means for scientific knowledge and the pursuit of truth itself. Now, another way to think about this is what you just did really and the reason this works. Professor Arthur Benjamin is showing me the joys of mathematical certainty using the number five. When you squared 65, I gave you the steps of it ends with 25 and you attach six times seven. But what was really happening was you were taking 65 and you were rounding it down five to 60, going up five to 70. 60 times 70 is 4,200. And then we added the number 25. Okay, so 4,200 plus 25 is 4,225. That's the process that

works so nicely when the number ends in five, but it also works for any number, this process. So for example, let's say, now do you know what 12 squared is? 12 times 12 is? 144. Okay, good. Here's how you would calculate that using the method I just described. You would start with 12, and you'd say, well, it's more convenient to work with 10, so I'm going to go down 2 to 10. I'll balance that by going up 2 to 14. So instead of doing 12 times 12, we're doing 10 times 14. And that's what? 140. That's almost the answer.

To get it exact, all you have to add is the square of the number you went up and down. And you went up and down 2. 2 squared is 4. And there's your answer. 140 plus 4 is 144. Amaze your friends, confound your enemies. That's right, exactly. Now, a great gambit in chess is hard to appreciate without a chessboard.

And likewise, the intricate beauties of number theory, like how to prove Fermat's last theorem, do tend to need at least a piece of paper to scribble on. That said, if there's one number that makes itself most convenient for purely mental mathematics, it's surely the number five. My favorite group of numbers are the Fibonacci numbers. They start with 1 plus 1 and 1. And when you add 1 plus 1, you get 2. And then you have 1 and 2.

You add those together, you get 3. Now with 2 and 3, you add those together to get 5. And then 3 plus 5 is 8. 5 plus 8 is 13. And it goes on forever. So starting at the beginning, the numbers are 1, 1, 2, 3, 5, 8 is 13, and it goes on forever. So starting at the beginning, the numbers are 1, 1, 2, 3, 5, 8, 13, 21, etc. 5 plays an important role in that, in that the fifth Fibonacci number is 5. So it has kind of a special, you know, the first Fibonacci number is 1,

and the fifth Fibonacci number is 5, and that's the only time that those numbers match up. There's no other thing. The 111th Fibonacci number is five and that's the only time that those numbers match up. There's no other thing. The 111th Fibonacci number is definitely not 111. Not 111. But what you can say is that every five Fibonacci numbers, the fifth, the 10th, the 15th, the 20th, the 25th, will always be multiples of five. They'll always be multiples of five.

And the only multiples of five will appear in these multiple of 5 places. So among the Fibonacci numbers, the only ones that are multiples of 5 are the 5th, 10th, 15th, 20th, and so on. That is really neat. I always thought of 5 as a friendly. It was like 11. You could multiply 11 and you'd always get those double digits at least up to 99. That's right. And 5 was friendly which which it was it was easy to learn your multiplication tables with five because they would always end in zero or five right it would go zero five ten

fifteen twenty twenty five thirty thirty five forty five and the reason the reason uh you have that nice property is that we work in base 10. And the reason we work in base 10 is we've got 10 fingers on our hands, right? That's why we chose to work in base 10. 5 goes into 10. 5 is a divisor of 10. And so that's going to make it have this nice regular pattern that you're not going to see with multiples of 7 or something. You know, it's just going to be a... Oh, by the way, speaking of hands, the Roman numeral for five is the letter V, right?

You ever wonder where that came from? It's because all the Roman numerals were chosen from hand signals. Like a five, if you put a five on your hand and you look between your index finger and your thumb, you'll see a V. Amazing. That's the v um and what's the roman numeral for 10 it's x and where does the x come from it could either be from this by crossing your arms and you know now you see the x or you take a v and an upside down v put them put them together, and you've got an x. You do.

So v plus v gives you x. That's 5 plus 5 equals 10. We're very glad that we're not using Roman numerals anymore, but that's where that all comes from. So that's just a little piece of 5 trivia. Yeah. Last one, seeing as you brought up multiplication tables, five trivia that yeah last one seeing as you brought up multiplication tables if i wanted to sum up all of the numbers in that multiplication table 10 by 10 is there an easy way yes

yeah i think the answer is 3025 and in five uh it doesn't in five. It's 55 squared. And in fact, here's the – this is perfect. This is perfect because you know 55 squared, right? It ends in 25 and five times six is 30. So it's 3,025. Now I'm going to try and explain to you why the numbers in the multiplication table add up to 55 squared. If you add up the numbers from 1 to 10, that adds up to 55.

There are shortcuts for seeing that, but trust me, numbers from 1 to 10 add up to 55. That's the first row of the multiplication table, right? What does the second row of the multiplication table add up to? These are the multiples of 2. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. Well, it's twice 55 because it's the same. It's the row you had above all multiplied by 2. So the second row adds up to 55 times 2. The third row adds up to 55 times 3. The last row adds up to 55 times 10. Now, if I add 55 times 1 plus 55 times 2

all the way up to 55 times 10, that's 55 times the quantity 1 plus 2 plus 3 all the way up to 10. But we know 1 plus 2 plus 3 up to 10 is 55. So that's 55 times 55, which is 3,025. Hey, let's hear it for fives. Squaring numbers that end in five.

I can't believe you asked me that question. Well, thank you so much. I really feel a lot stronger in my position on having chosen five after speaking with you. Thanks for joining me. My pleasure. Let's do it again. So fives' musical personality and fives' mathematical personality both relate in different ways to the human body. In the one case to our inner ear's sense of proportion, in the other case to our fingers and thumbs and toes. Do you want to do it in French? Here's the numbers in French probably more than English. You want to do it in Chinese?

Mm-hmm. One, two, three, four, five. All right, you're done. Okay. Sherry Ho and Jonathan Hoff are scientists. She's doing a PhD in epidemiology. He's a data scientist and former statistics teacher. I'm at their apartment in Montreal with their baby daughter, Vera. Her name literally means faith in the Slavic languages it comes from.

Somewhat relevant to our conversation, linguists believe that there's an older meaning of Vera, which probably existed in a language spoken before Slavic or Latin or Greek, and that meaning was truth. But back to numbers, Sherry and Jonathan both have strong views on a connection between the number five and the pursuit of truth. I think there's a risk of throwing the baby out with the bathwater. Like, again, if you treat it as the thing that it's always meant to be, which is as one of the many measures to understand the level of confidence you have in your data, then it can stay. But stop treating five like magic.

It's not just arbitrary. It's just not always appropriate for what you're doing. I remember when I learned it, it was like, oh, yes, five. If it's above five, it's not good. If it's below five, it's good. And I think very smart people feel that way. I mean,

Sherry and I have been lucky to work at tippity-tough universities, and lots of very smart people just see it as this truth value. I mean, I don't know, it's a nice number. It is nice. The idea that a number seeming to be nice has any relevance at all to the notion of scientific proof came as a bit of a shock to me as a non-scientist. The full explanation for Five's role here is dauntingly complex and, frankly, ill-advised. But that's the danger with Five. It tempts a person into thinking things are going to be easy. The first person who tried to warn me of how Five can lead us seriously astray was a statistician named Nathan.

I'm Nathan Tabak. I'm a professor in the teaching stream at the University of Toronto in the Department of Statistical Sciences. I originally went to Nathan with what I thought was a super basic question, one of those things you feel like you're already supposed to know. Why is it when newspapers release opinion poll results, like on how the Conservatives and the Liberals are doing or whatever, they always say something like, this result is considered true to within three percentage points, 19 times out of 20? In my opinion, they put that because they don't know what's going on the other

5% of the time. And statistics is all about studying variability. And the other 5% is, you know, what would have happened if you collected another sample of people, and then another sample of people, and then another sample of people, would the results have been the same? And that 19 times out of 20, it's saying, well, one in 20 times, it would be different. Well, 1 in 20 times it would be different. Okay. Why? Why would you say that?

Why 1 in 20? Why not anything else? Oh, I mean, you could set it to some other error limit or tolerance, but people have stuck with the 1 in 20 error rate. It's an error rate. It's an error in sampling. If you can imagine, you take a random sample of people, you ask for, you know, how many people would vote liberal in this next election? Well, if you randomly selected another sample, then you'd get a slightly different answer.

And then if you did it again, you'd get another answer, and then another answer, and then another answer. And so there'd be variability, even though you're asking the same questions, based on the random sampling. And is that because they have previously tried it 20 times and they find that the 20th time something weird happens? No, it's really they pick that. No, well, the problem is, is that they haven't done it 20 times. So they can only do it once. Because I I'm sure, for example, the CBC, when they do polls, they can only do them one time. You want to imagine what would happen if you did it hundreds of other times.

And this gives you a range of what might have happened if you've done it, let's say, 100 times. So in 95% of the time, then you get similar answers or like within that range that the poll said, you know, that plus minus whatever percentage, then 5% of the time, you would get something different. But you don't, but the trick is with statistics is you don't know if you have one of the 95 good ones, or the five bad ones. So you know what a bell curve is? Normal distribution.

Yay! That's great. Oh my gosh, you get a gold star. Thanks. Okay, so the bell curve is a normal distribution lots of things follow a normal distribution not in every population i think height usually follows a normal distribution a lot of things i think random things tend to and it just means that there's a big chunk in the middle that sort of group around an average value, and then it tails off on the either end to the lower high values. When you do hypothesis testing, for anyone out there who knows exactly what I'm talking about,

like, I am cutting some corners, but you're basically looking at where your sample falls on one of those bell curves. Jonathan is a friend of mine, and we were in the presence of his baby, so I felt perhaps overconfident that I was in a safe space here for asking about a devil in the details of statistical science. These five bad ones Nathan Tabak had tried to warn me

about. You don't know if you have one of the five bad ones. Five bad ones. Jonathan told me to imagine a scenario, a world in which the average height of Canadian men is six feet tall. In reality, the average is five foot ten. We didn't know this at the time. So in this other world, Canadian men are six feet tall. In reality, the average is five foot ten. We didn't know this at the time. So,

in this other world, Canadian men are six feet tall, and the reason we're talking about height is... Height is often normally distributed. Therefore, when you plot everyone's heights on a graph, you'll see something like the bell curve shape. Now, imagine we've come to suspect that men in Quebec are taller than other Canadians. That would be an interesting fact. If we could prove this, the news media would be all over it. We'd get fame and funding. But here's the technique for proving claims of that sort.

Instead of hypothesizing that Quebecers are taller, Jonathan and I would test the null hypothesis, which is they aren't. I made my bell curve based on my assumption that Quebecers are six feet tall. Jonathan goes out and measures the height of one male Quebecer chosen at random. Wait a second, this guy's six foot nine. Did Jonathan just discover that Quebecers are, on average, a nation of giants? Quebecers are, on average, a nation of giants. Nope, it's probably just a tall person that Jonathan landed on by picking a single data point at random from under the bell curve. It's just too likely that it came from a population that is six feet tall.

So he goes on. He measures 40 Quebec men, and they average six foot two. That's two inches more than the national average in this scenario. But Jonathan still hasn't proved anything very interesting yet. My 40 Quebecers average six foot two, and I do some math to just calculate a probability that they came from a population that's six feet tall. And if the probability is too low, then I'll say, oh, Quebecers are probably not from a population that's six feet tall they're probably from a population that's on average a little taller i love how you're telling me this like i'm a baby and yet i still don't understand okay yeah yeah so i i want to test an idea yeah and so i quantify that idea so the idea is quebecers are six feet

tall i measure some some Quebecers. That little distribution of my sample, I calculate the probability that they could have come from the population of six feet tall on average people. Basically, is my hypothesis that on average the people of Quebec is six feet tall true? Right. That is the hypothesis testing.

Okay, now we just measured 40 people, and let's say they average to six foot two. They average the hypothesis testing. Okay, now we just measured 40 people, and let's say they averaged to 6'2". Average 6'2", I do my math, I get this p-value at the end, this is the thing you're after. Let's say p equals 0.5. So there's a 50% chance that my sample of Quebecers came from a population that is on average 6 feet tall. That probability is too high. I'm not going to rule out that Quebecers on average are six feet tall. But let's say P is 0.001.

The probability is very, very low that that sample came from an overall population that averages six feet. And so I'm going to say, oh, OK, so there's a strong probability all of Quebec are not six feet and probably taller. I like how you just sort of waved a wand over, like you do some math. Yeah. This math involves comparing the size of your sample to the total population of, in this case, Quebec men, and comparing that with the shape of the bell curve.

Anyway, the point is, what we're trying to look for is how often our number of random data points could naturally be expected to fall under the tail of the bell, around a point at least as extreme as where they actually just landed. The smoke clears and you get a p-value of 6%. Yeah. The smoke clears and you get a p-value of 6%, you'll say, well, it's not good enough evidence to say that Quebecers on average are not 6 feet tall because the probability is too high. But if the p-value is 4%, you know, below 5%, then you'll say, okay, that's strong enough evidence to say Quebecers probably are not six feet tall on average. Someone decided that 0.05 is the threshold.

If it's below that, then it's real. If it's not, it's false. And as I think Jonathan was alluding to, it's arbitrary. It's 0.04 versus 0.06 versus 0.05. Like, the measure that I think is supposed to give you a sense of your data has evolved into a target. So people would say, if it's lower than that, then it's worthy. Like, not just your result is worthy, but that your whole experiment is worthy. Yeah, it wasn't always the measure. When it first evolved, experiment is worthy. Yeah, it wasn't always the measure when it first evolved, it was one tool of many to assess the quality of your data and work. But as Sherry said, it's become the goal now. And that's actually an unhealthy thing in the sciences, in part because it's arbitrary. And also think about it. If you say, okay, you have a one in 20 chance of being hit by a car on any given day. Like that's really high. Yeah. Like you have a 1 in 20 chance of being wrong.

Like, that is pretty high. But then to have this number to be like, yes, now it's true, seems even more silly if you think about it. You have, like, for every 20 paper that you publish, one will just be, like, drawing the wrong conclusion. Just the number 5 is burned in your head as the truth threshold. Below five, it is truth. And I'm above five, and it's nothing. And that's silly.

It may be silly, but this feeling about the importance of the number five runs right through the past century of science. In 1925, Ronald Fisher wrote a book on statistical methods. It became so popular that he's now known as the godfather of statistics. He's also known as a proud eugenicist. That's a different story. Fisher understood that even the most careful minds doing the most careful work can jump to the wrong conclusions. He also understood you don't have infinite time and resources. You've got to draw a line somewhere. In his book, he wrote that he personally felt by setting our acceptable doubt level at 5%, we would, quote, not often go

astray. His gut instinct became extremely influential, and it led to a serious headache for the publishers of scientific research. Hello, I'm Stavroula Kusta. I'm the chief editor of Nature Human Behavior, a general science journal that covers all disciplines that have something to say about human behavior and selectively publishes research that is of high significance to the scientific community. Like Jonathan and Sherry, Stavroula Kusta has strong

opinions about how people use the famous 5% cutoff point when it comes to that crucial bit of statistical jargon, the probability value. P-values are the bane of our existence as editors. Several authors only report the test result and the p-value and nothing else. It's obvious that that tells you very little. Say you have a scientist who's interested in whether a new Alzheimer's drug works for slowing down memory decline. And she takes two groups of Alzheimer's patients. One group takes the drug, the other doesn't take the drug. Results come in, numerical difference is there. The scientist runs her significance test, and lo and behold, it is significant. However, that's one little bit of information. What we're more interested in is how big is the difference.

Drugs have side effects. So you want to take a drug only if you're going to see a measurable difference in your memory. For instance, if you're an Alzheimer's patient, if the difference is minuscule, even if it's statistically significant, it's practically meaningless. statistically significant, it's practically meaningless. How big the difference is and how certain you are about your estimates is arguably much more important than whether you have statistical significance that may, in practical terms, mean very little. Why would someone send you an article for your journal and not tell you this other stuff? Because it's a matter of convention. Convention is probably the worst thing.

And you'd expect scientists, being the smartest people around, would have figured out how to work around conventions and update their conventions to better ways of working. Journals used to prioritize for publication positive studies, given that you can't say very much about null results. So only studies that found a difference ended up being published. The ones that didn't find a significant difference ended up in the file drawers. And the file drawer problem in science is huge, because the true answer to questions that really matter to people may be in the file drawers, not in what was published. Things have started changing. Scientists are idealists. They want to be scientists because they want to find the truth. However, the conventions around using p-values,

competitiveness for publication, and a lot of other incentives were working against finding truth. I admit, as the show's appointed cheerleader for the number five, I wasn't sure I should bring up its role in this sordid tale of the five bad ones and the p-values and the publication bias. It doesn't sound like an argument for calling five the greatest number of all time. And yet, even if this all arises from an instinct to see five as trustworthy, we can't really say it's five's fault. It's not the fault of the math, but it's the fault of people. This is Professor Andrew Gelman. And I teach statistics and political science at Columbia University in New York.

Andrew puts the blame on that great human flaw, hubris. I think people want a level of certainty that they can't really have. They'd love to say, when you've reached the 5% level, that you can now act as if you know that something works or that you know something's effective. And in real life, it doesn't work that way. If you look at enough things, you're likely to see something that would be statistically significant. If something reaches this 5% level, it's likely to be large, because if it's small, it won't be statistically significant. That's one reason why effects tend to be overestimated

and one reason why things don't always replicate so well. Can you take me through that point just one more time? I'm not sure we'll get it. Well, okay. So I'll give you an example. There was a study done a few years ago of early childhood intervention. The parents of four-year-olds are given certain parenting tips, and then the kids are followed up 20 years later to see how they were doing. It was an economic study, and they looked at the earnings of the people, and the kids who were in the intervention group had 42% higher earnings than the kids not in the intervention group. If that had been 35%, it wouldn't have been statistically significant.

So, that study was so noisy, it would have to be at least 35% or 40% to be statistically significant in the first place. So, condition on this result being reported, it was going to be huge. So, then what they said was, well, we found this statistically significant effect, and it was huge, but it had to be huge. Otherwise, it wouldn't have reached the threshold. There kind of is a resolution to that, which is to report all the outcomes that you can look at and look at a series of studies over time rather than a single study. But when you pick out just one, that can create this headline effect. And then obviously, that can create this headline effect. And then obviously, the news media and even the scientific establishment can contribute to that by getting all excited about things.

So I guess the problem with the 5% is just if people, sometimes people become overconfident, which is, again, not the fault of the math. Five is a Janus character. On the one side, the face of ease and convenience, the natural middle point. But then there's the other half of it. Science needed somewhere that felt right for the meeting place between our two irreconcilable realities, the unknowable and our need to know. Or to put it more dramatically, the terrifying infinite randomness surrounding our puny lives and the comforting almost certainty we feel as our intellect makes meaning and sense to live by. We had to pick a number to represent that thin line. Without quite knowing

why, we chose five. I think it's, I don't know, it's a nice number. It's nice. Yeah. You're listening to We Give You Five, part of our series on the greatest numbers of all time. This episode was produced by Tom Howell. Thanks to Rebecca Hennessy and her band, Makeshift Island, as well as to all of our guests. My name is Clayton Fisher, and I'm a clinical audiologist. Next up in our series, Nine, a figure of synchronicity. You can hear the complete series of ideas documentaries about the greatest numbers of all time. Check out our podcast feed. Technical production, Danielle Dupal.

Our web producer is Lisa Ayuso. Senior producer, Nikola Lukšić. Greg Kelly is the executive producer of Ideas. And I'm Nala Ayyad. For more CBC Podcasts, go to cbc.ca slash podcasts.