## Posts tagged ‘average’

### Mr. Consistency, Khris Davis

If you flipped four coins, the probability of getting exactly one head would be 0.25.

But the probability of doing that four times in a row is much lower, somewhere closer to 0.0039, or about 1 in 250.

Now, imagine flipping 100 coins four times, and getting the same number of heads each time. The odds of that happening are only slightly better than impossible. In fact, if every person *in the entire world* were to flip 100 coins four times, it would still be highly unlikely that this would ever happen.

That’s how rare it is, and it gives you some idea of what Major League Baseball player Khris Davis just pulled off. The Oakland Athletics outfielder just finished his fourth consecutive season with a batting average of .247. That’s right — the same average four seasons in a row.

Davis had some advantage over our coins, though. For starters, he wasn’t required to have the same number of at-bats every year. Moreover, batting averages are rounded to three decimal places, so his average wasn’t *exactly* the same during those four years; it was just really, really close:

**2015**: .24745 (97 hits in 392 at-bats)**2016**: .24685 (137 in 555)**2017**: .24735 (140 in 566)**2018**: .24653 (142 in 576)

How could something like this happen? According to Davis, “I guess it was meant to be.“

Perhaps it *was* predestination, but I prefer to put my faith in numbers.

Empirically, we can look at the data. From 1876 to present, there have been 19,103 players in the major leagues. The average length of an MLB career is about 5.6 years, which means that an average player would have about three chances to record the same batting average four seasons in a row. It’s then reasonable to say that there have been approximately 3 × 19,103 = 57,309 opportunities for this to happen, yet Khris Davis is the only one to accomplish this feat. So experimentally, the probability is about 1 in 60,000.

Theoretically, we can look at the number of ways a player could finish a season with a .247 batting average. In 2007, the Phillies’ Jimmy Rollins recorded an astounding 716 at-bats. That’s the most ever by a Major League Baseball player. So using a sample space from 1 to 716 at-bats, I determined the number of ways to achieve a .247 batting average:

- 18 hits, 73 at-bats
- 19 hits, 77 at-bats
- 20 hits, 81 at-bats
- 21 hits, 85 at-bats
- 22 hits, 89 at-bats
- 36 hits, 146 at-bats
- …
- 161 hits, 652 at-bats
- 161 hits, 653 at-bats
- …
- 177 hits, 716 at-bats

And, of course, there are the examples above from Davis’s last four seasons.

It’s interesting that it’s not possible to obtain a batting average of .247 if the number of at-bats is anywhere from 90 to 145; yet it’s possible to hit .247 with 161 hits for either 652 or 653 at-bats. I guess it’s like Ernie said: “That’s how the numbers go.“

All told, **there are 245 different ways to hit .247** if the number of at-bats is 716 or fewer.

That may sound like a lot, but consider the alternative: there are 256,441 ways to **not** hit .247 with 716 or fewer at-bats.

So, yeah. No matter how you look at it, what Davis did is pretty ridiculous. Almost as ridiculous as what happened to Saul…

Saul is working in his store when he hears a voice from above. “Saul, sell your business,” the voice says. He ignores it. His business is doing well, and he’s happy. “Saul, sell your business,” the voice repeats. The voice goes on like this for days, then weeks. “Saul, sell your business.” Finally, Saul can’t take it any more. He finds a buyer and sells his business for a nice profit.

“Saul, take your money, and go to Las Vegas,” the voice says.

“But why?” asks Saul. “I have enough to retire!”

“Saul, take your money to Las Vegas,” the voice repeats. It is incessant. Finally, Saul relents and heads to Vegas.

“Saul, go to the blackjack table and bet all your money on one hand.”

He hesitates for a moment, but he knows the voice won’t stop. So, he places his bet. He’s dealt 18, while the dealer has a 6 showing. “Saul, take a card.”

“What? The dealer has…”

“Saul, take a card!” the voice booms.

Saul hits. He gets an ace, 19. He sighs in relief.

“Saul, take another card.”

“You’ve got to be kidding me!” he pleads.

“Saul, take another card.”

He asks for another card. Another ace, 20.

“Saul, take another card,” the voice demands.

But I have 20!” Saul shouts.

“TAKE ANOTHER CARD, SAUL!”

“Hit me,” Saul says meekly. He gets another ace, 21.

And the voice says, “Un-fucking-believable!”

### The Beer Paradox

From Gene Weingarten’s recent column, “Rhymes Against Humanity,” in the January 28 edition of the *Washington Post Magazine*:

An infinite number of mathematicians

Walked into a bar on one recent night,

And, under the strangest of barroom conditions,

What followed quite nearly became a big fight.“I’ll have a pint,” said the first to the ’tender.

“I’ll have a half,” said the next fellow down.

“I’ll have a quarter,” said the third (no big spender).

“Give me an eighth,” said the next, like a clown.The bartender fumed and grew suddenly pale

Then, calmly, he turned and he went to the spout

Drew up two pints, set them down at the rail.

Said, “Enough of this nonsense — you all work it out.”

This is an MJ4MF original, though like Gene’s, it’s based on a stale, old joke:

With my head in an oven

And my feet on some ice,

I’d say that, on average,

I feel rather nice!

What other classic math jokes can be easily converted to poems? Or have already been?

### Why Joe Doesn’t Think He’s Average

Today is an average day, with 182 days left in the year, and 182 days in the rear-view mirror.

I know a lot of average jokes:

When my stats teacher said that I was average, she was just being mean.

With my head in a fire

And my feet on some ice,

I’d say that, on average,

I feel rather nice.Two men are sitting in a bar when Mark Zuckerberg walks in. One of the men says to his friend, “How awesome! On average, everyone in this bar is a billionaire!”

The last joke highlights the issue with using the arithmetic mean when the distribution would be more meaningful. Let’s assume that one person in the bar has a net worth less than $10,000, two people have net worth between $10,000 and $100,000, and nine people have net worth between $100,000 and $1,000,000. (These are reasonable estimates for the distribution of net worth in the U.S., by the way.) Then a hypothetical histogram with a logarithmic scale showing the net worth of all the people in the bar might look something like this:

The average net worth of all 13 people in the bar is over $1,000,000,000 — actually, it’s over $4,000,000,000, because Zuckerberg’s net worth is around $60 billion — but only one of them actually has that much money.

In general, describing data with its average is a terrible idea…

If you’re an “average” person, then you’re a 5’9” (10%) male (50%) with brown eyes (55%) and straight (55%), black hair (85%) who wears size 10.5 (US) shoes (20%). You have 25 teeth in your mouth (30%) — including 4 wisdom teeth (80%) — normal color vision (95%), and O+ blood (40%), but you don’t have dimples (75%). You don’t have hitchhiker’s thumb (75%) or a bent little finger (95%), either, but you can roll your tongue (80%). You have an innie belly button (90%), loops in your fingerprints (65%) instead of whorls or arches, and attached earlobes (65%), and when you interlace your fingers, your left thumb rests atop your right thumb (55%). You sleep 6.5 hours per night (15%), smoke 800 cigarettes per year (10%), and consume 2 alcoholic drinks per week (30%). You’re 29 years old (2%), eat 3 servings of fruits and vegetables a day (20%), get 70 minutes of cardio exercise per week (25%), and have a body mass index (BMI) of 25 (15%).

And thanks to the artistic styling of Paul Wrangles at Sparky Teaching, the average person might look a little something like this:

The numbers in parentheses represent the percent of the world population that has the given characteristic. Admittedly, they’re WAGs; I grabbed each statistic from a random location on the web, and I have absolutely no data to back up any of these claims. Moreover, they’re not very precise; I rounded each to the nearest five percent, because a greater level of precision might give the appearance that they’re somehow more accurate.

That said, I don’t think they’re horribly wrong, either, and even if they’re slightly off, they’ll still serve my point. Which is this: Though this description captures an “average” person, it’s pretty far from representing a typical person. The probability that such a person actually exists is only about 1 in 3,500,000,000.

So if you read the description above and thought, “Hey, that’s me!” then you should feel pretty special, indeed — there is likely only one other person in the world with those same characteristics.

The characteristics for the average person used above are sometimes the **mean** for the category (height, shoe size) and sometimes the **mode** (eye color, fingerprints). Both of these measures of central tendency are known as averages, as is the **median** (which I used at the start of this post when claiming that today is an average day).

Life expectancy is another one of those situations where the average provides misleading — or, at least incomplete — information.

Today, the infant mortality rate worldwide is just under 5%, and life expectancy is 71 years. A very simplistic model for this data is to assume that 19 out of 20 people will live to age 75, but 1 out of 20 will die during their first year of life. This model is clearly wrong, but as George Box said, “All models are wrong, but some are useful.” Check out the math with this model:

This model is useful, because it shows that the 95% of people who survive infancy can expect to live to age 75.

Now compare that to the middle ages, when the infant mortality rate was a staggering 30%, and life expectancy was 35 years. Again using a simplistic model, 7 out of 10 people would live to age 50, while 3 out of 10 would die before they reached the age of 1. The math looks like this:

So, there’s a problem with using the average to talk about life expectancy, because the distribution in the middle ages was badly skewed by so many childhood deaths.

If we compare life expectancy now to the middle ages using the average of the entire population, it’s a distorted picture. But when we remove the deaths as a result of infant mortality, it’s a little less bleak: those living past age 1 today have a life expectancy of 75 years; those living past age 1 in the middle ages had a life expectancy of 50 years. The scales are still tipped heavily in our favor, but it doesn’t seem quite as drastic as a ratio of 71 to 35.

To put this in perspective, the life expectancy in 1950 was just under 50 years. Most of the increase in life expectancy has actually happened in the last century; during the last 70 years, longevity has increased by more than 20 years.

How typical are you? How long will you live? I have no idea, but I do know this: Half of the people you know are below average.

### A Father’s Day Gift Worth Waiting For

Alex made a Father’s Day Book for me. Because the book didn’t make it on our trip to France, however, I didn’t receive it until this past weekend. It was worth the wait.

The book was laudatory in praising my handling of routine fatherly duties:

I loved when you took me to Smashburger.

I appreciated when you helped me find a worm.

I love when you read to me at night.

I love when I see you at the sign-out sheet [at after-school care]. It means I can spend time with you.

But my favorite accolade — surprise! — was mathematical:

I liked the multiplication trick you taught me. Take two numbers, find the middle [average], square it. Find the difference [from one number to the average], square it, subtract it. (BOOM! Done!)

Priceless.

The trick that I taught him was how to use the difference of squares to quickly find a product. For instance, if you want to multiply 23 × 17, then…

- The average of 23 an 17 is 20, and 20
^{2}= 400. - The difference between 23 and 20 is 3, and 3
^{2}= 9. - Subtract 400 – 9 = 391.
- So, 23 × 17 = 391.
**BOOM! Done!**

This works because

,

and if you let *a* = 20 and *b* = 3, then you have

.

In particular, I suggested this method if (1) the numbers are relatively small and (2) either both are odd or both are even. I would not recommend this method for finding the product 6,433 × 58:

- The average is 3,245.5, and (3,245.5)
^{2}= 10,533,270.25. - The difference between 6,433 and 3,245.5 is 3,187.5, and (3,187.5)
^{2}= 10,160,156.25. - Subtract 10,533,270.25 – 10,160,156.25 = 373,114.
- So, 6,433 × 58 = 373,114.

Sure, it works, but that problem screams for a calculator. The trick only has utility when the numbers are small and nice enough that finding the square of the average and difference is reasonable.

Then again, it’s not atypical for sons to do unreasonable things…

Son: Would you do my homework?

Dad: Sorry, son, it wouldn’t be right.

Son: That’s okay. Can you give it a try, anyway?

I’m just glad that my sons understand math at an abstract level…

A young boy asks his mother for some help with math. “There are four ducks on a pond. Two more ducks join them on the pond. How many ducks are there?”

The mother is surprised. She asks, “You don’t know what 4 + 2 is?”

“Sure, I do,” says the boy. “It’s 6. But what does that have to do with ducks?”

### Math for Figger Filbert*

A well-known problem:

A man walks 1 mile south, 1 mile east, and 1 mile north. He arrives at the same place where he started, and then he sees a bear. What color is the bear?

The answer, of course, is white. It’s a polar bear. These three moves will let a person return to the same place if he starts at the North Pole. (The person could also return to the same place if he starts at an infinite number of points near the South Pole, too. He could start at a point so that when he walks 1 mile south, he is at a point such that the east-west circle on which he is standing has a circumference of 1 mile. Then, he can walk 1 mile east to return to the same spot. Finally, he can walk 1 mile north, and he’s back where he started. Then again, he could also start at a point so that he can walk 1 mile south to a point where the circumference of the east-west circle is 1/2 mile, do that loop twice, then walk 1 mile north. Or find points where the cicumference is 1/3 mile, 1/4 mile, 1/5 mile, etc. You get the idea. However, since there are no bears in Antarctica, the answer to my original question is still correct.)

Two points about this:

- In answer to the question, “Are there polar bears in Antarctica?” there is only one correct answer:
*Only if they are bipolar.* - I really don’t care to receive silly comments about how a bear trapper could capture a grizzly and take him to Antarctica, or how a brown bear might mistakenly meander north to the Arctic Circle.

Here is a similar question:

A man runs 90 feet, turns left, runs another 90 feet, turns left, runs another 90 feet, and turns left. He is now headed home, and two men with masks are waiting for him. Who are they?

If you don’t know the answer to this riddle, remember that today is the first day of the World Series. My prediction? The Rangers will win easily. It’s not really a fair fight. I mean, members of the Lone Star State’s law enforcement agency with opposable thumbs and automatic weaponry versus defenseless birds? Seriously, if the Rangers don’t win, then we need to seriously reconsider the theory of natural selection.

If you watch the first game of the World Series tonight, remember to enjoy the game. Please don’t get caught up trying to figure out if it converges or diverges.

Here are a few baseball-related math puzzles:

- A baseball player has four at-bats in a game. At three different times during the game, his batting averages for the entire season (rounded to three decimal places) have no digits in common. What was his average at the end of the game?
- During a little league game, the visiting team scored 1 run per inning, and the home team scored 2 runs per inning. What is the final score of this seven-inning game?
- During the first half of the season, Derek batted .100, but his average was .300 during the second half of the season. Similarly, Alex batted .200 the first half of the season and .400 the second half of the season. Both players ended up with the same number of total at-bats, yet Derek had a higher batting average for the entire season. How is this possible?

* *Figger Filbert* is a term for baseball fans who are obsessed with statistics. Such fans are easily identified; they will make statements like, “Did you know that Albert Pujols is batting .275 when facing married pitchers in suburban ballparks that only sell popcorn on the mezzanine level?” It is a synonym for *number nut*.

### Stuck in the Middle With You — July 2

Today is an average day, exactly halfway between the beginning and end of the year.

Benoit Mandelbrot, the father of fractal geometry, often said he was born in Poland and educated in France — making him German, on average.

*Average* is something upon which hens lay their eggs. For instance, “My hens lay four eggs a week on average.”

When she told me that I was only average, she was just being mean.

A statistician with his head in the freezer and his feet in the oven will say that, on average, he feels fine.

Three statisticians go hunting. When they see a duck flying overhead, two of them take a shot. The first fires six inches over the duck; the second fires six inches under the duck; and, the third excitedly exclaims, “We got it! We got it!”

### Analyzing My Favorite Game

I’ve discussed my favorite game before, which is played as follows:

- On a piece of paper, everyone playing writes down a positive integer.
- Show your number to a neighbor (for verification purposes only).
- The winner is the person who wrote down
*the smallest integer not written by anyone else*.

I recently used this game with a group of 32 people at the end of a presentation. The first round was a sample round only, and folks didn’t know the rule for determining a winner before choosing their number. (People often find the rule confusing, so I often do a first round where I don’t tell folks the rule until *after* everyone has written down a number. Then I give the rule, and we determine the winner to provide an example of the rules in action.)

We played six rounds. I gave the winner of the third and sixth rounds a copy of *Math Jokes 4 Mathy Folks*; the other rounds were just for fun.

I’ve always been curious about the strategy that folks use when playing this game, so I asked folks to record their numbers for each round, and then I collected their choices. Geek that I am, I analyzed the results, and I thought I’d share them with you. (Don’t you feel special?)

In Round 1, choices were all over the charts. This is to be expected, since folks had no idea why they were choosing a random number. Choices ranged from 3 to 99, with a mean of 15.9.

After the rules were revealed, though, things got more interesting. The charts below show the choices during Rounds 2‑6. (Horizontal axis is the number chosen; vertical axis is the number of attendees who chose the number.)

Some observations about these results:

- The maximum number chosen by any player decreased in each round.
- The average value chosen for Rounds 2‑6 was 8.5, 7.5, 5.3, 7.7, and 6.4, respectively. It’s interesting that the average decreased to 5.3, then shot back up to 7.7. This might be explained by the trend in winning values. The winner in Rounds 2 and 3 chose 2. In Round 4, at least three players chose each of the numbers 1‑4, and the winner chose 5. Players may have assumed that too many players were tending to choose low numbers, so they chose slightly higher numbers in Round 5.
- No fewer than five players chose the number 7 in every round.
- Interestingly, the winner who chose 1 in Round 5 was also the winner who chose 1 in Round 6. She went to the well twice — and it paid off!

A few days later, I ran the same experiment with a different group of 35 people. The results were slightly different.

- The average value chosen decreased in every round, as follows: Round 1, 11.96; Round 2, 7.54; Round 3, 7.53; Round 3, 4.88; Round 4, 4.84; and Round 6, 4.70. As with the previous group, the average took a big dip from Round 3 to Round 4. However, unlike the previous group, the average did not shoot back up in Round 5.
- No fewer than four players chose the number 4 in every round, and it was chosen by 10 players in two different rounds.
- Even after learning the rules by playing a practice round in Round 1, several folks chose surprising numbers in Round 2. Among them: 100; 102; 1,000; 1,900; and 10
^{100}.

I hope you enjoyed this diversion. We now return you to our regularly scheduled programming…