## Posts tagged ‘graph’

### Guess the Graph

The bar graph below was created because of a recent discussion with my wife. The title and axis labels have been removed. **Can you identify the data set used to create the graph?** I’ll give you some hints:

- The data set contains 32 elements.
- It’s based on a real-world phenomenon from this year.
- The middle five categories account for 81% of the data.
- The special points marked by A, B, and C won’t help you identify the data set, but they will be discussed below.

Got a guess?

No clue? Okay, one more hint:

- The vertical axis represents “Teams.”

Still not sure? Final hints:

- Point A represents the lowly J-E-T-S, who are currently winless.
- The region outlined by B shows that 26 teams have from 3 to 7 wins.
- Point C on the graph represents my Pittsburgh Steelers, whose record is a perfect 10‑0. (It’s my hope that I’ll still be able to gloat on Friday morning, after the Steelers host the Ravens on Thanksgiving night.)

This graph was generated while discussing the current standings in the NFL with my wife, who speculated that there seemed to be a lot of really good teams and a lot of really bad teams this year. The horizontal axis represents the number of wins. As it turns out, the distribution above is somewhat typical at this point in the season. At the end of most seasons, about 2/3 of the teams finish a 16-game season with 5 to 10 wins. It may be a little unusual that there are 8 teams with 7 wins, but it’s not statistically cray-cray.

If you’ve read this far, then you may enjoy these other math-related football trivia questions:

- Describe two ways in which an NFL game can end with a score of 2‑0.
- What’s the greatest score that cannot be attained by scoring only touchdowns (7 points) and field goals (3 points)?
- Express the ratio of width:length of a football field. For length, include the end zones.
- What are the only positions allowed to wear single-digit uniform numbers?
- During a typical broadcast of an NFL game, approximately what percent of the time is spent actually playing football (as opposed to commercials, half time, or just milling around between snaps)?

Happy Drinksgiving! And, go Stillers!

—

Answers

- A game can end 2‑0 if one team scores a safety and the other team doesn’t score at all. It can also end 2‑0 if one team forfeits before either team has scored, by league rule. (In high school and college, a forfeit is officially recorded as a 1‑0 loss.)
- 11 points. Any point total above that is (theoretically) possible. Below that, it’s not possible to score 1, 2, 4, 5, or 8 points.
- A field is 53 1/3 yards wide and 120 yards long. In feet, that’s 160:360, which can be reduced to 4:9.
- Quarterbacks and kickers.
- According to several analyses, 11 minutes of a three-hour broadcast is spent actually playing. That’s about 3%. Sheesh.

### Guess the Graph on Square Root Day

Today is **Opening Day** in Major League Baseball, and 13 games will be played today.

It’s also **Square Root Day**, because the date 4/4/16 transforms to 4 × 4 = 16.

With those two things in mind, here’s a trivia question that seems appropriate. **Identify the data set used to create the graph below.** I’ll give you some hints:

- The data set contains 4,906 elements.
- It’s based on a real-world phenomenon from 2015.
- The special points marked by A, B, and C won’t help you identify the data set, but they will be discussed below.

Got a guess?

- The horizontal axis represents “Distance (Feet)” and the vertical axis represents “Frequency.”

Still not sure? Final hints:

- Point A on the graph represents Ruben Tejada’s 231-foot inside-the-park home run on September 2, 2015.
- Point B on the graph represents the shortest distance to the wall in any Major League Baseball park — a mere 302 feet to the right field fence at Boston’s Fenway Park.
- Point C represents the longest distance to a Major League wall — a preposterous 436 feet to the deepest part of center field at Minute Maid Park in Houston.

Okay, you’ve probably guessed by now that the data underlying the graph is **the distance of all home runs hit in Major League Baseball during the 2015 season**. That’s right, there were 4,906 home runs last year, of which 11 were the inside-the-park variety. The distances ranged from 231 to 484 feet, with the average stretching the tape to 398 feet, and the most common distance being 412 feet (86 HRs traveled that far). The data set includes 105 outliers (based on the 1.5 × IQR convention), which explains why a box plot of the data looks so freaky:

The variety of shapes and sizes of MLB parks helps to explain the data. Like all math folks, I love a good graphic, and this one from Louis J. Spirito at the Thirty81Project.com is both awesome and enlightening:

Here are some more baseball-related trivia you can use to impress your friends at a cocktail party or math department mixer.

**Who holds the record for most inside-the-park home runs in MLB history?**

*Jesse Burkett, 55 (which is 20 more than he hit outside-the-park)*

**Which stadium has the tallest wall?**

*The left field fence at Fenway Park (a.k.a., the “Green Monster”) is 37 feet tall.***Which stadium has the shortest wall?**

*This honor also belongs to Fenway Park, whose right field wall is only 3 feet tall.***Although only 1 in 446 home runs was an inside-the-park home run in 2015, throughout all of MLB history, inside-the-park home runs have represented 1 in ____ home runs.**

158**Name all the ways to get on first base without getting a hit.**

*This is a topic of much debate, and conversations about it have taken me and my friends at the local pub well into the wee hours of the morning. I have variously heard that there are 8, 9, 11, and 23 different ways to get on base without getting a hit. I think there are 8; below is my list.*

*(1) Walk*

*(2) Hit by Pitch*

*(3) Error*

*(4) Fielder’s Choice*

*(5) Interference*

*(6) Obstruction*

*(7) Uncaught Third Strike*

*(8) Pinch Runner*

**What is the fewest games a team can win and still make the playoffs?**

*39. The five teams in a division play 19 games against each of the other four teams in their division. Assume that each of those teams lose all of the 86 games against teams***not**in their division. Then they could finish with 39, 38, 38, 38, and 37 wins, respectively, and the team with 39 wins would make the playoffs by winning the division.**Bases loaded in the bottom of the ninth of a scoreless game, and the batter hits a triple. What’s the final score of the game?**

*1-0. By rule, the game ends when the first player touches home plate.***In a 9-inning game, the visiting team scores 1 run per inning, and the home team scores 2 runs per inning. What is the final score?**

*16-9. The home team would not bat in the bottom of the ninth, since they were leading.*

### Making Heads or Tails of Randomness

Which of the four graphs below have a collection of truly random dots?

To put a little distance between the question and the spoiler below, let me interject with a couple jokes about randomness and probability.

From the ANZSTAT listserv:

If you choose an answer to this question at random,

what is the chance you will be correct?

- 25%
- 50%
- 60%
- 25%

Had enough? Okay, then, on with the show!

The 25 dots shown in each graph were randomly selected, but four different methods were employed:

**Completely Random**: The coordinates of each dot were selected completely at random, independent of the other dots.**Random-ish**: The area was divided into a 5 × 5 grid of squares, and one point was then randomly selected within each square.**Dartboard**: I placed a 5″ × 5″ square on a dartboard and threw darts at it until I hit it 13 times. I then turned the square 90°; and threw darts until I hit it 12 more times. (I’m a reasonable dart player, but yes, there were quite a few darts that landed outside the square.)**Eli’s Choice**: I gave my seven-year-old son a 5″ × 5″ square and asked him to draw 25 random points.

Perhaps those descriptions will better help you determine which graph contains truly random dots. Care to take another look?

No? Okay.

**Graph C** was the most fun; it was **Dartboard**. I despise dumb-ass probability problems like, “What’s the probability that a randomly thrown dart will land inside the square and outside the circle?” What kind of person throws darts randomly? That’s just stupid, not to mention irresponsible. You’ll put someone’s eye out! Yet throw them randomly I did. And I’m happy to inform you that none of the walls around my dartboard were harmed in the making of this graph.

**Graph A** was **Random-ish**. Although there is some white space in the upper right and two dots are somewhat close in the bottom middle, generally the dots are well spaced.

**Graph D** was **Eli’s Choice**. Like Random-ish, Eli was rather careful to make sure that none of the dots were too close to one another. I divided the square into 25 unit squares after he drew his dots, and there was a single dot in 23 of them. There are some definite linear tendencies.

By process of elimination, you now know that **Graph B** was **Completely Random**. The function **=5*RAND()** was used to choose each coordinate for all 25 points in an Excel spreadsheet, and then they were graphed as a scatterplot.

So, what’s the point?

Graph D was generated by a seven-year-old, but it’s not terribly different from what would be generated by an average adult. What most people think of as random is usually anything but. Truly random data usually has clusters within it. For instance, flip a coin 200 times, and there is an 80% chance that there will be a run of at least 6 consecutive heads or 6 consecutive tails within your data.

For graph B, the clustering that happens in the upper left and with the two dots that touch in the lower right is completely typical when data is generated at random. As humans, we often try to ascribe some special meaning to these occurrences. For instance, when there are ten homicides in one week in a city that normally only has two homicides per week, headlines will declare “Murder On The Rise,” and the mayor will hold a press conference on what the police are doing to address the problem. But there’s nothing to be done. This kind of grouping is normal. A few weeks later, when there are zero homicides in one week — which is also typical — there will be hardly a notice.

If you didn’t correctly choose graph B, then perhaps you’re random-averse.

Or, more likely, you’re just normal.

### A Gridiculously Clever Blog Post

Do you know what the following graph represents?

*Sine on the dotted line.*

If you tell that joke to the right audience, you’ll likely hear a *triggle*. (If you tell it to the wrong audience, you’ll likely hear the sound of tomatoes whizzing past your head.)

*Triggle* is a portmanteau, a combination of two or more words and their definitions.

trigonometry + giggle= triggle

In a similar vein, when the expression

13 + 5 · 0 – 4

is simplified to

13 – 4,

you might say that it has suffered from *zerosion* — the removal of a term because of multiplication by zero.

The following portmanteaux may be useful for your next math discussion.

**bi·sect·u·al**

*adjective*

attracted to both halves of an angle

**grid·ic·u·lous**^{1}

*adjective*

inviting derision on the coordinate plane

**cha·rad·i·us**

*noun*

a segment from the center to the circumference based on false pretenses

**bi·zarc**

*noun*

an unusual curve

**graph·ish**

*adjective*

diagrammatically disreputable

**sub·line**

*adjective*

inspiring awe in only one dimension

**trig·a·ma·role**

*noun*

a complicated and annoying trigonometric process, such as verifying that

cot *x* + tan *x* = sec *x* · csc *x*

^{1} It came to my attention after the publication of this post that Gridiculous is (a) a trivia game developed for Windows 8 and (b) an HTML5 responsive grid boilerplate (though the link to the site seems not to be working).