Political Polls and margin of error

At the time of my writing this, it is about a week until the election. It's Trump vs. Clinton - Duel of the Century. I thought I would look into political polls as a mathematics application this week. Specifically, let's look at what is called the margin of error.

I looked at a couple of what I think are reputable websites. However, they seemed to not quite get this concept. For example, something like this was stated by a couple of sites:

A poll states that candidate A is at 52% with a margin of error of +/- 3%. This means the candidate could actually be polling anywhere from 49% to 55%.

Unless I've been lied to in my past math classes, I believe this is wrong information. This is a common misconception, but I didn't think I would find news agencies writing this.

He is what I believe is the correct scoop. Polls usually have a confidence level. Part of the confusion is when CNN, NBC, etc mention their polls, they don't talk about this. Anyway, for most polls it is 90%. So, in actuality, a much truer fact is that there is a 90% chance that that candidate A is between 49% and 55%. She (or he - I'll stick with "she" the rest of the way so I don't have to mention both genders each time. Why "she" rather than "he"? I flipped a coin. Seriously.) is probably in that range, by she can't be certain of that.

You can never be certain of polls. Common sense tells you that you can't have absolute certainty. If there are millions of voters in the U.S., and your survey covers a few thousand, how do you know you didn't just happen to survey only ones that are against candidate A. Yes, unlikely, but it could happen. So if a poll states A is ahead of B, 57% to 42% with a margin of error of 5%, it's all over, right? No, it isn't. It's not looking good for B, but it's not all over.

We see surveys during election years a lot, but we see them often at other times without knowing it. The government's unemployment reports, bestselling books, the top TV shows for the week are all done by random sampling of a relatively small sample.

Students could figure out the margin of error. It goes like this:

Margin of error = z x squareroot(p(1-p)/n). The z-value is based on how accurate you want your poll result to be. You would have to look that up. The value of p is your polling result and n is the number in your sample. (Oddly, the number in your total group, whether it is the entire U.S., the state of Oregon, or your bowling league, has nothing to do with the answer.)

Common sense tells us that there is a trade-off. The more exact you want to be, the wider your interval is going to end up being. I might be able to state, from a recent survey of adult males, that I am 90% certain the average height of all adult males is between 5'7" and 5'11". One the other hand, if I want to be 99.99% certain, I might only be able to state that the average height is between 3' and 8'. You gain in certainty and you lose in precision.

Let's try one out.

We polled 1,000 people. Of those, 560 said they would vote for Candidate A. So, she is polling at 56%. We want to be 90% certain of the range her number would actually land in. Looking up the 90%, we find a z-value of 1.645.

1.645 x squareroot(.56(1-.56)/1,000) = .0258. If we round it to 2.5%, she is 90% sure of her actual number being between 53.5% and 58.5%.

Just for fun, here are some other possibilities.

Suppose we chose a confidence level of 95%:

95% corresponds to z = 1.96, so
1.96 x squareroot(.56(1-.56)/1,000) = 3.1%, giving a range of 52.9% to 59.1%

Suppose we take our original example and assume we surveyed twice as many people:
1.645 x squareroot(.56(1-.56)/2,000) = 1.8%, giving a range of 54.2% to 57.8%

I was right. That was fun.

Standard Deviations and Baseball

Its World Series time and I feel compelled to stick with a baseball theme this week. I've considered this application since I was not much more than a child. I wasn't sure how the math on it would work, and I'm still not certain, but I thought it would be worth exploring.

Batting averages are the ratio of hits to times at bat. So getting one hit in four times up to bat gives a batting average of .250.

It would make sense that the overall batting average in baseball might vary over the years. Things have changed since it started in 1869. There used to be no night games. That is mostly because the electric light hadn't been invented yet. Night games have made it harder for hitters. Although, they've outlawed spit balls. That has made it easier.

Does it all even out? Apparently not. There used to be quite a few batters that hit .400 or better for a season. No one has done that in the past few decades, though. I've wondered if there a way to even things out mathematical. I've seen some attempts at this.

I found a person's website that has the major league batting average for each season. Over a century worth, it is at .263. The highest year ever was 1894 when it was .309. So maybe a player that year could have their batting average dropped by .046 (.309 - .263 = .046). Similar adjustments could be made for players of each year.

Not a bad idea. I've seen other similar methods. However, I've thought that some measure of variance should come into play. I've had a theory that the standard deviation of the batting average statistics have been going decreasing over the years. So, there were more .400 hitters in the past, far above the league average, but I would guess that back then there were also more hitters far below the league average.

Why might that be? Now there are scouts going to colleges, high schools, Japan, Dominican Republic, etc. looking for possible talent. In the early days, they took what they could get. It wasn't necessarily the best baseball talent. Someone might come in from the coal mines, look pretty good, and you sign him to a contract. Over the years the process has improved.

To take a shot at that proving my theory, I used a website that showed the league average for each year. I then entered twenty years worth of yearly batting averages and found the standard deviation. Its not perfect, but I think it kind of backs me up. Here we go:

1871-1900  Standard deviation = 15.91
1901-1920  Standard deviation = 10.66
1921-1940  Standard deviation = 7.38
1941-1960  Standard deviation = 3.76
1961-1980  Standard deviation = 7.91
1981-2000  Standard deviation = 5.84
2001-2012  Standard deviation = 5.15

So to really do this right, I probably should find the standard deviations of each individual year using each player, rather than using the year as a whole. However, that seemed like a lot of work, so I settle for this. I bet there is some data base that has all the averages and the capability of adjusting the mean averages and the standard deviations for each year and adjusting each player's batting average accordingly. It won't be me, but somebody should take that on.

No hitters

Sorry, but I can't help but go back to baseball stats for the next couple weeks. It is playoff time for baseball, so I can't really help it.

Clayton Kershaw had a no hitter going for a while a couple days ago. No hitters are pretty rare. I got to thinking that you could maybe estimate the chances of a no hitter. Let's say a team would normally bat 0.250 against you. That is, they would get a hit every four times at bat. What are your chances of a no-hitter? You need to retire 27 batters (3 in each of the 9 innings). The probability you retire the first batter is .75. The probability of retiring the second batter is 0.75 x 0.75. The probability of having a no hitter in just the first inning is 0.75 x 0.75 x 0.75 or 42.2%.

For the whole game, the probability of a no-hitter would be (0.75)^27 = 0.000423. Unlikely.

You can give up walks or have batters reach on errors and still have it count as a no-hitter. I don't think we need to take that into account, though, as they do not count as official at-bats anyway. I had to think about that a bit, but I'm pretty sure I'm right on that.

Then I thought about estimating how many there should be in a season or any given period of time. I went back to 1998 because that is the last year major league baseball added teams. Since then, to the present day, there have been 30 major league baseball teams. With 162 games for each team, from 1998 to 2016 (19 years) there have been 162 x 30 x 19 = 92,340 save opportunities. If we use the above probability, the number of no-hitters during that time would be: 92,340 x 0.000423 = 39.1 saves.

How many have there actually been? 49. Keep that number in mind. We'll compare other outcomes to that.

So, not bad. In fact, a lot closer than I thought it would be.

The big question mark in all this, I think, is the batting average. The overall major league average is a little higher than this, maybe 0.260. Doing the math again would give an estimate of 27.2 (lower than the aforementioned 39.1).

But maybe we shouldn't be talking about the league average. You would figure the type to get a no-hitter is a better than average pitcher. And in fact, looking at the list of those that have thrown no-hitters shows some of the best pitchers of the past 20 years - Jake Arrieta, Max Scherzer, Cole Hamels, Clayton Kershaw, and Justin Verlander. (And there have been some pitchers that had some talent, but also a good amount of luck on their side that day of their no hitter.)

So maybe the correct batting average would be 0.240 -- 55.9 no-hitters.

Or maybe a batting average would be 0.230 -- 79.6 no--hitters.

Anyway, for those somewhat interested in the topic of baseball this was an interesting math application on baseball and probabilities.

Running Pace

I'm running in a race in about a month. Its a half marathon, which is 13.1 miles. I think I can make it, but I'm not absolutely certain of that. Being in shape for a race is not really enough of a challenge for me. The only reason I'm doing it is for the scenery. It goes across the Golden Gate Bridge. Twice, in fact. So that will be an adventure in itself. I have run two other races in the past that are highlights in terms of the races themselves. One was in Knoxville, Tennessee that finished on the 50 yard line of the University of Tennessee stadium. The other was a half marathon in Indianapolis whose course included one lap (2.5 miles) on the Indy 500 track.

Anyway, on to math. Since I'm not 100% sure I can even finish, I'm definitely not sure what pace I should try to run. There was a predictor in my latest copy of Runner's World Magazine. They gave a way to predict your pace in various races by looking at times for shorter distances. I thought - good application.

The had predictors for the 5K, 10K, half marathon, and marathon. Since it applies to my situation, I'll just use the one for the half marathon.

• The Workout - "Race" a 10K at 80 percent effort. 
• The Formula - Take your 10K time in minutes (for example, a 55:30 is 55.5) and add 0.93. Multiply the result by 2.11.
• When - Three to five weeks before race day.
• Why - A 10K is great because it has that endurance aspect of a half marathon but doesn't require you to run too much so close to race day,

Yes, they could have condensed things quite a bit by using an equation rather than an explanation.

So the "formula" is f(x) =  2.11(x+0.93), 

To take their example of 55:30, you would have a half marathon time of f(55.5) = 2.11(55.5+0.93) = 119.067 minutes or 1 hour 59 minutes 4 seconds.

To put a little more algebra into this, we could say that we are hoping to run the half marathon in one hour 50 minutes. What kind of 10K would predict that kind of time?

Answer:  110 = 2.11(x+0.93), so  x = 51.203 or 51 minutes 12 seconds.

The other three races; 5K, 10K, and marathon; have different, but similar formulas, and would be great for Algebra I classes.

Morse Code

I saw something about Morse code and thought it might be an interesting topic as a mathematics application.

First, some background.

Samuel Morse was born in 1791. He attended Yale, graduating in 1810. He aspired to be a painter. I didn't realize he did of this other career until I read about his paintings in David McCollough's book, The Greater Journey: Americans in Paris. Here is his portrait of President James Monroe.

He lost his wife and both parents in a three year span. As an escape, he went to Europe. During this time he made some contacts that led to led to the invention of Morse Code.

It didn't catch on for a few years. A U.S. congressman showed interest and a test was done with a wire stretching from Washington D.C. to Baltimore. He successfully asked, "What hath God wrought" and the rest is history.

It relies on a series of dots and dashes. They can be communicated with electronic impulses or light impulses. It was very important, but began to fall out of favor with the invention of Bell's telephone in which actual words could be used instead of a code for spelling out words. It is still used in various areas, including signal lamps by the coast guard. Those without speech can use the tapping of Morse code to communicate. But for the most part, it is found in history books.

SOS, for example, is ...---... How many are combinations of dots and dashes are needed to cover the alphabet? This could be found use the fundamental counting principal (If there are "m" ways to do one thing, and "n" ways to do another, there are "m x n" ways to do both.)

• Using one symbol means a dot or a dash could be used - two choices.
• Two symbols means there are 2 x 2 = 4 ways.
• Three symbols means there are 2 x 2 x 2 = 8 ways.
• Four symbols means there are 2 x 2 x 2 x 2 = 16 ways.

Since there are 26 letters in our alphabet, this still isn't enough. We could use five symbols, but that makes it more cumbersome. I can be done, though, by using one, two, three, or four symbols. Since 2+4+8+16 = 30. That is plenty to cover the whole alphabet.

If we need more that just words - digits, or symbols like ? and ;, we are going to need more. So for them, we need to use 5 symbols. How many possibilities would that give us?

Two to the fifth power is 32, and that means we have 2+4+8+16+32 = 62 possibilities. That gives us enough for 26 letters, 10 digits, and 26 more symbols beside.

Where Are We Going to Use This?

I was browsing through and found one of those site where someone asks a question and others respond with their thoughts. This one asked about where advanced math gets used in real life. I thought, "Hey, that is right up my alley", so I checked it out.

I thought I would include it here in my blog. The responses are quite interesting. I just pasted them in as is, so there might be some grammatical or logical errors. That is just what makes it interesting. The original website was at https://www.physicsforums.com/threads/what-is-advanced-level-mathematics-used-for.500513/

So, here is what people had to say:

I barely understand the bare basics of algebra, my math skills are abysmal. But what applications do advanced mathematics such as stochastic calculus and linear algebra have? Other than in physics, science, and engineering, what other things can advanced-level mathematics be used for? How about in daily life?

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It can enhance your abstract thinking ability. Not directly useful for anything in daily life.

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I fully disagree, how do you think barcodes were invented? A few uses of Linear Algebra: Codabar system Digital image compression Calculating life expectancy Modelling population growth Profit maximization Universal Product Code Lots more. If you want a thorough discussion of exactly how they are used, then just open up some linear algebra books, or do a Google search. Higher math isn't just solving puzzles (In fact, that's not really what math is). People don't just do mathematics to improve their thinking abilities; it certainly helps, but it has many applications. Keep in mind that mathematics need not be applied to anything. Just because you can't use a result of mathematics (at first) for anything practical does not make it useless.

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There is great deal of uses in physics (also era involves some Chemistry) and engineering. In general relativity a lot of linear algebra and calculus is required. In engineering, mostly calculating some basic mechanics problem. Other than these, Economics uses great deal of calculus to model the market which is very important. You can search more on financial mathematics (not accounting~~boring).

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Also, I've heard that matrices can be used to balance out chemical reactions in chemistry. I haven't actually done that myself, but I was happy to know that there was an easier way than what I did in my first year chemistry class!

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Oh sure it's useful in all sorts of professions. But not daily life, and many well paid jobs don't need it either. But the trouble is, when you're 16 you don't quite know what you'll be doing in 10 years time, and by then it's too late to learn so easily. 

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Everything has its applications. Of course, applications of some fields are more obvious than others.

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Mathematics models the natural world. So your question is nearly identical to ...... What use is it to learn English ? Furthermore, Logic, one of the foundations of Mathematics, is the link between The Arts & The Sciences. All art forms (nearly) seek to communicate. How better to make your case than with clear precise easy to understand logic ? Be it painting, screenplay, poem, courtroom summation or a novel. Some of the very the best lawyers were good at Math. That is one of the reasons they excel at the Law. Mathematics underlies nearly everything you see around you. But it will not guarantee a good life. That is the province of religion and moral philosophy.

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I think balancing equations is simply simultaneous equations, I mean for rather complicated equations. Matrices are just simple forms of simultaneous equations, they just save your paper and ink. Of course there are many ways of balancing equations, but many of them might not work for all cases. Also, matrices are very useful in doing statistics, though I haven't learn much of statistics, I heard of something called covariant matrix that is used for complicated systems. And statistics can be applied to many areas. May be you can look more on that.

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How does that even matter? Just because some random CEO doesn't use his knowledge of basket-weaving doesn't make basket-weaving useless. That's a really poor argument. Why does everything have to be immediately useful in daily life, and by daily life, I assume you mean eating, breathing, sleeping, and no more. It seems to me that you think that if you don't use something every day, or can't use it to make lots of money, then it's useless. As said many times in this thread, mathematics is all about logical and abstract thinking; it's basically a form of creativity. Now tell me, how useless are the former?

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When people ask me what the use is for mathematics, I always respond with the following poem by Morris Bishop: There's a tiresome young man in Bay Shore. When his fiancee cried, "I adore the beautiful sea". He replied, "I agree, it's pretty, but what is it for?"

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He never said higher math was useless in the general context, simply useless in daily life. There's nothing wrong with the validity of his statement since we take the meaning of "useless" in every day conversation as "generally useless" rather than "completely useless". However, there are few professions that are useful in daily life, such as cooking, etc. so the statement, although basically true, is misleading.

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Cryptography is a pretty big one.

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I prefer this: A math professor, a native Texan, was asked by one of his students: "What is mathematics good for?" He replied: "This question makes me sick! If you show someone the Grand Canyon for the first time, and he asks you `What's it good for?' What would you do? Well, you kick that guy off the cliff!"

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Yes, I was one of those people who said that. But I think the OP was concerned about direct applications. Like when you would want to sit down with a pen and paper to write an integral or perform a matrix operation. For most people the answer would be never in their life. Us mathy types would think about it all the time when we hear news stories or write on internet forums, but that's not normal people. If you aren't inclined to analyze things for fun, then knowing how to integrate won't make you do it.

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*Science *Physics *Engineering *Computer programming *Genetics and other fields of biology *Chemistry *Business accounting/finance and economics But people other than physicists, scientists, and engineers wouldn't have any real use for any advanced maths. 

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I believe that advanced-level mathematics (especially pure maths) is for fun and for appreciating the beauty. I am currently at high school and love learning math (college-level math), but find the maths at high school very dull and boring. The current education system just turns maths into a very systematic work. For example, when we learn Pythagoras Theorem, after teaching the theorem itself, we are told how to (1) Find the length of the hypotenuse if the sides are given (2) Find the length of one side when the hypotenuse and one of the sides are given while the teacher can just let us find the way of doing it ourselves with the original theorem. Even with the exercises provided, the questions are divided into parts about the first type of problem and second type of problem. It is just plain stupid (sorry for being a bit too rude).

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Finance, cryptography...

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There is a method to this madness. True you can derive these different formula's from the original theorems each and every time you need to use them. But, you will need to apply these things from time to time as you move into higher level work and it's helps if you have a lot of these very basic things like trig formulas memorized (at least somewhat memorized) such that you don]t have to go back and derive them each and every time you need them. Of course there is nothing wrong with learning how to use the theorems and understand their meanings to derive the formulas or, given a formula prove it's validity based on the theorem(s). When i was in high school (and freshman college) many of the more fundamental courses omitted the proofs or simply glossed over them. But I never felt comfortable, I always preferred working through the proofs and, thinking of other approaches I could take to them.

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You must distinguish between these two questions: "What do people often use advanced mathematics for" and "What can advanced mathematics be used for". People who have training in advanced mathematics, and a certain knack for applying it, can apply it to almost any subject, even art and literature. Statistically, the people who know advanced mathematics tend to be engineers, physicists etc. so that's where you most often see advanced mathematics applied. If an artist or literary historian happened to be an expert in differential equations, they might well be able to apply it to their field of study. However, they might not find many of their peers able to understand or appreciate their work.

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Because of it's sometimes mind-boggling complexity and difficulty, it is difficult to wrap your mind around that level of math without having to have an IQ of 140+ Math is despised by most high school and college students. Only the people with high IQ's (a tiny percentage of the population) tend to excel in, and take further interest in it. 

Baseball Distances

I stumbled onto an interesting website. It had baseball statistics and had some stats I didn't know they even kept track of.

It gives numbers on average flights of batted balls for each player. It's interesting to look at as a math application. I tried it out using formulas and didn't get the quite the same answer. However, the trajectory equations don't account for the air resistance encountered. And, of course, I might have just done the math wrong. More on this after I let you know how far off I was.

The categories were "Average Launch Speed", "Average Distance", "Average Velocity", "Average Launch Angle", and "Average Height". For example Evan Longoria (no relation to the actress) had:

• Average Launch Speed: 92.17 miles per hour
• Average Distance: 248.2 feet
• Average Launch Angle: 14.39 degrees
• Average Height: 46.06 feet

I assume Launch Speed could be found with a radar gun. Launch Angle perhaps by camera, although it seems like that would depend on where the camera is in relation to the camera. Ideally, the camera would be pointed perpendicularly to the ball's trajectory, I would think, but that wouldn't always be the case.

So, I wondered if I could compute what they had for Average Distance. I'm guessing that is how far the ball before it hits the ground. But what if Evan hits a line drive and is caught? It went a certain distance, but would have gone farther without the fielder there? Anyway, here we go.

First I figured I need to get its average speed into feet per second to match with the other categories.

92.17 miles per hour = 286,657.6 feet per hour = 135.183 feet per second

I then used the formula:  y(t) = h + (vsinA)t -16t^2.

I'll assume an height of the ball when making contact with the bat to be 5.5 feet. I want to see how long it takes to hit the ground (y(t) = 0).

0 = 5.5 + 135.183sin(14.39)t - 16t^2

Using the quadratic formula, this game me two answers, the positive one being 2.25 seconds. 

Then I used this to find how far it went with x(t) = v(cosA)t = 135.183(cos(14.39))2.25 = 294.62 feet

According to that website (http://m.mlb.com/player/446334/evan-longoria) the distance is only 248.2 feet. 

I was ready to call this a big old fail. But, perhaps not. Like I mentioned before, I'm not sure how they figure balls that are caught before they land or balls that bounce off the outfield fence. And are those distances found by observation of where the ball seems to land? Air resistance slows down the ball quite a bit. They say that the Colorado Rockies in mile-high Denver is the easiest place to hit home runs because of its thin atmosphere. The math equations assume a vacuum, so the formula would give a greater distance. 

So, maybe my math is all right. Regardless, it's a nice math application.