The Math In Life
Mathematics is intrinsic to nature. Moreover, it is at the base of all human behaviour and an intuitive reflection of human thinking. We apply a number of mathematical tools in our daily life, without even realising it. A science of common sense, math touches our lives in a deeply profound manner.
How Not To Be Wrong by Jordan Ellenberg, explains the way mathematicians think, how math is deeply woven in our lives, how to use it to solve the different problems we encounter, and most importantly, how we can avoid making mistakes while applying the tools of math in life. It shows the path to finding correct solutions and how not to be wrong!
The Science Of Common Sense And Not Being Wrong
People often find math challenging. More often than not, most people think that the complicated problems they were expected to solve in school were irrelevant in the real life. However, it is a fact that one never stops using math. Math is often the basis of finding solutions to different problems in life. Only, it is not called ‘math’.
Mathematics is the science of common sense and not being wrong.
During WWII, a number of American planes returned from Europe, ridden with bullet holes. It was however observed that the planes’ fuselage had more bullet holes than the engine.
The obvious suggestion from military advisers was to strengthen the fuselage with better armour. However, one young mathematician suggested strengthening the engine armour. He was able to see what the military advisers did not.
He deduced that the planes that did not survive and come back were the ones that were hit on the engine. He deduced that with stronger engine armour, more planes would survive. Their myopia of focussing on what survived rather than on improving areas that were the actual problem is a logical-mathematical error called ‘survivorship bias’. While the problem might not have presented itself as a mathematical one, it was.
Furthermore, mathematics is derived from common sense and a reflection of what we already know intuitively. For example, everyone knows that 5+2 is the same as 2+5. It is so obvious, that it becomes difficult to explain. The understanding that one has of this from childhood, is a reflection of the basic definition of math being commutative, where for any value of a and b, a+b = b+a.
Simplifying With Linearity
While learning math, everyone follows one basic principle – make a difficult problem easier, and then solve the easier problem, hoping that it is close enough to the difficult problem.
In geometry, things with straight lines are considered linear, whereas curves are nonlinear. Tough problems can be broken down into easier ones by assuming they have linearity.
From the perspective of an ant that is walking around a big circle, it would think that it is walking in a straight line. If one was to zoom in closer on a part of the curve of the circle, it would look like a straight line, thus showing that a curve of a circle is actually similar to a straight line bent at very slight angles.
If one had to measure the area of the circle, one could insert a square in the centre of the circle, with each corner touching the circle. Then one can measure the area of the square. Next, one can insert a number of polygons in the remaining areas, calculate the area of those polygons, and ascertain an approximation of the circle’s areas by just using straight lines.
The idea is to simplify the problem with linearity.
Linearity is often used in statistics. A mathematical relation can be seen even in calculating voting preferences or salary levels. For example, the statistic that shows, for every extra $10,000 a person makes, the person is 3% more likely to vote republican. This is an example of Linear Regression and is often used to compare how different observations can be related (like voting preference and salary level).
Linearity helps in finding approximations of any data taken as a whole and making it simpler to understand and apply in real life.
How Probability Theory Helps
Theories are built based on data collected through observations. However, drawing conclusions from observational data can be precarious as it can also come about by chance.
For example, a neuroscientist, in 2009, showed pictures of people to a dead fish. The fish’s brain activity showed that it responded correctly to the emotions displayed by the people in the photos. While the whole experiment was a gag, the scientist proved that research data can come about by chance.
When brain scans are performed, scientists divide the scans into thousands of pieces called voxels, corresponding to different parts of the brain. These scans, even if it is of a dead fish, will measure some amount of ‘noise’ in every voxel. Considering the number of voxels that are created, the chances of even one voxel responding to any stimulus are very high.
However, it is not always clear that the observational data has been acquired by chance. Hence the Probability Theory comes into use.
Consider a scientist is testing a new drug to cure a disease. He uses a mathematical tool called the null hypothesis significance test. In this tool, the scientist will first start with a null hypothesis – an assumption of what will happen in the first test conducted. The null hypothesis of the first test will be that the drug does nothing at all.
Then, the scientist will consider the deviations of the data observed in the experiment. The scientists will also consider the probability of the data coming by chance. This probability is called its p-value. Considering p = 00.5, if the probability is lesser, then the data is considered significant, statistically.
If the data is statistically significant, then the scientist can be 95% sure that the new drug has the desired effect.
Probability, Expected value, And Risk
Probability doesn’t ascertain the future outcome in an uncertain situation. But it can help in understanding what one can expect.
For instance, the probability theory can be applied to understand a likely outcome when one places a bet. If a person buys a lottery, he can use probability by determining its expected value. Thus to calculate the lottery ticket expected value, he will have to consider each and every possible outcome, multiply the chances of each and every outcome with the ticket value, given that outcome, and add up the results.
Let us say the ticket has only 2 outcomes – winning and losing. At a cost of $1 a ticket, there are 10 million tickets in circulation, and the winning ticket is worth $6 million. Thus based on calculations, the ticket expected value is 60 cents, and on average, the person should expect a loss of 40 cents every time he plays the lottery.
The concept of expected value is applied while pricing life insurance or stock options.
Expected value, however, doesn’t consider risk. For example, given an option, would one rather receive $50,00 or place a bet where the odds are either to lose $100,00 or gain $200,000?
Here, while the expected value in both cases is the same, the outcome of losing in the second option is far worse than not placing a bet at all. Expected value tends to hide the fact that a bet also has a tangible risk.
Thus a risky investment is a good bet only if one has the ability to cover probable losses, and one should always consider the risks in any bet before placing it.
The Regression Effect
The regression effect states that if a variable produces an unlikely outcome, the next outcome will tend to be closer to the mean.
Consider the trend where a novelists second book doesn’t garner as much success as the first book does. This phenomenon can be attributed to the regression effect, for anything that involves randomness. For example, while short people have short children, the children of very short people arent likely to be very short. Instead, their height will be closer to average.
The height of a person depends on many factors including genes. Factors such as eating habits, health, exercising preferences and even chance affect height. It is thus highly unlikely that the external factors affecting the children mirror those of the parents.
The regression effect is, however, not recognised often. In 1976, the British Medical Journal published a paper mentioning that bran helps regulate the digestive system. However, participants with a fast digestion rate on one day showed a slower rate the next day, and vice versa for those with slow digestion rates.
The effects of regression proved that the effects of bran on digestion might not be remarkable after all. Thus researchers have to be careful and consider the effects of regression, especially for any biological phenomena.
Linearity Of Regression
We have seen that Linearity of Regression can be used to understand the correlation between variables. While it is an important tool in statistics, it cannot be used for every set of data. Used incorrectly, it can produce results that can mislead the whole observation.
The linear regression of any data set can be plotted on a graph. Then, one finds the line that passes closest to all the reference points plotted. This method only makes sense if all the plotted points are already generally linear.
For instance, the curve path a missile follows when fired seems like a line if one segment of that curve is zoomed upon. This can be used to predict where the missile would be in a few seconds, but it isn’t possible to clearly indicate where it would land, as linear regression doesn’t account for the curvature of its path. Zooming out from the path segment, its curvature indicates nonlinearity and can’t be described by one line anymore.
Applying linear regression to nonlinear observations will produce incorrect results. In 2008, the Journal Obesity published a paper that claimed all Americans would be obese by 2048.
What the researchers of the paper had done was to apply the concept of linear regression to a graph that plotted the percentage of obesity against time. The graph crossed 100% at 2048, and estimated that 109% of Americans would be obese by 2060!
Thus, trends such as obesity cannot be plotted linearly as they tend to produce curved graphs over longer periods of time.
Incorrect Probability Calculations And Misused Data
Professor John Ioannidis, in 2005, published a paper “Why Most Published Research Findings are False.” The paper had made some valid points.
The paper stated that chance can lead to insignificant observations passing a statistical significance test. For example, while considering the genetics of a person while calculating the probability of the person getting schizophrenia, it is sure that some genes get passed on, but it is unclear which genes do.
Out of about 100,000 genes, 10 could be related to schizophrenia. Moreover, even with the most commonly used significance test of 95%, about 5% of the genes will pass by mere chance. Calculating the numbers, 5% amounts to 5,000 genes, which isn’t a result specific enough to mean anything significant.
The paper also stated that the results of unsuccessful studies don’t get published, hence resulting in disproportionate attention given to the results of successful tests. Consider that 20 labs, conduct tests to check if green jellybeans can cause acne. Of these 20, 19 labs don’t find any significance to the claim and don’t publish their results. But that lab that finds successful statistically significant results is more likely to publish the study.
The paper also stated that scientists tend to tweak the data to be able to get statistically significant results.
If 95%certainty is needed for a study to be even considered as statistically significant, and if gets a 94% certainty for the experiments conducted, it would be deemed insignificant for just one percentage point.
In such cases, researchers are known to tweak the data by just a little so that the study can be deemed statistically significant. This happens not because the researchers have ill intentions to cheat, but because they deeply believe in their hypothesis.
The Problem With “Public Opinion”
How is a vague concept such as public opinion measured? While polls are the tools used to measure public opinions, there are problems with the accuracy of the results.
Public opinion is based on what a majority thinks. Considering that every individual can have contradicting opinions and these change from time to time, the results will also change, thus invalidating the precious “public opinion.”
For example, CBS News, in Jan 2011, claimed that according to a poll conducted, 77% of the respondents thought that the best way to address the Federal Budget is to cut spending. However, a month later, Pew Research conducted a poll where people were asked their opinions on 13 different categories of government spending. The results in 11 of the categories showed that people actually supported spending.
Another issue is of majority.
The rule of majority works when respondents have only 2 options to choose from. When there are more options, the responses get divided differently, leading to completely different results.
For example, in a poll conducted in October 2010, about 52% of the respondents opposed the U.S. Affordable Care Act, while 41% supported it. however, when the responses were broken down into detail, only 37% wanted the health care bill repealed, 10% wanted the law weakened, and 15% wanted to leave the bill as it was. About 36% opined that it should be expanded to change the system even more.
Such scenarios can be seen in elections as well. During the 2000 Presidential election, while George Bush had 4885% of the Florida vote, Al Gore had 48.84%, whereas Ralph Nader had 1.6%.
This clearly indicated that Bush was the winner. However, it can also be said that all those who voted for Nader would have prefered Gore instead of Bush if they hadn’t had another choice. It indicated that out of a hundred per cent, 51% did not want Bush to win, which would have led to a completely different scenario.
Mathematics and mathematical ideas can be applied to daily life. They can be used to draw accurate conclusions of unmeasurable concepts. The mathematical tools that can be used, are derived from common sense and one learns how to not make mistakes, whether it is to find the probability of winning the lottery, getting life insurance or even researching a cure for an illness!