Weber -Fechner laws in psychometrics

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The exponential model of memory retention

If you were to ask someone about the relationship between maths and love, they'd probably tell you about how much they hate it…(I swear the joke sounded better in my head). Anyway…:

The Mathematics of Marital Conflict, 1998 study by John Gottman, Catherine Swanson, and James Murray.

50% of married American couples end in divorce, and some researchers did a study on why this happens by putting couples together and letting them talk while they analysed their conversations, facial expressions, and many other factors and then made a mathematical model that predicts whether or not they’d end in divorce.

1. Summary of the process

They computed for each conversation turn. Behaviour was considered positive if it correlated positively with marital happiness and negative otherwise. The Rapid couples interaction scoring system (RCISS) and the Specific Affect Coding System (SPAFF) was used to create numerical values of the behaviours. The sequence of scores were \(W_t\,H_t,W_{t+1},H_{t+1}\).\(H_t\) is the husband variable at time t, and \(W_t\) is for the wife (so it's: wife talks, husband responds, wife responds, and so on.)

$$ W_{t+1}=a+r_1W_{t}+I_{HW}(H_t) ... (1) $$ $$ H_{t+1}=a+r_2H_{t+1}+I_{WH}(W_t) ... (2) $$

\(a\), \(b\) and \(r\) are the parameters estimated from the data. \(r\) is the emotional inertia, which is the tendency of the person to remain in the same state over a period of time. \(a\) and \(b\) is the initial uninfluenced level of positivity minus negativity each spouse brings to the interaction. The \(I_{HW}\) and \(I_{WH}\) are the influence functions. Once the \(r_1\),\(r_2\), \(a\) and \(b\) are estimated from the data, (\(a\)+\(r_1W_t\)) and (\(b\)+\(r_2H_t\)) are then subtracted from equations (1) and (2) respectively, and the remainder will be the influence functions which were then plotted.

2. Analysis of the graphs

There are 2 types of graphs for both husband and wife (one type is for a couple that ends in divorce and one type is for a couple that has a lasting marriage), but we'll focus on the 2 types of husbands in a couple since the graphs are the same for the wives.

Graphs A and B represent the influence the husbands have on their wives; what the husband does is the x-axis, and the wife's reaction is the y-axis. In graph A, the husband has no influence on her until he does or says something at P or -P, then he gets either a big positive or a big negative reaction. And in B, the husband in a different couple, gets a little reaction for everything he does, either positive or negative, so the growth is linear. Before we move on, in your opinion, what couple do you think would have a longer lasting marriage?



3. Conclusions from the findings

They used the term “threshold of negativity” , which is the point at which negativity has an impact on the partners immediately following the behaviour. Apparently, a low threshold like in B, means couples end up happy and stable. It is the "negativity detection effect", spouses responding to negative behaviour when it is less escalated.

With a couple of graph A, they accept negativity. Setting their threshold for a response at a higher level or more negative level, then blowing up when they've had enough, if the negativity threshold is really high, it means either the husband or wife has to become very aggressive to get a proper reaction from their spouse in an argument, which is a high risk factor for divorce, meanwhile couples in graph B who see a problem and immediately bring it up before it escalates too high usually have a longer lasting marriage, it was compared to how it's better to go to the hospital immediately when you notice a symptom and not when a disease gets worse. The bible verse Ephesians 4:26 ‘’Do not let the sun go down while you are still angry’’; was stated to support this. On the contrary, though some studies have suggested that couples can cultivate an empathetic response to their partner's negativity, but they didn't say anything about that leading to a longer lasting marriage.

Computed Tomography

Global positioning system

4 satellites are actually needed because there's a time offset between our phones and the atomic clocks used by satellites. So, a 4th satellite and some calculus are used to calculate the offset time. Satellites are placed in a way that we can access at least 4 from wherever we are, which is pretty cool. Einstein's theory of relativity also plays a factor since satellites orbit the earth at really fast speeds and experience less of the earth's gravity, which affects their atomic clocks. These conditions are, however, integrated to the computers that do these calculations, thus accounting for the rates of the atomic clocks. If this wasn't done, there'd be an added error of 10km every day, which is crazy to think about. Mathematicians, engineers, and scientists are the real unappreciated heroes of our modern world, honestly.

Median Voter Theorem

the median voter theorem states that political parties that cater to the median or average person in a group of people are more likely to win, because if for example a party tries to cater too much to the rich people they might lose voters from the poorer group and if they cater to the poor they'll lose votes from the rich, but if they cater to the middle persons though they'll get votes from middle people, some votes from rich people and some from the poor who are willing to compromise vs a party that only caters to and gets votes from one side. It's not just rich and poor that play a factor in who the middle person is, but also a wider range of ideologies and/or policies. I learnt about this little theory in second year economics, and it also applies to businesses too, sometimes even movies. The model has some obvious flaws, which I won't delve into, but you can go and learn more about it if you're interested.

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The golden ratio

The golden ratio, also known as the divine proportion, in simple terms, is a ratio between 2 numbers which is an irrational number approximately equal to 1.618, denoted by \(\phi\). But where does the number 1.618 come from?

Suppose we have a rectangle with dimensions \((a+b)\times a\), then we draw another rectangle of the same proportion and same length as the previous rectangle inside said rectangle, then we have another rectangle of dimensions \(a\times a\), We can keep repeating the process to infinity, but for now let’s focus on the first 2 rectangles. The ratios of the 2 rectangles will be: $${{a+b}\over a}={{a}\over b}=\phi$$ $$1+{{a}\over b}=1+{{1}\over \phi}$$ $$1+{{1}\over \phi}=\phi$$ $$\therefore \phi^2-\phi-1=0$$ and using the quadratic equation we get: \(\phi\)=\({1+\sqrt{5} }\over 2\)=\(1.6180339887...\)

Relationship between the Fibonacci and the Golden ratio.

Before we get into the relationship between the Fibonacci sequence and the Golden ratio, let's talk about the Fibonacci sequence. What is the Fibonacci sequence?

The Fibonacci sequence is a recursive sequence where each number is the sum of the two previous numbers in the series. For example, look at the first 10 terms of the sequence: $$ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... $$ Notice how 8=5+3(the 2 numbers before 8),13=8+5(the 2 numbers before13) and so on. This can be mathematically recursively defined as: $$ F_0=0\\ F_1=1\\ F_{n}=F_{n-1}+F_{n-2}\ (where\ n\ \in\mathbb{N} ,n\geq2)$$ and also, non-recursively defined as $$F(n) = { 1\over {\sqrt5}}\Biggr[({1+\sqrt{5} \over 2})^n-({1-\sqrt{5} \over 2})^n\Biggr](where\ n\ \in\mathbb{N})$$ The second formula is obtained by applying the first formula in conjunction with the power series and performing some algebraic manipulations, which I will not elaborate on here. Back to the first 10 terms of the Fibonacci shown above, notice how \(13 \over8\)=1.625, \(21 \over13\)=1.615, and \(34 \over21\)=1.619? The ratios between the 2 consecutive numbers of the Fibonacci sequence converge to the golden ratio \(\phi\) the bigger the Fibonacci sequence grows in mathematical terms: $$ \lim_{n \to \infty}{F_{n}\over F_{n-1}}= { {1+\sqrt{5} }\over 2}=1.6180339887...$$

The golden ratio and beauty:

The golden ratio has been observed to appear in numerous places in nature and is considered visually pleasing by many. As a result, artists over the years have sought to incorporate this beauty by applying the golden ratio in their artworks.

My Graph Art

These are some of the artworks I’ve made using the online graphing calculator Desmos. Not the greatest pieces of Desmos art ever created, but I had a lot of fun making them, and I am very proud of them.

Convergence

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In mathematics, you see this with sequences and functions as they approach a limit; a convergent function/sequence will just keep getting closer and closer to the convergent point as the domain values increase or decrease. This is a formal mathematical definition of convergence:

A sequence \(a_{n}\) converges to limit \(L\) if and only if for every \( \varepsilon > 0 \) there exists a natural number \(N\) such that \(\forall_{n\epsilon N}\) :$$n >N \Longrightarrow \ |a_n-L|<\epsilon$$ this is denoted by \(lim_{n \rightarrow \infty }a_n=L\).\(a_{n}\) tends to \(L\) as \(n\) tends to infinity, which basically describes what is happening in the following graph

One thing that makes mathematical convergence possible is limits in calculus, because they help us make use of the concept of infinity. Limits and convergence are the backbone of numerical analysis. Numerical analysis helps us solve problems that are unsolvable by conventional means, by using approximations that are so close to the answer they're practically the same as the answer. Numerical methods include the Gauss elimination method, the Taylor series (which can approximate any function as a polynomial), the Riemann sum(for integrals), the Finite difference method (for differential equations), the Newton-Raphson method, the Euler method, Runge-Kutta methods, the Gradient descent, and many others.

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Since these methods are algorithmic, they require a lot of iterations to get accurate results, but thanks to computers, this is made easier. Computers are particularly very good at equations like these if programmed correctly. Back in 1943-1946, the ENIAC(Electronic Numerical Integrator and Computer) was developed. It was built to solve numerical problems(among other things ) to help with the war and calculate ballistic firing tables quickly. The ENIAC could do 5000 calculations per second; smartphones nowadays can do more than 100 billion calculations per second.

Convergence is useful in many branches of mathematics, like statistics and probability, with the Central limit theorem and the law of large numbers, which states that the larger the sample, the more the results converge to the true value. It's also useful in computer science and machine learning with the gradient descent algorithm. In Economics there is a theory that states that poor countries converge to richer countries. Developing countries have the potential to grow at a faster rate than developed countries because of diminishing marginal returns. Rich countries have already developed their science and technologies, so poor countries can learn just from them, right? Well…that’s not always the case for a variety of reasons, but we’re talking about convergence, not economics! (Image source: Introduction to Economic Growth, 3rd Edition. Norton & Company.)

There's a lot more convergence phenomena in our world, like convergence in animal evolution, where some animals have developed similar traits despite having no common ancestor with that specific trait. E.g., butterflies, birds, and bats both have wings but no common ancestor with wings, maybe pigs will really fly someday, who knows. There is also Media convergence, which is the process by which multiple media technologies are brought together into one computerised device.

Another one is Convergence insufficiency, which happens when the nerves that control our eye muscles don't work the right way. Which means in the absence of convergent insufficiency, our eyes are always converging when we look at things.

There is also convergence in research, where a development in one research field leads to a development in many other fields.
As you can see, Convergence is everywhere!!! Are there any other convergence phenomena you have noticed in our world? Let me know if there is.

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Equilibrium

Equilibrium is when opposing forces or influences are balanced, if you did economics you probably recognise the equilibrium as the point where demand and supply are equal, the definition applies to that too as the forces of demand and supply are balanced at that point, Equilibrium is very prominent in economic theories, not only in demand and supply (there's a lot of examples of it in the economics section), it is also referred to as the steady state. In physics, it's when an object is at rest, isn't moving, and has zero acceleration. There's stable and unstable equilibrium. Stable equilibrium implies the object in equilibrium will converge back to the equilibrium if it is displaced a little, like a spring when extended, and unstable equilibrium implies that a little displacement induced will result in the divergence of the object, like a ball on top of a hill if pushed.(www.researchgate.net/figure/Stable-and-unstable-equilibria-In-a-stable-equilibrium-a-small-disturbance-caused-by_fig1_261758142)

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Statistical Misconceptions

Statistics has been the backbone of decision-making throughout history, not used only by governments, research facilities, and big corporations, but also by us as individuals. Before you buy a product, statistics like the rating of the product on the internet and phrases like "Recommended by 90% of experts” can sway your decision to buy the product. Before you choose which university to go to, statistics like “100% of people who graduate from this institution are employed” , and “there is only a 50% chance of completing this degree in record time” , can also influence your decision. But in many cases, phrases like these, when looked at face value, can be misleading at best and outright false at worst. As such, it is very crucial to understand the statistical traps, we as people are prone to fall for. Interpreting data and statistics accurately is the first step towards making informed decisions. In this section, we will discuss 12 of those statistical traps.

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1. Correlation ≠ Causation

Correlation shows how strongly related 2 values are; in statistics, it is expressed numerically by the correlation coefficient(r). A positive correlation means that 2 values increase and decrease at the same time, and a negative correlation means that when one variable is increasing, the other is decreasing and vice versa. However, just because 2 events are correlated, doesn’t mean one caused the other.

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2. Sampling Bias or Selection Bias

Samplingis the process of selecting a smaller group as a representation of a larger group . This makes gathering data and making inferences about an entire group a lot simpler. Sampling bias, on the other hand, occurs when certain members of the group are more likely to be selected than others . This often leads to false inferences about the larger group due to the data being unfairly skewed.

What leads to sampling bias is when researchers use their judgment to select participants, when researchers select individuals or elements that are easiest to access or reach, flawed random selection methods, and when people with specific characteristics are more likely to agree to take part in a study than others.

Sampling bias teaches how important it is to consider how a study was conducted and not just go with the percentage numbers alone.

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3. Survivorship Bias

Survivorship bias is when we only assess the successful visible subgroup as the entire group.

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4. False Causality Fallacies

A false cause fallacy occurs when someone falsely assumes that one thing caused another. The fallacy comes in many different forms, most notable ones being: Post Hoc Ergo Propter Hoc, Cum Hoc Ergo Propter Hoc and Non causa pro causa

Examples:

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5. Will Rogers paradox

Will Rogers paradox occurs when moving a data point from group 1 to group 2 raises the average of both groups.

6. Regression Toward the Mean

Regression toward the mean is a statistical phenomenon that, if one variable is very high or very low on the first measurement, then it will tend toward the mean on its second measurement. Regression toward the mean is possible due to randomness. When you get an outlier when measuring a variable, it is often because other one-time random events were also at play, events that will not be at play when you measure the variable again. Hence, it will return closer to the mean.

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Regression toward the mean fallacy

Regression toward the mean fallacy occurs when people attribute different reasons for things that happened due to a normal regression towards the mean. For example, The same student who got a high score on the previous test scored lower. the teacher might say it’s because the student did not study as much, when they did, just that odds were not in their favour this time around. or people might claim the athlete who finished in record-breaking time got arrogant and did worse in the next round, when that’s also not the case.

Such false attributions can lead to unnecessary policy or businesses model changes when companies and states don’t take Regression toward the mean into account when making statistically based decisions.

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7. Hawthorne Effect or participant reactivity

Coined from a Harvard study conducted in 1924-1932, The Hawthorne effect is a phenomenon where people act differently when they are aware that they are being observed. As such, the observed behaviour may not represent the normal behaviour leading to wrong conclusions when conducting experiments.

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8.Simpson's Paradox

Simpson's paradox occurs when groups of data show one trend, but this trend is reversed when the groups are combined.

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Example

	Male	Female	Acceptance per course
Course 1	\(\frac{1712}{2744} = 62\%\)	\(\frac{468}{588} = 80\%\)	\(\frac{2180}{3332} = 65\%\)
Course 2	\(\frac{676}{2638} = 26\%\)	\(\frac{820}{3082} = 27\%\)	\(\frac{1496}{5720} = 26\%\)
Acceptance per gender	\(\frac{2388}{5382} = 44\%\)	\(\frac{1288}{3670} = 35\%\)

Look at the diagram above. The total application acceptance of men is 44%, which is higher than the respective 35% for women. However, if we break things down by each course, you'll find that women scored higher than men in each course, 80% and 27% vs 62% and 26% for Courses 1 and 2 respectively. This is what is known as the Simpson's paradox, the skew is caused by the fact that more women applied a highly competitive course with with low acceptance rates for everyone. This is based on the UC Berkeley Gender Bias controversy (1973) where the university was accused of gender bias until the data was broken down by departments.

The Simpson's Paradox teaches us to look at data from every angle before making conclusions.

9. McNamara Fallacy

The McNamara Fallacy happens when you only focus on factors that can be numerically represented or measured, quantitative factors, while ignoring qualitative factors, those that cannot be accurately numerically represented or measured, which in many cases are just as important, if not more. Named after U.S. Secretary of Defence Robert McNamara, who relied heavily on easily quantifiable metrics, like enemy body counts, to measure success during the Vietnam War (1955-1975), ignoring crucial unquantifiable factors like the morale of the Vietnamese people and the political instability within the U.S., leading to a strategic failure despite the US having overall higher metrics in many categories.

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10. Base rate fallacy (base rate neglect)

The Base rate fallacy Is a statistical fallacy where people ignore the base rate or original probability when making judgments about the likelihood of an event, it stems from failure to apply Bayes' theorem, a probability theorem which describes the probability of an event based on conditions that are related to the event. \(P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}\).

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11. Gambler's Fallacy

Suppose you have 10 blue balls and 5 red balls in a bag, without looking inside the bag, what is the probability of reaching inside and taking out a red ball? it’s \(5\over 15\) =33.33% because there are 5 red balls and a total of 15 balls. Now suppose you do take out the red ball, and now there are 4 red balls. Now, what is the probability of taking out the red ball again? \(5\over 15\)=26.66%, there is now a lower probability compared to the first attempt. This is because the first and second events of taking the ball out of the bag are dependent on each other; in probability, these are called dependent events.

Now, on the other hand, suppose you have a coin, and you flip it, what is the probability it will be heads? It's \(1 \over2\)=50%, and if you flip it again, after getting heads, what is the probability of getting heads again? still \(1 \over2\)=50%. This is because these 2 events are independent of each other, even if you flip the coin 10 times and get heads, the probability of getting heads the next time you flip is still the same.

Gambler's Fallacy

The Gambler's fallacy is the belief that because something has happened more frequently than usual, it’s now less likely to happen in the future, and vice versa. This stems from confusing independent events with dependent events.

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12. Multiple comparisons fallacy (multiple testing problem)

Every statistical test or experiment is likely to contain some error or unexpected finding because nothing is perfect, as such, the more you make conduct tests, the more errors you’re likely to run into. The multiple comparisons fallacy occurs when researchers take such errors to be significant findings, even though they appeared by chance and are very few compared to the rest of the findings