History Of Mathematics

Data Analysis & Probability and Statistics

Mark Twain

“Facts are stubborn things, but statistics are more pliable.”

Why You Need to Study Statistics

Other Developments In Statistics

???: No one knows when the concept of chance first came about but evidence has been found from a multitude of ancient archaeological digs that things were used to generate random events. Items called astragli, the heel bones of sheep and other vertebrates, are theorized to be used for religious ceremonies and gambling. Astragli have six sides, are not symmetrical, and are usually found with their sides engraved and numbered. (Abrams)

3800 BC: The first known census was conducted by the Babylonians around this time. There is evidence to support this and that in addition to population data, they also collected data on food supplies. (Census)

Pre-2000 BC: Pottery dice have been found in Egyptian tombs dating before this time. Loaded dice have also been found leading to believe that while they would have struggled to master the concepts of probability during this time, they did master the art of cheating. (Abrams)

2: The oldest existing census is from this time. It comes from the Han Dynasty in China and is believed to be very accurate. (Census)

1525: Gerolamo Cardano became obsessed with games of chance. He formalized what is now called the classical definition of probability: If the total number of possible outcomes, all equally likely, associated with some actions is n and if m of those n result in the occurrence of some given event, then the probability of f that event is m/n. This would become the most basic of principles in probability. Cardano wrote a book on this in 1525, but it was not published until well after his death in 1663. (Abrams)

1654: Blaise Pascal and Pierre Fermat are the two who are credited with founding mathematical probability. They are responsible for solving the problem of points, which was also known as the problem of equitably dividing the stakes when a fair game is halted before either player has enough points to win. This problem had been discussed for several centuries before the year 1654, when they solved it, but they were the first two to solve the problem and give a solution to which we in the present consider to be a correct response. They also focused on research of other fair odds in games of chance. (Shafer)

1657: Christian Huygens, who published their works in his De ratiociniis in ludo aleae, made the ideas of Pascal and Fermat popular. (Schafer) This became the standard book on probability for nearly 50 years (Abrams)

1713: James Bernoullis published his Ars conjectandi. This laid the philosophical foundations for broader applications. He is responsible for bringing probability into mathematical theory and formulated rules for combining the probabilities of arguments. His theorem later became known as the law of large numbers, presented by Poisson. At this point gambling, on games and lives, was still the primary source of new ideas in probability theory during the first half of the eighteenth century. (Schafer)

1812: Pierre-Simon Laplace published his Theorie analytique des probabilites containing the first central limit theorem. This would pave the way for Russian mathematicians to develop this theorem further. This also led into problems with discrete variable being solved with calculus. (Normal)

1900: P.L. Chebyshev and his students Andrei A. Marleov and Alexandr Lyapusov further develop the central limit theorem into how it is known today. “For almost any distribution, if enough data is taken, the normal distribution can be used to calculate probabilities. (Soloman)

Mathematics Supports Religion

In 1670, Pascal used the theory of probability to support religious belief. “God is, or he is not. At the far end of an infinite distance a coin is being spun which will come down Heads or Tails. How will you wager? You must wager. It is not optional. There is an infinitely happy life to gain, and a chance of gain against a finite number of chances of loss, and what you stake is finite. It is no use saying that whether we gain uncertain, while what we risk is certain…” Basically, the reward for believing in God is infinite, while what we give up in worldly pleasures is finite. Therefore, however small the probability of God’s existence is, the gain from belief is greater than the gain from disbelief. (Soloman)

Chaos Theory

Jurassic Park (2/10) Movie CLIP - Chaos Theory (1993) HD

Connections To The Classroom

“Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” This would become known as “The Butterfly Effect,” a major idea in Chaos theory. Chaos theory, as presented by Edward Lorenz, is a very important development in the world of probability and data analysis. While the majority of probability and statistics focuses in on determining the possibility of outcomes, chaos theory instead looks at systems that are so complex, that they can seem random, hence the chaos. Chaos theory deals with non-linear phenomena that are seemingly impossible to control or accurately predict such as turbulence, weather, stock markets, brain states, etc. (What). In a simple system, say a deck of cards, the probability of drawing a king of hearts and then drawing a king of diamonds can be easily computed. However, when looking at a complex system such as weather, the calculations must be so precise and have so many variables to account for, that it becomes impossible to accurately predict weather (Chaos). This becomes especially true considering the accuracy to which variables must be measured. For example, measuring a variable to three decimal places versus six could produce a drastically different outcome. This theory became such an important development in probability and statistics because before this theory, people believed that events were linear, meaning, “things were caused by other things” (Chaos). It has been found that some systems tend to go from a chaotic state, to an ordered state and further study into chaos theory can lead into the unpredictable, becoming at least more predictable. Chaos theory could lead to very interesting discussion topics and ideas from students. Students can research different types of complex systems and then try to determine some probability from that system or discuss things that could influence the outcome of a system. This would be a great way to showcase dependent events. While it is nice to show dependent events in things such as a deck of cards or a bag of marbles, to really push the boundaries of student learning and understanding, more in depth and sophisticated examples need to be worked with. While these complex systems would be vastly more complicated from the “bag of marbles” example, it does provide actual real world application of what this type of learning leads into.

Calculus and Statistics

Uses and More Connections

Advancements in statistics can go hand in hand with calculus. In addition to using calculus to find the area under a graph of z- and t-scores, calculus can also be used to formulate predictor equations using already gathered data. When teaching calculus, one can incorporate a lesson on z- and t-scores and show how the probabilities are calculated with the associated scores by calculating the area under the graph. This will give further context as to how the associated scores are related to their probabilities and what they actually mean. Calculus can also be used to develop predictor equations, such as population models. In order to construct population models from given sets of data and time, one must be able to understand integration. Then, by applying knowledge of limits, models can be developed that will help control population and will never exceed the maximum sustainable population as long as the model is followed. Then, by taking the second derivatives of these population models, we can find the half-life of the population. So, a teacher could create a project of a lesson that combines the uses of calculus, data analysis, and statistics by developing projects such as population models.