Available courses

(MDM4U) Mathematics of Data Management

Course Introduction

Statistics is the scientific application of mathematical principles to the collection, analysis, and presentation of data. Statisticians contribute to scientific enquiry by applying their mathematical and statistical knowledge to the design of surveys and experiments; the collection, processing, and analysis of data; and the interpretation of the results.

Statistics is a word with two meanings. Most people are aware of the mundane definition of statistics as a collection of data, such as baseball statistics or statistics the government collects during a census. There is little awareness of the more important, broader definition of statistics as a branch of academics--some would say a branch of mathematics--and so to much of society the act of doing statistics is merely the collection and presentation of data for informational or persuasive purposes.

The larger definition of statistics is a discipline concerned with the analysis of data and decision making based upon data. It can also be used to spot trends or isolate causes. Statistics is based upon a solid edifice of mathematical theorems proven through unassailable laws of logic. In theory, statistics works every time. We shall discuss the inherent problems with statistics in due course, because as many people know, statistics can be misleading.

Statistical analysis and decisions are based upon the notions of probability, the study of measuring how chance affects certain events or outcomes. One of the simplest probability problems is flipping a coin. The probability of getting "heads" is one-half, or fifty percent. This is an example of a theoretical probability, and all theoretical probabilities are concerned with finding ways to count all possible outcomes to some hypothetical experiment. Some thought was given to these types of problems in the sixteenth century but modern theories of probability can be said to date to a correspondence between Pierre de Fermat and Blaise Pascal in 1654.

Statistics as we know it and use it was developed beginning around 1893 and continues to be refined to this day. It is a very large subject area mostly applied in practical ways to data for which there is no theoretical probability. For example, pharmaceutical firms must test new drugs before they can be put on the market. Groups of people are given these drugs and compared to similar groups not taking these drugs. How can we predict an outcome? We can't, so we measure the outcome and determine through methods of statistics--based on probability--whether the results were significant or not.

Statistics has been used much more extensively in the business world since the end of WWII. A little known statistician named W. Edwards Deming was sent to Japan to help the Japanese rebuild their country. He taught them the value of statistics in monitoring quality, as well as establishing a set of management principles known as Total Quality Management (TQM). (Several others did similar things for the Japanese by Deming is now the most famous.) Eventually the Japanese became very competitive and renown for their quality products, and Deming's ideas, which always included a role for statistics, have spread throughout this country and others.

The field of statistics provides the scientist with some of the most useful techniques for evaluating ideas, testing theory, and discovering the truth.

From medical studies to research experiments, from satellites continuously orbiting the globe to ubiquitous social network sites like Facebook or MySpace, from polling organizations to United Nations observers, data are being collected everywhere and all the time. Knowledge in statistics provides you with the necessary tools and conceptual foundations in quantitative reasoning to extract information intelligently from this sea of data.

Who needs statistics in the 21st century? Anyone who wants to be able to look critically at numerical information and not be misled. Anyone who has problems to solve, problems they won't be able to solve until they find out a little more about the world and how it operates. Such problems include finding ways to make a business more profitable right through to improving living standards and fighting cancer. Investigative questioning, designing ways to collect data to answer those questions, collecting data, and making sense of what that data says to produce reliable answers - this is the subject matter of statistics.

We live in an information age. Computers allow us to collect and store information in quantities that previously would not even have been dreamed of. What is this information? It might be costs, values, sales volumes, measurements, ratings, distances, prices, percentages, counts, times, or market shares. But raw, undigested data stored on computer disks is of no use until we can start to make sense of it. Statistics is the human side of the computer revolution, an information science, the science (and art!) of extracting meaning from seemingly incomprehensible data. In your future life and career, you will need to be able to make good use of such information to make sound decisions.

The study and practice of statistics is exciting. In one week, a practicing statistician may help to design an experiment to evaluate the effects of a new treatment for a disease, analyze a set of data gathered by an ecologist, and help a freight carrier to study work processes to find ways of making the company more profitable.


(MHF4U) Advanced Functions

This course extends students’ experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden your understanding of rates of change; and develop facility in applying these concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students taking the Calculus and Vectors course as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.


MCV4U Calculus and Vectors

This course is divided into two components. Calculus and vectors. While many people believe that calculus is supposed to be a hard math course, most don't have any idea of what it is about. The good news is that if you remember your algebra and are reasonably good at it then calculus is not nearly as difficult as its reputation supposes. The word "calculus" comes from "rock", and also means a stone formed in a body. People in ancient times did arithmetic with piles of stones, so a particular method of computation in mathematics came to be known as calculus.

Calculus was developed out of a need to understand continuously changing quantities. Newton, for example, was trying to understand the effect of gravity which causes falling objects to constantly accelerate. The speed of an object increases constantly every split second as it falls. How can one, for example, determine the speed of a falling object at a frozen instant in time, such as its speed when it strikes the ground? No mathematics prior to Newton and Leibnitz's time could answer such a question, which appeared to amount to the impossibility of dividing zero by zero. The solution to this type of issue came to be known as the derivative. Derivatives are slopes of particular lines called tangent lines, and the reader may recall that slope of a line is a concept from Descartes' graphing.

Calculus provides the foundation to physics, engineering, and many higher math courses. It is also important to chemistry, astronomy, economics and statistics. Medical schools and pharmacy schools use it as a screening tool to weed out weaker aspirants under the assumption that people who are unwilling or unable to handle the rigors of calculus stand little chance of surviving the hard work of studying medicine or pharmacology.

On the other hand, a vector is a quantity or phenomenon that has two independent properties: magnitude and direction. The term also denotes the mathematical or geometrical representation of such a quantity.

Examples of vectors in nature are velocity, momentum, force, electromagnetic fields, and weight. (Weight is the force produced by the acceleration of gravity acting on a mass.) A quantity or phenomenon that exhibits magnitude only, with no specific direction, is called a scalar . Examples of scalars include speed, mass, electrical resistance, and hard-drive storage capacity.

Vectors can be depicted graphically in two or three dimensions. Magnitude is shown as the length of a line segment. Direction is shown by the orientation of the line segment, and by an arrow at one end. The illustration shows three vectors in two-dimensional rectangular coordinates (the Cartesian plane) and their equivalents in polar coordinates.

The main reason to study vector is that nearly everything in mathematical modeling is a vector in one way or another, and frequently the vector space operations are things you want to apply. Often enough, people try to model something and end up trying some manipulation that turns out to be equivalent to vector addition or scalar multiplication, but they only figure out later on what the tool was that they were looking for: vectors. So, teachers try to equip you in advance for those times of need by making you carry around vector spaces in your toolkit.

To throw out some random examples of vectors: stock indices (the index, not the value of the index), still images, videos, photoshop transforms, audio signals, radio transmissions, probability densities of about anything, heat distributions, quantum wave functions, magnetic fields, in fact almost anything called a distribution, differential equations, markov processes, processing filters, tracking state spaces, GPS solutions, survey responses on scales, etc, etc. All are vectors. Most of them have norms or metrics. Many have angles and inner products.

Even 1 dimensional vectors have some interest, although they look just like plain old numbers, because they explain why real numbers have signs, i.e. directions.

Students who enjoy intellectual stimulation and the power of abstract thinking tend to enjoy the beauty of calculus the most, but there is much to appreciate for those who are looking for powerful tools which to understand and create in the physical world.

Finally, a good reason to take calculus is that you will be more competitive and have more career opportunities. Many people avoid demanding challenges; those willing to face them head on tend to go much further in life.


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