Title: Introduction: Probability
Course: UGQuants (SSPS) > Section: Probability

Introduction to the module on probability.

In this module, we look at inferential statistics, or how to make statements and claims about populations on the basis of small samples drawn from them. This means, we assess the extent to which we can confidently generalise the findings from our particular sample to a wider population. To understand the logic of how we do this, we look in some detail at the properties of the normal curve and standard normal distribution. You may find this module difficult at first. Indeed you may find it necessary to read through it carefully a few times. Don’t worry: you are in very good company. The logic of what is explained here took several hundred years to develop, and a broad consensus on it was only reached in the early 20th century.

Like much social science, in essence inferential statistics are simple once you have grasped the basic principles. These are difficult to follow at first since they seem profoundly counter-intuitive. It is worth persevering, however, since the logic of inferential statistics gives us powerful tools to study the world by analysing only very small parts of it. In essence, this module is about how we can make meaningful statements about several million people (say, the population of an entire region or country) on the basis of studying only about a thousand. Without the time and resource savings inferential statistics offer us, most social science would simply not be possible (imagine how many people it would take to interview a whole population within a limited amount of time!).

But to make these savings, we need to understand the logic of generalising from the results we obtain from our sample to the characteristics of the population from which it was drawn. If we do not do this carefully, we will produce findings that are irrelevant or, worse, that are simply wrong. If you like gambling, or playing cards, then you will soon feel at home in inferential statistics, since they are based on theories of probability or chance. In what follows, then, we'll be thinking quite a lot about tossing coins: a simple but useful example of an action whose outcome is quite unpredictable for any given individual toss but very predictable for a very large number of them.

A useful place to start is chaos. If everything in the world, natural or social, varied randomly or occurred purely by chance, then we would live in a state of absolute disorder that was neither knowable nor predictable. There would be no regularities in the past that we could remember and describe to help us understand the present, nor would we have any way of conceptualising or predicting anything about the future.

Life, however, is not completely random or chaotic: it has order. This is easiest to see in the natural world. We know that if we throw something up in the air, it will – unless something intervenes - come down again. We know that if we hold a mirror up to a source of light, the mirror will reflect it. We know that a cube of ice immersed in hot water will melt. We know that each day, at dawn, the sun rises (unless you live in Scotland).

These kinds of order can be thought of as the existence of regular or stable relationships between variables. For example, in the case of water, if the variable temperature takes the value 10 degrees Celsius, then we can confidently predict that the value of the variable state of matter will be 'liquid'. Likewise, at 110 degrees Celsius the value would be ‘gas’.  Physics describes some important relationships in terms of 'Laws', which are better thought of as well-established theories: such as Newton's laws of motion.

Like our theories about the social world, such theories are falsifiable (that is to say, they are open to criticism, testing and retesting, and thus improvement) and have indeed changed as new evidence has been gathered and new theories developed. Thus Einstein's theories about time and space led to a revision of Newtonian physics and later the development of quantum physics in which the ‘Laws’ are quite different.