Differences between Probability and Statistics

Probability and Statistics are related areas of mathematics which concern themselves with analyzing the relative frequency of events. Still, there are fundamental differences in the way they see the world:

• Probability deals with predicting the likelihood of future events, while Statistics involves the analysis of the frequency of past events.
• Probability is primarily a theoretical branch of mathematics, which studies the consequences of mathematical definitions. Statistics is primarily an applied branch of mathematics, which tries to make sense of observations in the real world.

Both subjects are important, relevant, and useful. But they are different, and understanding the distinction is crucial in properly interpreting the relevance of mathematical evidence. Many a gambler has gone to a cold and lonely grave for failing to make the proper distinction between Probability and Statistics.

This distinction will perhaps become clearer if we trace the thought process of a mathematician encountering her first craps game:

• If this mathematician were a Probabilist, she would see the dice and think Six-sided dice? Presumably each face of the dice is equally likely to land face up. Now assuming that each face comes up with probability 1/6, I can figure out what my chances of crapping out are.
• If instead a Statistician wandered by, she would see the dice and think “Those dice may look OK, but how do I know that they are not loaded? I’ll watch a while, and keep track of how often each number comes up. Then I can decide if my observations are consistent with the assumption of equal-probability faces. Once I’m confident enough that the dice are fair, I’ll call a probabilist to tell me how to play”.

In summary, probability theory enables us to find the consequences of a given ideal world, while statistical theory enables us to to measure the extent to which our world is ideal.

Why and ?

So the remaining question is why we have and. To solve it, we need to figure out: what would happen if we do not have and. We know that Statistical techniques are employed in almost every phase of life:

• Surveys are designed to collect early returns on election day.
• Consumers are sampled to provide information for product preferences.
• Research physicians conduct experiments to determine the effect of various drugs and controlled environmental conditions on humans.
• Engineers sample a product quality characteristic and various controllable process variables.
• Economists observe various indices of economic health over a period of time.

But, why do they have to do that? Or they do all things for what purposes? Clearly, the appropriate answers might be: * Surveys are designed to collect early returns on election day to forecast the outcome of an election. * Consumers are sampled to provide information for predicting product preferences. * Research physicians conduct experiments to determine the effect of various drugs and controlled environmental conditions on humans in order to infer the appropriate treatment for various illnesses. * Engineers sample a product quality characteristic and various controllable process variables to identify key variables related to product quality. * Economists observe various indices of economic health over a period of time and use the information to forecast the condition of the economy in the future.

And that’s where Probability comes.

In a nutshell, Probability and Statistics are similar to two sides of one coin. These two subjects always go hand in hand and thus you can’t study one without studying the other.