When we talk about statistics, it’s unavoidable to link it to probabilistic tools, most of the time. But which are the aspects that differentiate the two sciences?
By definition, Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data, while Probability is the measure of the likelihood that an event will occur.
The probability theory is widely used to draw inferences about the expected frequency of events. So the main difference between probability and frequency is that Frequency is the number of occurrences of a repeating event per unit time, while Probability calculates which is the most positive chance for that to happen.
When probability is used, it includes the Combinatory logic or Probabilistic calculus, which is a notation to get rid of the need for quantified variables in mathematical logic.
So at first glance we’re inclined to think the main difference between Statistics and Probability is analyzing concrete data in the first case, and guessing the data behaviour by assumptions in the second one.
But even if it may look like both Statistics and Probability are based on completely different approaches to the studied data, a combining version ot the two can be commonly used in a study.
For instance, there are two main statistical methods that are used in data analysis: descriptive statistics – which summarize data from a sample using indexes such as the mean or standard deviation – and inferential statistics – which draw conclusions from data that are subject to random variation. As for the latter, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.
In fact, a connection between the two lies within the law of large numbers:
In probability theory, the law of large numbers (LLN) is a theorem that explain the result of performing the same experiment a large number of times. According to the law, “the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed”.
It’s clear how this probabilistic theory outputs a result which has to be calculated in a statistical (average) way in order to give some construable informations.
So, in the end we can say that through Calculus, Probability provides some useful tools for the statistics to implement, but there’s also the case in which a probabilistic calculus needs a statistical application to conclude useful results. These two mathematics branches may look different – and they actually can be! – but they’re nevertheless linket to each other.
Rambling things 