In probability, we solve forwards. It starts with an exact and well-defined rule. We use this rule to predict what will happen in the future. We don't know the results, but probability allows us to quantify what should happen.
In statistics, we solve backwards. It starts with the end results, known as data. We take this data and try to determine what well-defined rules created it.
Any statistical prediction should have some margin of error. That's because we are using data to estimate some unknown value.
If we insisted on a range that covered 100% of possible errors, the range is potentially unlimited. So we pick some threshold of error we are willing to tolerate.
One tool for setting the threshold is assuming multiple samples would form a "normal curve". The height of the curve at a particular point indicates proportionally how many observations will fall there.
We must place a "threshold" on when we start to think data looks suspicious, but it is dependent on both judgment and context. And this threshold is not the same as the probability our conclusion is correct.
Deception in statistics can be quite intentional. They are subject to researchers wanting to convey data in a certain way.
Deception can also be unintentional. A researcher may be earnest about their hypothesis, but they still interpreted the results of their experiment poorly.
Alternatively, sometimes the statistics themselves can deceive. It depends who gathers the data and how they do it.
Statistics cannot be done in a mathematical vacuum. Usually it involves real-life knowledge about the circumstances. This can be as important as any sort of calculation.
When looking at graphs or charts, it is important to inspect the axes and labels. When analyzing data, it is important to remember the data source and how the data was collected. This can help you recognize deceptive statistics. It may seem, at this point, that statistics can only serve to confound. However, you'll find statistics can be used to pierce deception rather than just create it.
A basic understanding of probability and statistics will help you understand the world better and make better predictions. Making better predictions leads to a better future for you and others.