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Along with probability, upon which it is mostly based, statistics is a mathematical discipline which provides techniques for drawing conclusions from observed data. It is heavily relied upon in most scientific fields (experiments and observational studies), as well as in business and industry (marketing, operations management, economic analysis, quality control) and government (public polling, policy analysis).

Statistics can be divided into two large subfields, descriptive statistics and inferential statistics:

Descriptive statistics 
Numerical and graphical summaries of data — i.e., "charts and graphs". These are more in the public eye (think USA Today), but they can be used by unscrupulous sorts to mislead ("lies, damned lies, and statistics").
Inferential statistics 
Probability-based analysis used to infer something about a larger, mostly unobserved, population based on what is seen in a sample from that population. While the results of statistical analyses are often reported in the mainstream media (medical studies and the like), the details of the statistics behind them are usually left out and can often only be found in academic journals.

Significance and inference

A central idea in inferential statistics that is widely misunderstood is statistical significance. In an experiment to test, say, whether a new drug to treat a disease is more effective than the old, standard treatment, the degree of improvement the new drug provides is said to be statistically significant if it is so large as to be unlikely to have occurred by chance alone.

To be more specific, if one sees any improvement using the new drug over what is expected (or observed) using the standard treatment, there are two possible explanations:

  1. There is actually no overall benefit to the new drug (i.e., if given to the entire population of people having this disease), and the (sample) results simply occurred "by chance" because of individual differences in how people respond to medical treatments (i.e., we just "happened" to get a sample of individuals that responded better than the average member of the population).
  2. There is a benefit to the new drug over the old (in the population), and the (sample) results are simply reflecting this fact.

The larger the amount of improvement actually observed in the sample (or the larger the sample size used in the experiment), the less convincing explanation #1 becomes, and the more convincing explanation #2 becomes. Statistics gives a way of calculating the probability that explanation #1 could be true (it is technically a conditional probability, assuming there is no overall benefit, and is called a p-value). The smaller this probability is, the more likely explanation #2 is the correct one.

Note, by the way, that just because an "effect" (the benefit of using the new drug over the old, in this case) is large enough to be statistically significant, that doesn't mean it is actually large in an absolute, real-world sense. For example, the new drug might only give a small benefit (that is nonetheless significant because of the sample size used in the experiment) that is outweighed by considerations of cost or side-effects.

In any case, accepting explanation #2 as correct even though one has not actually tested the drug on every member of the population, makes the conclusion an inference and not a logical deduction. The decision could be wrong, even if the results are highly (statistically) significant. Explanation #1 could, in fact, be the truth. This is why any conclusion based on the statistical analysis of data is fundamentally subject to error (the kind of error being discussed here is usually called a "type I error" or "alpha error"). By carefully controlling for other sources of error (experimenter bias, data collection errors, etc.), and performing a valid statistical analysis, one can quantify the probability of making such an error — something that ad hoc and most other unscientific explanations cannot hope to achieve.

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