What is a Statistic?

from: https://controls.engin.umich.edu/wiki/index.php/Basic_statistics:_mean,_median,_average,_standard_deviation,_z-scores,_and_p-value#Mean_and_Weighted_Average

In the mind of a statistician, the world consists of populations and samples. An example of a population is all 7th graders in the United States. A related example of a sample would be a group of 7th graders in the United States. In this particular example, a federal health care administrator would like to know the average weight of 7th graders and how that compares to other countries. Unfortunately, it is too expensive to measure the weight of every 7th grader in the United States. Instead statistical methodologies can be used to estimate the average weight of 7th graders in the United States by measure the weights of a sample (or multiple samples) of 7th graders.

Parameters are to populations as statistics are to samples.

A parameter is a property of a population. As illustrated in the example above, most of the time it is infeasible to directly measure a population parameter. Instead a sample must be taken and statistic for the sample is calculated. This statistic can be used to estimate the population parameter. (A branch of statistics know as Inferential Statistics involves using samples to infer information about a populations.) In the example about the population parameter is the average weight of all 7th graders in the United States and the sample statistic is the average weight of a group of 7th graders.

A large number of statistical inference techniques require samples to be a single random sample and independently gathers. In short, this allows statistics to be treated as random variables. A in-depth discussion of these consequences is beyond the scope of this text. It is also important to note that statistics can be flawed due to large variance, bias, inconsistency and other errors that may arise during sampling. Whenever performing over reviewing statistical analysis, a skeptical eye is always valuable.

Statistics take on many forms. Examples of statistics can be seen below.

Now that we've discussed some different ways in which you can describe a data set, you might be wondering when to use each way. Well, if all the data points are relatively close together, the average gives you a good idea as to what the points are closest to. If on the other hand, almost all the points fall close to one, or a group of close values, but occassionally a value that differs greatly can be seen, then the mode might be more accurate for describing this system, whereas the mean would incorporate the occassional outlying data. The median is useful if you are interested in the range of values your system could be operating in. Half the values should be above and half the values should be below, so you have an idea of where the middle operating point is.