Math and Arithmetic

Statistics

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INFERENCES

Any calculated number from a sample from the population is called a 'statistic', such as the mean or the variance.

I believe this question is asking what a population is in statistics. A "population" is simply the group from which a sample is drawn to conduct statistics. It is often wrongly assumed that "population" refers to all people, at least in the social sciences, or all of whatever the research is studying in other fields. For example, suppose you wanted to know the average length of time that teenagers watch television. You would obviously not be able to ask every teenager. So you would collect data from a random sample of teenagers. Thus, your population would be teenagers. However, some caution is necessary. If you collected only from teenagers in the United States, you could not really say that your population is all teenagers in the world. You would probably describe the population as US teenagers. In this case, then, a responsible researchers should make an effort to collect data from different states. Expanding this, you can see how the issue becomes difficult. Some researchers maintain that if you collect only data from say, sophomores at one university (more common than you think), your population can only be all sophomores at that university. Other researchers are more liberal in expanding the definition of the population, and the debate is ongoing. What is important, however, is that the sample reasonably reflects the population. Thus, the boundaries of the population are rarely so wide as to include "all" or "every" of anything. But they must still be wide enough that the research has some importance beyond just the sample, which is the ultimate goal of both science and statistics. Where the line is drawn is often a decision that you have to make and justify yourself in your project.

The term "descriptive statistics" generally refers to such information as the mean (average), median (midpoint), mode (most frequently occurring value), standard deviation, highest value, lowest value, range, and etc. of a given data set. It is a loosely used term, and not always meant to contrast with inferential statistics as the question implies. But in the context of the question, descriptive statistics would be information that pertains only to the data that has actually been collected. In the case of an instructor calculating an average grade for a class, for example, the collected data would most likely be the only point of interest. Thus, descriptive statistics would be enough. However, it is more common for a researcher to use a sample of collected data to make inferences and draw conclusions about a larger group (or "population") that the sample represents. For example, if you wanted to know the average age of users of this site, it would be unrealistic to question every singe user. So you might question a small sample and then extend that information to all users. But if you found the average age in your sample to be 40, you could not immediately assume that 40 is the average for all users. You would need to use inferential statistics to calculate an estimate of how accurately your data represents the larger group. The most common way to do this is to calculate a standard error, which will produce a range within which the population average most likely (but not definitively) lies. Therefore, in the simplest description (inferential statistics are also a part of much more powerful tests outside of this answer), descriptive statistics refer only to a sample while inferential statistics refer to the larger population from which the sample was drawn.

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