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Statistics is defined as a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data. However, in statistics, analyzing a data helps one to be able to estimate a population parameter and form generalizations about a population that may be finite or infinite using a sample data.

Summarizing Information in Populations and Sample

For an infinite population it is impossible to outline all the attributes of a particular population and so summarizing a data and using graphs or numerical techniques enables one to understand a population. To do this one usually needs a sample data, which is a subset of the population. The measurements may be obtained from the sample using a random sampling technique where every member has an equal chance of being chosen. These sample results is used in constructing a frequency or relative frequency to help generate a graph like a histogram that can explain some characteristics of the population. Alternatively, graphs like pie chart and bar graph can also be constructed for a qualitative or a non-numerical data. The measurements obtained are used in finding summary statistics like the mean, median, variance and standard deviation and calculate probabilities of the sampling distribution. Similarly, for a finite population, a sample can be chosen with replacement so that the selection of one member does not affect the probability of selecting another member thereby making it independent. This can be achieved by choosing a random digit from the random number table. Using results from a sample in such a population produces a statistics like the sample total which is an unbiased estimator and can enable us estimate the population total. Sampling with replacement is an unfit and an unrealistic approach for a lot of sampling cases so applying the sampling without replacement technique where the selection of one item influence the probability of the next item selected is an ideal way of getting a much better estimator. However, a sampling with replacement estimator may have a lower variance than that of a sampling without replacement estimator. Besides, many other methods and designs can also be applied to determine good estimates.

Reference

Merriam-Webster Dictionary. Retrieved from https://www.merriam-webster.com/dictionary/statistics

Scheaffer R.,  Mendahall W., Ott R., Gerow K.(2012). Elementary Survey Sampling, Seventh Edition