Variability in statistics là gì

Abstract

This paper attempts a brief account of the history of sample measures of dispersion, with major emphasis on early developments. The statistics considered include standard deviation, mean deviation, median absolute deviation, mean difference, range, interquartile distance and linear functions of order statistics. The multiplicity of measures is seen to result from constant efforts to strike a balance between efficiency and ease of computation, with some recognition also of the desirability of robustness and theoretical convenience. Many individuals shaped this history, especially Gauss. The main contributors to our story are in chronological order, Lambert, Laplace, Gauss, Bienayme, Abbe, Helmert and Galton.

Journal Information

The aim of Statistical Science is to present the full range of contemporary statistical thought at a technical level accessible to the broad community of practitioners, teachers, researchers, and students of statistics and probability. The journal publishes discussions of methodological and theoretical topics of current interest and importance, surveys of substantive research areas with promising statistical applications, comprehensive book reviews, discussions of classic articles from statistical literature, and interviews with distinguished statisticians and probabilists.

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The purpose of the Institute of Mathematical Statistics [IMS] is to foster the development and dissemination of the theory and applications of statistics and probability. The Institute was formed at a meeting of interested persons on September 12, 1935, in Ann Arbor, Michigan, as a consequence of the feeling that the theory of statistics would be advanced by the formation of an organization of those persons especially interested in the mathematical aspects of the subject. The Annals of Statistics and The Annals of Probability [which supersede The Annals of Mathematical Statistics], Statistical Science, and The Annals of Applied Probability are the scientific journals of the Institute. These and The IMS Bulletin comprise the official journals of the Institute. The Institute has individual membership and organizational membership. Dues are paid annually and include a subscription to the newsletter of the organization, The IMS Bulletin. Members also receive priority pricing on all other IMS publications.

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Since standard deviation, mean, median, and mode are all absolute data on statistical samples, they do not permit a direct comparison of variation between samples with different means or different units of measurement.

One way to obtain a measure of variation that has no units is to divide the standard deviation [s] by the mean

[

], and multiply by 100 to give a percent. This quantity is called the coefficient of variation [CV], and can be used to compare methods that give different units.

For example, the coefficient of variation for two different glucose methods would be calculated as shown below after the mean and standard deviation for each method has been established. The hexokinase method has

= 99 mg/dL, and s = 8.0 mg/dL. The orthotoluidine method has = 105 mg/dL, and s = 12.5 mg/dL.

From these CV's we would conclude that the hexokinase method is relatively more precise because it has a lower CV.

Sampling > Sampling Variability

What is sampling variability?

Sampling variability is how much an estimate varies between samples. “Variability” is another name for range; Variability between samples indicates the range of values differs between samples.

Sampling variability is often written in terms of a statistic. The variance [σ2] and standard deviation [σ] are common measures of variability. You might also see reference to the variability of the sample mean [μ], which is just another way of saying the sample mean differs from sample to sample. Sampling variability only refers to a statistic [i.e. a number generated from a sample]—never a population.

Variability and Sampling Error

A closely related term [almost a synonym] is sampling error. An error in sampling isn’t a mistake — it’s a measure of how much a value differs from the “true” value. Let’s say the true weight of a population is 150 lbs. You take a sample and find the mean weight for the sample is 151 lbs. The 1 lb difference is an “error.” If you sample again, you might get different mean weights of 148 lbs, or 150.5 lbs, or 153 lbs. The different errors — 1/2 lb, 1 lb, 2 lbs, 3 lbs — are a reflection of the variability between your samples, or sampling variability.

Variability and Sample Sizes

The “perfect” sample size is practically impossible to find.

Increasing or decreasing sample sizes leads to changes in the variability of samples. For example, a sample size of 10 people taken from the same population of 1,000 will very likely give you a very different result than a sample size of 100.

There is no “perfect” sample size that will give you accurate estimates for the sample mean, variance and other statistics. Instead, you take your best “guess” — using standardized statistical procedures [see: Finding the sample size]. In general, estimates will change from sample to sample and will probably never exactly match the population parameter.

References

Lodico, G. et al. [2010]. Methods in Educational Research: From Theory to Practice.

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