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Estimating central tendency from a single spot measure: A closed-form solution for lognormally distributed biomarker

Joachim D Pleil1, Jon R Sobus1

  • 1a Human Exposure and Atmospheric Sciences Division, NERL/ORD, U.S. Environmental Protection Agency , Research Triangle Park , North Carolina , USA.

Journal of Toxicology and Environmental Health. Part A
|September 3, 2016
PubMed

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Measures of central tendency are tools used in biostatistics to identify the average or center of a dataset. They offer a single representative value for understanding and summarizing data distribution.
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The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median. The words "mean" and "average" are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean" and "average" is technically a center location. However, in practice among non-statisticians,...
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In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
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The central limit theorem, abbreviated as clt, is one of the most powerful and useful ideas in all of statistics. The central limit theorem for sample means says that if you repeatedly draw samples of a given size and calculate their means, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. In other words, as sample sizes increase, the distribution of means follows the normal distribution more closely.
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