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Related Concept Videos

Median01:08

Median

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Besides mean, the median is a widely used measure of central tendency. Typically, median is defined as the central or middle value of a data set, measured by arranging the data elements in an increasing or decreasing order. Since this middle value is not affected by the precise numerical values of the outliers or fluctuations, it is insensitive to them. Hence, in cases where a data set may have outliers or the extreme values are not known, the median is a better measure of the central tendency...
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Wilcoxon Signed-Ranks Test for Median of Single Population01:14

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The Wilcoxon signed-rank test for the median of a single population is a nonparametric test used to evaluate whether the median of a population differs from a specified value. Unlike parametric tests, it does not require data to follow a normal distribution, making it suitable for non-normal or small samples. The test begins by calculating the difference (d) between each observation and the hypothesized median. The absolute values of these differences are ranked in ascending order, with ties...
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Sign Test for Median of Single Population01:20

Sign Test for Median of Single Population

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In general, the sign test serves as a nonparametric method to test hypotheses about the median of a single population when the data does not follow a known distribution. This simplicity makes it particularly useful for small sample sizes or when the assumptions of parametric tests cannot be met. The process begins with identifying a null hypothesis, typically stating that the population median equals a specific value. The alternative hypothesis could be that the median is either not equal to,...
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Wilcoxon Rank-Sum Test01:21

Wilcoxon Rank-Sum Test

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The Wilcoxon rank-sum test, also known as the Mann-Whitney U test, is a nonparametric test used to determine if there is a significant difference between the distributions of two independent samples. This test is designed specifically for two independent populations and has the following key requirements:
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Kruskal-Wallis Test01:19

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The Kruskal-Wallis test, also known as the Kruskal-Wallis H test, serves as a nonparametric alternative to the one-way ANOVA, offering a solution for analyzing the differences across three or more independent groups based on a single, ordinal-dependent variable. This statistical test is particularly valuable in scenarios where the data does not meet the normal distribution assumption required by its parametric counterparts. Kruskal-Wallis test is designed typically to handle ordinal data or...
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Microsoft Excel: Median, Quartile range, and Box Plots01:29

Microsoft Excel: Median, Quartile range, and Box Plots

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In Microsoft Excel, calculating the median, interquartile range, and creating box plots can help understand the distribution of your data.
Median and Quartile Range: The median is calculated using the formula `=MEDIAN(range)', which provides the middle value of your data set. Quartiles divide your data into four equal parts. To find the first and third quartiles, use ‘=QUARTILE(range, 1)' and ‘=QUARTILE(range, 3)', respectively. The interquartile range (IQR), which...
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A Univariate Inequality for Medians.

Clifford Spiegelman1

  • 1National Bureau of Standards, Washington, DC 20234.

Journal of Research of the National Bureau of Standards (1977)
|September 27, 2021
PubMed
Summary

This study introduces a novel inequality for medians, extending Karamata's theorem on majorization. The findings offer new insights into the mathematical properties of median-based inequalities.

Area of Science:

  • Mathematical Analysis
  • Probability Theory

Background:

  • Karamata's theorem is a fundamental result in the theory of majorization.
  • Majorization theory has broad applications in various fields, including statistics and economics.

Purpose of the Study:

  • To establish a new inequality for medians.
  • To demonstrate the relationship between this new inequality and Karamata's theorem.

Main Methods:

  • The study utilizes techniques from inequality theory.
  • The derivation involves concepts related to majorization and statistical functions.

Main Results:

  • A novel inequality specifically for medians has been derived.
  • This median inequality is shown to be an analog of Karamata's theorem.
Keywords:
concaveconvexinequalitymajorizationmedian

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Conclusions:

  • The established median inequality provides a new tool for mathematical analysis.
  • The results contribute to the understanding of majorization theory and its extensions.