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Sampling is a technique to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. The sampling method ensures that samples are drawn without bias and accurately represent the population. Because measuring the entire population in a study is not practical, researchers use samples to represent the population of interest.
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The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...
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It isn't easy to measure a parameter such as the mean height or the mean weight of a population. So, we draw samples from the population and calculate the mean height or mean weight of the individuals in the sample. This sample data acts as a representative measure of the population parameter. These sample statistics are known as estimates. 
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Sampling is a crucial step in analytical chemistry, allowing researchers to collect representative data from a large population. Common sampling methods include random, judgmental, systematic, stratified, and cluster sampling.
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Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first, find the median or second quartile. The first quartile, Q1, is the middle value of the lower half of the data, and the third quartile, Q3, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:
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Median Estimation with Quantile Transformations: Applications to Stratified Two-Phase Sampling.

Fatimah A Almulhim1, Hassan M Aljohani2

  • 1Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia.

Entropy (Basel, Switzerland)
|December 24, 2025
PubMed
Summary
This summary is machine-generated.

New quintile-based median estimators improve accuracy and robustness in stratified sampling. These methods offer better precision and effectiveness for practical median estimation, especially with skewed data.

Keywords:
Monte Carlo simulationauxiliary informationbiasmean squared errorsmedian estimationquantile transformationsrelative efficiencystratified two-phase sampling

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Area of Science:

  • Statistics
  • Survey Methodology

Background:

  • Traditional median estimators often assume normality and are sensitive to outliers.
  • This sensitivity limits their reliability in real-world applications with non-normal or skewed data.

Purpose of the Study:

  • To introduce novel quintile-based median estimators.
  • To enhance accuracy and robustness in stratified two-phase sampling.
  • To improve the efficiency of median estimation using auxiliary data.

Main Methods:

  • Utilized transformation methods within a stratified two-phase sampling framework.
  • Developed quintile-based median estimators.
  • Derived bias and mean squared error (MSE) expressions via first-order approximations.
  • Evaluated estimator efficiency using MSE.

Main Results:

  • Proposed estimators demonstrated superior performance in simulations under skewed distributions.
  • Analysis on real population datasets confirmed the effectiveness of the new methods.
  • The quintile-based estimators achieved higher precision and effectiveness compared to existing approaches.

Conclusions:

  • The novel quintile-based median estimators are robust and accurate for practical applications.
  • These estimators provide a more effective alternative for median estimation, particularly in stratified sampling scenarios.
  • The methods enhance the utility of auxiliary data and perform well with heterogeneous strata.