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Related Concept Videos

Distributions to Estimate Population Parameter01:26

Distributions to Estimate Population Parameter

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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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Estimating Population Mean with Unknown Standard Deviation01:22

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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.
William S. Gosset (1876–1937) of the...
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Estimating Population Mean with Known Standard Deviation01:16

Estimating Population Mean with Known Standard Deviation

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To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean is the point estimate of the unknown population mean μ.
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Estimating Population Standard Deviation01:26

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When the population standard deviation is unknown and the sample size is large, the sample standard deviation s is commonly used as a point estimate of σ. However, it can sometimes under or overestimate the population standard deviation. To overcome this drawback, confidence intervals are determined to estimate population parameters and eliminate any calculation bias accurately. However, this only applies to random samples from normally distributed populations. Knowing the sample mean and...
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Sampling Distribution01:12

Sampling Distribution

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Given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example...
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Extraction: Partition and Distribution Coefficients01:14

Extraction: Partition and Distribution Coefficients

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The distribution law or Nernst's distribution law is the law that governs the distribution of a solute between two immiscible solvents. This law, also known as the partition law, states that if a solute is added to the mixture of two immiscible solvents at a constant temperature, the solute is distributed between the two solvents in such a way that the ratio of solute concentrations in the solvents remains constant at equilibrium.
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Determination of Aggregate Surface Morphology at the Interfacial Transition Zone ITZ
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Efficient estimation of smooth distributions from coarsely grouped data.

Silvia Rizzi, Jutta Gampe, Paul H C Eilers

    American Journal of Epidemiology
    |June 18, 2015
    PubMed
    Summary

    Ungrouping binned data improves analysis accuracy and comparability. This new method effectively refines histograms, even with wide, open-ended intervals, by assuming a smooth underlying distribution.

    Keywords:
    grouped datapenalized composite link modelsmoothingungrouping

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    Area of Science:

    • Statistics
    • Data Analysis
    • Biostatistics

    Background:

    • Binned data often requires ungrouping for detailed analysis.
    • Coarse bins and differing grouping methods hinder accurate comparisons.
    • Open-ended intervals in histograms obscure crucial tail data.

    Purpose of the Study:

    • To develop a versatile method for ungrouping histogram data.
    • To enable more accurate analysis and comparison of binned data.
    • To effectively handle wide and open-ended intervals in statistical distributions.

    Main Methods:

    • A composite link model with a smoothness penalty was employed.
    • Penalized likelihood maximization using an iteratively reweighted least-squares algorithm.
    • Akaike's Information Criterion was used for smoothing parameter selection.

    Main Results:

    • The proposed method successfully ungroups binned data under a smooth distribution assumption.
    • Demonstrated effectiveness in simulation studies and practical examples.
    • Proper handling of wide, open-ended intervals was achieved.

    Conclusions:

    • The method offers a versatile solution for ungrouping histograms.
    • Suitable for various applications, including disease incidence rates and life tables.
    • Extensible to estimate rates with grouped event counts and exposures.