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Related Concept Videos

Variance01:15

Variance

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 The deviations show how spread out the data are about the mean. A positive deviation occurs when the data value exceeds the mean, whereas a negative deviation occurs when the data value is less than the mean. If the deviations are added, the sum is always zero. So one cannot simply add the deviations to get the data spread. By squaring the deviations, the numbers are made positive; thus, their sum will also be positive.
The standard deviation measures the spread in the same units as the...
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One-Way ANOVA: Equal Sample Sizes01:15

One-Way ANOVA: Equal Sample Sizes

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One-Way ANOVA can be performed on three or more samples with equal or unequal sample sizes. When one-way ANOVA is performed on two datasets with samples of equal sizes, it can be easily observed that the computed F statistic is highly sensitive to the sample mean.
Different sample means can result in different values for the variance estimate: variance between samples. This is because the variance between samples is calculated as the product of the sample size and the variance between the...
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One-Way ANOVA: Unequal Sample Sizes01:15

One-Way ANOVA: Unequal Sample Sizes

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One-way ANOVA can be performed on three or more samples of unequal sizes. However, calculations get complicated when sample sizes are not always the same. So, while performing ANOVA with unequal samples size, the following equation is used:
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Variation01:19

Variation

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An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation, which is the square root of variance.
When independent and dependent variables are plotted on a scatter plot, the slope of a line is a value that describes the rate of change between the two...
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Estimating Population Mean with Unknown Standard Deviation01:22

Estimating Population Mean with Unknown Standard Deviation

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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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Testing a Claim about Standard Deviation01:19

Testing a Claim about Standard Deviation

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A complete procedure to test a claim about population standard deviation or population variance is explained here.
The hypothesis testing for the claim of population standard deviation (or variance) requires the data and samples to be random and unbiased. The population distribution also must be normal. There is no specific requirement on the sample size as the estimation is based on the chi-square distribution.
As a first step, the hypothesis (null and alternative) concerning the claim about...
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Improvement in variance estimation using transformed auxiliary variable under simple random sampling.

Hameed Ali1, Syed Muhammad Asim1, Muhammad Ijaz2

  • 1Department of Statistics, University of Peshawar, Peshawar, Pakistan.

Scientific Reports
|April 6, 2024
PubMed
Summary
This summary is machine-generated.

This study introduces an efficient ratio estimator for population variance by transforming an auxiliary variable. This novel approach significantly boosts estimation efficiency, outperforming existing methods in simulations.

Keywords:
Auxiliary variableMean square errorPercentage relative efficiencyPopulation variance

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Area of Science:

  • Statistics
  • Survey Methodology
  • Statistical Inference

Background:

  • Accurate estimation of population variance is crucial in statistical analysis.
  • Traditional ratio estimators rely on auxiliary variables but can be limited in efficiency.
  • Transforming auxiliary variables offers a potential avenue for improving estimation precision.

Purpose of the Study:

  • To develop a novel and efficient ratio estimator for population variance.
  • To investigate the impact of transforming auxiliary variables on estimation efficiency.
  • To theoretically and empirically validate the performance of the proposed estimators.

Main Methods:

  • Formulation of a new ratio estimator using a transformed auxiliary variable.
  • Derivation of the theoretical properties of the proposed estimators.
  • Empirical and simulation studies to compare performance against existing estimators.

Main Results:

  • The transformation of auxiliary variables leads to a substantial gain in efficiency.
  • The newly developed estimators demonstrate superior performance compared to existing ones.
  • Both theoretical derivations and simulation results confirm the enhanced efficiency.

Conclusions:

  • The proposed ratio estimator using a transformed auxiliary variable is highly efficient.
  • This novel approach provides a valuable improvement for estimating population variance.
  • The findings are supported by rigorous theoretical analysis and practical simulations.