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Variance

Variance

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 data.

Variation

Variation

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...

One-Way ANOVA: Equal Sample Sizes

One-Way ANOVA: Equal Sample Sizes

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...

One-Way ANOVA: Unequal Sample Sizes

One-Way ANOVA: Unequal Sample Sizes

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:

Testing a Claim about Standard Deviation

Testing a Claim about Standard Deviation

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...

One-Way ANOVA

One-Way ANOVA

One-way ANOVA analyzes more than three samples categorized by one factor. For example, it can compare the average mileage of sports bikes. Here, the data is categorized by one factor - the company. However, one-way ANOVA cannot be used to simultaneously compare the sample mean of three or more samples categorized by two factors. An example of two factors would be sports bikes from different companies driven in different terrains, such as a desert or snowy landscape. Here, two-way ANOVA is used...

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Related Experiment Video

Updated: Jul 15, 2026

Decomposing the Variance in Reading Comprehension to Reveal the Unique and Common Effects of Language and Decoding

Decomposing the Variance in Reading Comprehension to Reveal the Unique and Common Effects of Language and Decoding

Published on: October 11, 2018

Assessing the adequacy of variance function in heteroscedastic regression models.

Lan Wang¹, Xiao-Hua Zhou

¹School of Statistics, University of Minnesota, 224 Church Street SE, Minneapolis, Minnesota 55455, USA.

|May 9, 2007

Summary

This study introduces a new nonparametric test to validate parametric variance models in heteroscedastic regression. The proposed kernel-smoothing test offers a robust alternative for analyzing heteroscedastic data without assuming error distributions.

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Area of Science:

Statistics
Econometrics
Biostatistics

Background:

Heteroscedastic data, where variance is not constant, are common in various scientific fields.
Current methods often assume a specific parametric form for the variance, which may not always be appropriate.
Checking the adequacy of these parametric variance assumptions is crucial for reliable regression analysis.

Purpose of the Study:

To develop a new nonparametric test for assessing the validity of parametric variance function specifications in heteroscedastic regression.
To provide a method that does not require specifying the distribution of the random errors.
To evaluate the performance and applicability of the proposed test.

Main Methods:

A kernel-smoothing type nonparametric test statistic is proposed.
Asymptotic distribution of the test statistic under the null hypothesis is derived.
A bootstrap algorithm is suggested for approximating the test statistic's distribution in finite samples.
The test's power against various alternatives is investigated.

Main Results:

The proposed test statistic follows an asymptotic normal distribution under the null hypothesis.
The test demonstrates power against a broad range of alternative variance structures.
Simulation studies confirm the satisfactory performance of the test in finite sample sizes.
The test is successfully applied to a real-world radioimmunoassay dataset.

Conclusions:

The developed nonparametric test provides a flexible and reliable tool for checking parametric variance models in heteroscedastic regression.
The test's ability to avoid distributional assumptions for errors enhances its practical utility.
The bootstrap approach offers a viable method for practical implementation and inference.