Unequal Variance
Unequal Variance - Related Videos
01:16
Unusual Results
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2023
Unusual results are those that have a very low chance of occurring. Unusual results can be identified using probabilities and the range rule of thumb. In problems involving probability, unusual results can be observed in 2 instances – an unusually high number of successes or an unusually low number of successes. According to the range rule of thumb, any value above or below two standard deviations, 2σ from the mean, μ is considered unusual. Maximum unusual value = μ + 2σ Minimum unusual value...
01:25
Variance
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2024
Variance is a statistical measure that quantifies the degree of risk associated with an investment's returns by indicating how much the returns deviate from their expected value over time. It provides essential insights into the stability and predictability of an investment's performance. The variance calculation involves determining the mean return, which is the average return over a specified period, and then calculating the deviations of each return from this mean. These deviations are...
01:15
Variance
0 Views •
2023
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.
01:07
z Scores and Unusual Values
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2023
The z score is one of the three measures of relative standing. It describes the location of a value in a dataset relative to the mean. z scores are obtained after the standardization of the values in a dataset. The z score for the mean is 0. This score indicates how far a value is from the mean in terms of standard deviation. For example, if a data value has a z score of +1, the researcher can infer that the particular data value is one standard deviation above the mean. If another data value...
01:21
Friedman Two-way Analysis of Variance by Ranks
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2025
Friedman's Two-Way Analysis of Variance by Ranks is a nonparametric test designed to identify differences across multiple test attempts when traditional assumptions of normality and equal variances do not apply. Unlike conventional ANOVA, which requires normally distributed data with equal variances, Friedman's test is ideal for ordinal or non-normally distributed data, making it particularly useful for analyzing dependent samples, such as matched subjects over time or repeated measures from...
FAQs
Unequal variance can distort a pooled two-sample t-test because that test estimates a common variance from the groups and uses it in the standard error. If the groups have different dispersion, the pooled estimate may misrepresent sampling uncertainty, making the resulting p-values and confidence intervals unreliable. Welch’s t-test avoids this specific pooling assumption by adjusting its standard error and degrees of freedom.
Welch’s t-test replaces the pooled variance calculation with a standard-error adjustment that reflects the groups’ differing dispersion. It also modifies the degrees of freedom used for inference, rather than treating the two variance estimates as interchangeable. Consequently, the resulting test and confidence interval are calibrated to the unequal-spread situation more appropriately than the pooled two-sample approach.
Standard analysis of variance, like pooled comparisons, can produce unreliable standard errors, p-values, and confidence intervals when its equal-variance assumption is not appropriate. In regression, the corresponding issue is addressed with heteroscedasticity-robust methods, which are designed for settings where variability changes across groups, treatments, or measurement levels.
Graphical methods can reveal whether groups show noticeably different dispersion around their means, while formal methods provide a structured assessment of the same concern. Their purpose is to determine whether an equal-variance procedure is suitable or whether a variance-robust alternative is more credible. This assessment should occur before interpreting standard errors, p-values, or confidence intervals.
First examine group spread with graphical methods and, when appropriate, formal methods. If the analysis compares two group means, consider Welch’s t-test rather than automatically using a pooled test. For regression analyses, consider heteroscedasticity-robust methods. This sequence connects assumption checking with method selection and supports more credible statistical conclusions when variability changes across the data.
They are especially relevant whenever variability changes across groups, treatments, or measurement levels, because a method that ignores that change can make inferential summaries unreliable. Using Welch’s t-test or heteroscedasticity-robust regression helps align the analysis with the observed spread, supporting more credible conclusions about group comparisons and regression results.