Examples of pooled variance in the following topics:
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- it can be assumed that the two distributions have the same variance.
- Here, ${ S }{ x }_{ 1 }{ x }_{ 2 }$ is the grand standard deviation (or pooled standard deviation), 1 = group one, 2 = group two.
- Note that in this case ${ { s }_{ { \bar { X } }_{ 1 }-{ \bar { X } }_{ 2 } } }^{ 2 }$ is not a pooled variance.
- This is the formula for a pooled variance in a two-sample t-test with unequal sample size but equal variances.
- This is the formula for a pooled variance in a two-sample t-test with unequal or equal sample sizes but unequal variances.
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- Variance between samples: An estimate of σ2 that is the variance of the sample means multiplied by n (when there is equal n).
- Variance within samples: An estimate of σ2 that is the average of the sample variances (also known as a pooled variance).
- MS means "mean square. " MSbetween is the variance between groups and MSwithin is the variance within groups.
- As it turns out, MSbetween consists of the population variance plus a variance produced from the differences between the samples.
- MSwithin is an estimate of the population variance.
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- To calculate the $F$-ratio, two estimates of the variance are made:
- Variance between samples: An estimate of $\sigma^2$ that is the variance of the sample means multiplied by $n$ (when there is equal $n$).
- Variance within samples: An estimate of $\sigma^2$ that is the average of the sample variances (also known as a pooled variance).
- $MS$ means "mean square. " $MS_{\text{between}}$ is the variance between groups and $MS_{\text{within}}$ is the variance within groups.
- As it turns out, $MS_{\text{between}}$ consists of the population variance plus a variance produced from the differences between the samples.
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- The method of analysis of variance in this context focuses on answering one question: is the variability in the sample means so large that it seems unlikely to be from chance alone?
- The MSG can be thought of as a scaled variance formula for means.
- To this end, we compute a pooled variance estimate, often abbreviated as the mean square error (MSE), which has an associated degrees of freedom value dfE = n−k.
- The F statistic and the F test Analysis of variance (ANOVA) is used to test whether the mean outcome differs across 2 or more groups.
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- In such cases, we can make our t distribution approach slightly more precise by using a pooled standard deviation.
- The pooled standard deviation of two groups is a way to use data from both samples to better estimate the standard deviation and standard error.
- If s1 and s2 are the standard deviations of groups 1 and 2 and there are good reasons to believe that the population standard deviations are equal, then we can obtain an improved estimate of the group variances by pooling their data:
- Caution: Pooling standard deviations should be done only after careful research
- A pooled standard deviation is only appropriate when background research indicates the population standard deviations are nearly equal.
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- Other uses for the F distribution include comparing two variances and Two-Way Analysis of Variance.
- Comparing two variances is discussed at the end of the chapter.
- Next, calculate the variance of the three group means (Calculate the variance of 24.2, 25.4, and 24.4).
- Variance of the group means = 0.413 = $s^2_{\bar{x}}$
- Mean of the sample variances = 15.433 = $s^2_{pooled}$
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- Remember, we talked about every particular investment having an expected return and a variance.
- If you are managing a pool of assets, you want to get positive returns without being in danger of "losing your shirt. " The probability that one stock goes belly up is much higher than that the whole stock market does.
- In finance, systematic risk is the term associated with risk that can be diversified away by investing in a broader pool of assets.
- Proponents of passive management say the market knows best, and they seek a portfolio that has an underlying pool that mimics a benchmark index (think S&P 500).
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- When comparing two proportions, it is necessary to use a pooled standard deviation for the $z$-test.
- A $t$-statistic may be used for one sample, two samples (with a pooled or unpooled standard deviation), or for a regression $t$-test.
- $F$-tests (analysis of variance, also called ANOVA) are used when there are more than two groups.
- If the variance of test scores of the left-handed in a class is much smaller than the variance of the whole class, then it may be useful to study lefties as a group.
- The null hypothesis is that two variances are the same, so the proposed grouping is not meaningful.
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- The intraclass correlation can be regarded within the framework of analysis of variance (ANOVA), and more recently it has been regarded in the framework of a random effect model.
- The variance of $\alpha_j$ is denoted $\sigma_{\alpha}^2$ and the variance of $\epsilon_{ij}$ is denoted $\sigma_{\epsilon}^2$.
- One key difference between the two statistics is that in the ICC, the data are centered and scaled using a pooled mean and standard deviation; whereas in the Pearson correlation, each variable is centered and scaled by its own mean and standard deviation.
- This pooled scaling for the ICC makes sense because all measurements are of the same quantity (albeit on units in different groups).
- where ${ \mu }_{ x }$ and ${ \mu }_{ y }$ are the means for the two variables and ${ { \sigma }^{ 2 } }_{ x }$ and ${ { \sigma }^{ 2 } }_{ y }$ are the corresponding variances.
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- Compute the variance of the sum of two variables if the variance of each and their correlation is known
- Compute the variance of the difference between two variables if the variance of each and their correlation is known
- which is read: "The variance of X plus or minus Y is equal to the variance of X plus the variance of Y."
- The variance of the difference is:
- If the variances and the correlation are computed in a sample, then the following notation is used to express the variance sum law: