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You're making a transformation on $A$ and get another RV $X=f(A)$. This can be simple scaling or any other operation. The covariance between the newly transformed variable $X$ and $B$ will in general change, i.e. in general, we have$$\operatorname{cov}(A,B)\neq \operatorname{cov}(X,B)$$ A simple example would be scaling, $X=2A$:$$\operatorname{cov}(X,B)=\...


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Here simple examples, in which I focus on the terminology that seems to be confusing you. Suppose you have a sample of size 100 from the distribution $\mathsf{Norm}(\mu=200, \sigma=25).$ Then the standard deviation of a single observation is $SD(X_i) = \sigma = 25.$ The standard deviation of $\bar X,$ also called the 'standard error of the (sample) mean', ...


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As Glen mentioned, additivity of variances when samples are independent: VAR(ND) = NVAR(D), with VAR = S^2 then S(N*D) = SQRT(N)*S(D) In statistics, the standard error SE or standard deviation for a sample mean is given by the formula SE = S/SQRT(N) for the same reason as above, where N is the sample size and S is the standard deviation of the sample. ...


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You have various different standard deviations and standard errors there. Trying to unpick them, for given $p$: An individual trial has variance $p(1-p)$ and standard deviation $\sqrt{p(1-p)}$ The sum of $n$ independent trials has variance $np(1-p)$ and standard deviation $\sqrt{np(1-p)}$ The mean of $n$ independent trials has variance $\frac1n p(1-p)$ and ...


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This is a standard technique for normalizing distributions of data. It seems like this implies your data comes from the same family of distributions defined by the mean and std. dev. parameters.


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Once we have @ThorGirl's derivation for $ SE(\hat{\beta_1})^2 $ we can use that to derive the Standard Error for $ \hat{\beta_0} $ i.e, $ SE(\hat{\beta_0})^2 $ Note: If you are looking for a step-by-step explanation of @ThorGirl's answer take a look at this video. We are going to use the following assumptions / observations: 1) Each output $y_i$ is ...


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This really comes down to an issue of "dimensionality". Imagine that each of the five "difference" values is a vector (i.e., a line pointing positively or negatively with length equal to the difference value). Geometrically, the mean (of) difference value is obtained by putting these vectors end-to-end in a single dimension, to get the aggregate difference,...


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Gelman added a bit more detail to the explanation of the rescaling here. The -1/+1 recoding accommodates an interpretation of a 1 SD change to a change from 0 to 1 in the binary variable. If coded as 0/1, a change from 0 to 1 would correspond to a change of appx. 2 SD. With the recoding you can then rescale all inputs, whether binary og continuous, by ...


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Since your data are strictly positive (corresponding to times), and you are trying to estimate a ratio, then it is likely that a log-transformation will help you. Estimating the log-ratio is easier than estimating the ratio, because it is estimated by the difference of the logs of the two groups. So you could use for example a standard t-test to get the ...


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