Question about the use of the word "standard error" in several situations I'm now reading the book Computer Age Statistical Inference and I'm not sure what the standard error actually refers to in several cases.
In Page 9, the author defines for two data sets AML and ALL (they are chosen randomly from two large data sets) the two-sample t-statistic $t=\frac{\overline{\mathrm{AML}}-\overline{\mathrm{A} \mathrm{LL}}}{\widehat{\mathrm{Sd}}}$ and use this too find out whether their exists difference between the two original data sets. He says that "sd is an estimate of the numerator's standard deviation", but I cannot understand which standard deviation and what estimation he refers to.
Another one is that, say, we obtain a fitting curve (for example $y = ax + b$,etc) from a set of observed data $(x_i,y_i)$, then what does it mean by discussing the standard deviation of a point on the curve(this is mentioned in the Bootstrap Chapter)?
 A: Variance/standard deviation is a concept associated to a random variable. In your case,


*

*If we assume that one data point from AML comes from an independent sampling of a probability distribution, then the sampled mean $\overline{AML}$ is also a random variable, thus it (theoretically) has a standard deviation. Same goes for sampled mean $\overline{ALL}$ of ALL, and same goes for their difference $\overline{AML} - \overline{ALL}$. The text refers to this literal numerator.
As an observer however, we don't know exactly which probability distributions we are drawing from, and we only have observations. The best we can do is make estimations, and that's what the denominator intends to do.

*I can't find what you are referring to in the text, so here's a guess: maybe you are talking about linear regression. Again you need randomness somewhere for standard deviation to make sense. Let's assume the points $(x,y)$ are drawn based on the assumption that 
$$y = ax + b + \epsilon$$
where $a,b$ are fixed, and $\epsilon$ is a random error (for example, a Gaussian). Due to this random error, your "best-fit" straight line (whatever that means) $y \sim \hat{a} x + \hat{b}$ would likely not recover exactly $a,b$, but hopefully something close, as long as the error is "small". 
In other words, the "randomness" of the error would cause "randomness" in $\hat{a}, \hat{b}$ of your best-fit straight line, which then allows us to again talk about standard deviation.
