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Udacity gives two equations for standard error, and calls the second (pooled) one "corrected" without proof.

At 2:30, the narrator states that the following is the Standard Error for Independent Samples, after going through some kind of informal semantic explanation:

$$SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} $$

$s_1$ and $s_2$ are the sample standard deviations. Later in the lesson though, in a 14 second video, the narrator blithely replaces these individual sample variances with the pooled variance:

$$SE = \sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}} $$

The narrator specifically uses the word "corrected", implying that the first is biased...? Udacity has yet to use the word "biased" though, so perhaps I'm focusing too much on specific word choice.

Some googling led me to this video which states that the first SE formulation is the Satterthwaite approximation. It states that the Satterthwaite approximation may be always used instead of the pooled variant, which makes sense to me because it is more general.

Can I ignore the word "corrected" and always use the Satterthwaite approximation?

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They are the same thing.

Let's rephrase what we want to do here. We want to use two samples t-test for statistical testing, and we have two groups. Each group has it's own variance. The variance of the two groups can be anything. This makes your first formula, where you have s1 and s2. So far, we have not assumed anything other than the sample size of two groups is equal.

Now, we assume the variance of the two groups are equal. If they are assumed be equal, it can be proven that the pooled variance is an unbiased estimator. Since they are equal, we can set s1=sp and s2=sp (sorry, I don't know how to enter symbols). Next, we simply substitute the pooled variance into the first formula and this brings you to your second formula.

Summary:

  • Your second formula assumes equal sample size
  • Your second formula assumes homoscedastic (equal variance)

The Satterthwaite approximation is related to Welch's test, which assumes the variance unequal. If you can assume equal variance, the t-test becomes much easier.

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  • $\begingroup$ Doesn't $n_1$ and $n_2$ in both equations account for unequal sample size? As a side note, you enter symbols by surrounding with dollar signs, like this $n_1$. Look up "MathJax" for more info. $\endgroup$ – DharmaTurtle Sep 1 '16 at 20:11
  • $\begingroup$ @DharmaTurtle No. n1 and n2 are just symbols, they can be equal or not equal. $\endgroup$ – HelloWorld Sep 2 '16 at 1:21
  • $\begingroup$ Sorry, I wasn't specific enough in my previous comment; I am talking specifically about the first point of your summary. Don't both equations NOT assume equal sample size by having $n_1$ and $n_2$? If they assumed equal sample size, the equations would just use $n$, right? $\endgroup$ – DharmaTurtle Sep 2 '16 at 12:25
  • $\begingroup$ My previous question is a bit beyond the scope of this discussion, and gives Udacity too much credit. Two notes to future readers: 1) The assumption that $n_1 = n_2$ is explicitly stated here en.wikipedia.org/wiki/… 2) Many agree that the Welch test is more robust than the pooled test (despite it not strictly following the t-distribution) youtu.be/86ss6qOTfts?t=328 en.wikipedia.org/wiki/… daniellakens.blogspot.com/2015/01/… $\endgroup$ – DharmaTurtle Sep 7 '16 at 3:58
  • $\begingroup$ @DharmaTurtle Sorry I didn't see you previous message. n1 doesn't have to be equal to n2, but it's assumed be in Udacity. $\endgroup$ – HelloWorld Sep 7 '16 at 4:13

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