Ok, I don't know the exact answer to this, I suggest google might know, but if you consider the relationship between the variance and the standard deviation then also
I hope someone knows how the logic of this equation works.
In finding the standard error in a single sample t-test, why is n square-rooted? What does square rooting this number accomplish as a concept?
I don't have a problem using the equation, but I feel like a robot just doing it and not knowing why.
I want to understand.
I suppose at deepest level, we use so that everything is scaled properly. If you use something larger, like then as we will end up with our test statistic going to , regardless of whether null hypothesis is true or not, instead having a T-distribution.
Why does accomplish this? Well, because that is the rate at which the standard deviation of a sum increases. For example, if are independent from a normal distribution with mean and variance then so that the standard deviation is . So, we have the standard deviation of a sum increasing at a rate of - or equivalently the standard deviation of is decreasing at a rate of ; scaling by makes it so that our standard deviation (of the test statistic) is going neither to 0 nor to , but is staying constant.