In what way, does dividing by √N make sense in standard error mean formula, while calculating the z scores for multiple samples, and while describing the standard deviation of those means which is given as, Z=((x-μ))/□(σ/√N), where N is our sample size.
The assumption part is that the average of n values is less variable than any single observation from the population.
Suppose, a sampling from a population has variance σ2, i.e., V (Xi) =σ2. However, V (X¯) =σ2/n. So, the variance of X¯ does decrease as n increases, which matches the above case. Then it follows that the “standard error mean formula” as,
It is not difficult to understand that means are less variable than unit observations. For example, while trying to estimate the average weight of melons in a box, which may contain various sizes. On drawing one, we may get a big one or a small one.
Instead, if we do the same process multiple times, it is likely to get ones of different sizes, and the weight of their mean will be similar to the mean weight of the box. But from the above case, it does not tell that the exact relationship is V (X¯) =σ2/n for the variance or SD=σ/n for the standard error. For that, we need to do the math.
