This is a test version of Biostars. For the public version, visit https://www.biostars.org.
Why divide by the root n when constructing the u statistic

For a normal population, the following equation can be used to convert them to a standard normal distribution enter image description here

if, X ~ N(μ, σ^2).

When we apply hypothesis testing between a normal sample population and a normal population with a konwn σ^2, we use following equation to compute U or z: enter image description here

The question here is how to explain the difference in the denominator and why the latter is appropriate here. I noted some website (such as statisticshowto) explain this as following:

You always divide by sqrt(n). However, occasionally the square root of n sometimes equals 1 (making it just σ in the denominator. for example, if you are choosing one person and trying to figure out the probability their weight is under x pounds, then n=1. In other words, if you are calculating a z-score, you can always use sqrt(n). (https://www.statisticshowto.com/sigma-sqrt-n-used/)

But this looks like just a few handy tips to remember, right?

enter image description here

I would like to know the correct understanding of this issue, it would be better if it is more general and also rigorous.

Thank your for your attentions.

zhang

t-test statictic offtopic u-test
• 4,282 views • link
updated by Ram Ram • written by octpus616 • octpus616

1 answer

This is because you're computing the standard error of the mean, i.e. the standard deviation of the estimate of the mean. See the corresponding wikipedia article. In the first case, the standard deviation corresponds to the spread of the sample distribution, e.g. throw a die 1000 times and take the mean of all throws as the estimate of the population mean, whereas in the second case, this is the spread of the estimates of the mean, e.g. throw a die 10 times and take the average, repeat 100 times and estimate the population mean by taking the average of the 100 experiments.

• 0 views • link
written by Jean-Karim Heriche Jean-Karim Heriche

Hi @Jean-Karim Heriche，

Thanks for your reply, my English is not well, so thank you for understanding what I mean also.

The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution. so in fact, the standard deviation of X-bar is standard error of X(i) or samples. Is it right?In the latter expression, we are actually transforming the mean of the sample, so we need the standard error of the sample (σ/sqrt(n)) as the standard deviation of the mean (X-bar), in fact it is σ(X-bar) = σ/sqrt(n).

• 0 views • link
written by octpus616 • octpus616

Log in to answer this question.

More posts like this