Let $X_{1}, \dots, X_{n}$ be i.i.d. ramdom variables having mean $\mu$ and standard deviation $\sigma$. I wonder if the "studentized" $X_{i}$, the sample version of standardized $X_{i}$ where $\mu$ is replaced with the sample average and $\sigma$ is replaced with a sample standard deviation, admits a relatively canonical terminology in literature?

In my shallow opinion, the term "studentized" is informative but would cost some possibilities of confusion. So a terminology, if it exists, is sought.

The following is in response to a question raised in a comment below. To avoid introducing too many symbols, I described it in plain English. If this helps: If $$ \frac{X_{i} - \mu}{\sigma} $$ is called the standardized $X_{i}$, if $\bar{X}$ denotes the sample average, and if $s$ denotes the sample standard deviation under consideration, then I call $$ \frac{X_{i} - \bar{X}}{s} $$ the studentized $X_{i}$ above for reference ease.

  • $\begingroup$ Can you give a specific example, to explain what you exactly mean with these 'replacements', and with 'sample version of a variable $X_i $'. (note the value $\bar x / s_x $ is called the t-score, which contrasts with z-score, did you think about something like that?) $\endgroup$ – Martijn Weterings Nov 25 '18 at 16:44
  • $\begingroup$ @MartijnWeterings, Update is available. $\endgroup$ – Benicio Nov 25 '18 at 16:53
  • $\begingroup$ I am still a bit confused. Is this about computing a z-statistic or t-statistic (as used in z-test and t-test), or is this about normalization (I notice now you explicitly used the term 'standardized')? What confuses me (and the formula's don't solve this) are the random variables $X_1, ... , X_n$. Is $X_i$ a random variable (e.g. $X_i$ is the length of a person from population $i$) or is $X_i$ a specific member of a sample (e.g. $X_i$ is the length of person $i$ in the sample)? $\endgroup$ – Martijn Weterings Nov 29 '18 at 12:48
  • $\begingroup$ A new thing: it might be also useful to give examples/explanation where/how you wish to apply the term 'studentized'? If you use it in some context of defining confidence intervals (or anything else where distinction between the distributions normal/t is important) then it might be useful (e.g. see studentized residuals). If you use it a context like pca where one just seeks to roughly normalize the vectors then it will be confusing. $\endgroup$ – Martijn Weterings Nov 29 '18 at 12:59

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