Z critical vs Z Statistic

[Atin]

Hi David,

This certain confusion has cropped into my mind: Under what circumstances would I use Z critical (X - mean/std dev) against Z statistic (Sample mean - population mean/[population variance/square root(sample size)]). This is regarding P1.T2.71.3.

Please help!

Thanks much..

 

[David Harper CFA FRM]

Hi @Atin

Aside from possible, marginal technical (semantic) differences, I don't perceive much of a difference. Your "Z statistic" is a special case of "your Z critical" because [population variance/square root(sample size)] is the standard error of the sample mean which means that it is a standard deviation. Rather than (eg) a known population standard deviation or even just sample standard deviation, per CLT, it is the standard deviation of the sample mean. Your critical Z (aka, standard score http://en.wikipedia.org/wiki/Standard_score) is something than can always be computed ("a general case") whenever there is a mean and std deviation; it translates X into a Z variable with zero mean and unit variance. (it "imposes normality" when the data may not be normal!). Your "Z stat" similarly standardizes but I view it as a special case where you are standardizing the sample mean, I hope that helps,

 

[Atin]

Thanks much David for such a prompt response! So, I believe, only in case of sample mean I would use Z statistic else Z critical should serve the purpose.

 

[David Harper CFA FRM]

Hi @Atin

Yes, agreed: if it is a case of a test of the sample mean (as most of T2.71 are; and, indeed, this is arguably the most common test!) then your "Z stat" is appropriate. But I'd just add, for what it's worth:

• The only real difference is the source of your standard deviation (in the denominator), where CLT gives you you a SE in the case of sample mean. Otherwise, both formulas (to me) are essentially similar: they standardize the X or observed sample mean into a standard (or unit) normal variable (or student's t!). I view both of yours as "Z statistics," consequently
  
• "Z critical" or "critical Z" to me, connotes the lookup Z value based on some confidence. For example, 1.645 is famously our critical one-tailed "lookup" Z at 95% confidence. So, my view is that we compare a computed Z statistic to a lookup (critical) Z to accept/reject; e.g., how does our computed X.XX compare to the critical Z of 1.96 or 2.58 or 2.33 or 1.645. In this way, critical Z is already a standard (unit) normal variable. I just wanted to convey my impression on the semantics, I'm not sure these semantics are critical (pun intended!) although I think statisticians would take it more seriously

 

[Atin]

Thanks again! Quite clear, conceptually and semantically!