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# Z-Score Formula

#### equanimity

##### New Member
Hi All,

I have a question about the formula for Z-scores. I've seen the formula as:

a) Z-Score = (observation - mean) / standard deviation

and as:

b) Z-Score = (sample mean - population mean) / (standard deviation / SQRT(sample size))

Thus far, I've been using formula "a" above. When would we use formula "b"?

Thanks!

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
The b is used when we need to check whether the sample mean is closer to the actual population mean. Actually we don't have complete data set in the real world to calculate the population mean for e.g. for a company producing the screws of standard length we cannot measure the length of each and every screw but only infer the actual mean screw length using the sample of screws. we pick a sample of screw of size n and calculate the sample mean and the standard deviation(s) to infer the population mean (the mean length of all the screws produced).
for 95% CL, Z-Score =1.96= (sample mean - population mean) / (standard deviation / SQRT(sample size)) the standard deviation here is the population(all the screws produced) standard deviation we don't know it therefore we approximate it by using the sample standard deviation(s).
Thus, 1.96= (sample mean - population mean) / (s / SQRT(n))
=> population mean=sample mean +/- 1.96*(s / SQRT(n)) thus the mean lenght of all the screws produced must lie b/w sample mean +/- 1.96*(s / SQRT(n)).
thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @equanimity Consistent with @ShaktiRathore 's answer, I would just add that fundamentally both equations do the same thing: they standardize an observation (or sample mean); i.e., translate it into standard normal Z quantile. The latter is the special case when deminator is given by the standard error, which is a standard deviation but explicitly of a sampling distribution (in this case, the distribution of the sample mean, which is normal per CLT thus justifying the standardization in the first place). The hardest conceptual aspect of the difference, I think, is the following: both of these returns a Z-store which is simply a location (quantile) on the standard normal distribution, N(0 mean, 1 variance), while the former assumes the set of observations might be described (characterized) by a normal distribution, the latter is referring to the distribution of sample means. Thanks,