z=(x-mean)/(standard deviation of population distribution/square root of sample size)
T-score is for when you don't have pop. standard deviation and must use sample s.d. as a substitute.
t=(x-mean)/(standard deviation of sampling distribution/square root of sample size)
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If the distribution is Gaussian (or Normal) use z-scores. If it is Student's t, then use t-scores.
Z scores are used for standardized testing done by most school districts. It is the most common way of standardizing data. IQ scores can be standardized using z scores. The mean is 100 and the standard deviation is 15. You use the t score when the sample is small, <30 often. Many behavior ratings use t scores.
A z-score cannot help calculate standard deviation. In fact the very point of z-scores is to remove any contribution from the mean or standard deviation.
T-scores and z-scores measure the deviation from normal. The normal for T-score is 50 with standard deviation of 10. if the score on t-score is more than 50, it means that the person scored above normal (average), and vise versa. The normal for Z-score is 0. If Z-score is above 0, then it means that person scored above normal (average), and vise versa.
T scores are also standardized norm scores, where the mean value is 50 and standard deviation value is 10, in contrast to Z scores where mean value is "0" and standard deviation value is 1. -Rama Reddy Karri