What are the primary differences between z-scores and t-scores, and are they both considered standard scores? We are currently converting student test scores in this manner :   
( ScaledScore - ScaledScore Mean ) / StdDeviation ) * 15 + 100 

I was referring to this calculation as a z-score, I found some information that convinced me that I should really be referring to it as a t-score as opposed to a z-score.
My boss wants me to call it a "Standard Score" on our reports. Are z-scores and t-scores both considered standard scores?
Is there a well known abbreviation for "Standard Score"?
Can someone point me to a reference that will definitively solve this issue.  
 A: Most basic texts on statistics will define these as $z= \frac{\bar{x}-\mu}{ \sigma/\sqrt{n} }$ and $t=\frac{\bar{x}-\mu}{s/\sqrt{n}}$.  The difference is that $z$ uses $\sigma$ which is the known population standard deviation and $t$ uses $s$ which is the sample standard devition used as an estimate of the population $\sigma$.  There are sometimes variations on $z$ for an individual observation.  Both are standardized scores, though $t$ is pretty much only used in testing or confidence intervals while $z$ with $n=1$ is used to compare between different populations.
A: What you are reporting is a standardized score. It just isn't the standardized score most statisticians are familiar with. Likewise, the t-score you are talking about, isn't what most of the people answering the question think it is.
I only ran into these issues before because I volunteered in a psychometric testing lab while in undergrad. Thanks go to my supervisor at the time for drilling these things into my head. Transformations like this are usually an attempt to solve a "what normal person wants to look at all of those decimal points anyway" sort of problem.


*

*Z-scores are what most people in statistics call "Standard Scores".  When a score is at the mean, they have a value of 0, and for each standard deviation difference from the mean adjusts the score by 1.  

*The "standard score" you are using has a mean of 100 and a difference of a standard deviation adjusts the score by 15.  This sort of transformation is most familiar for its use on some intelligence tests.

*You probably ran into a t-score in your reading.  That is yet another specialized term that has no relation (that I am aware of) to a t-test.  t-scores represent the mean as 50 and each standard deviation difference as a 10 point change.


Google found an example conversion sheet here:


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*http://faculty.pepperdine.edu/shimels/Courses/Files/ConvTable.pdf
A couple mentions of t-scores here supports my assertion regarding them:


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*http://www.healthpsych.com/bhi/doublenorm.html

*http://www.psychometric-success.com/aptitude-tests/percentiles-and-norming.htm

*Chapter 5, pp 89, Murphy, K. R., & Davidshofer, C. O. (2001). Psychological testing: principles and applications. Upper Saddle River, NJ: Prentice Hall. 


A mention of standardized scores along my interpretation is here:


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*http://www.gifted.uconn.edu/siegle/research/Normal/Interpret%20Raw%20Scores.html

*http://www.nfer.ac.uk/nfer/research/assessment/eleven-plus/standardised-scores.cfm

*This is an intro psych book, so it probably isn't particular official either.  Chapter 8, pp 307 in Wade, C., & Tarvis, C. (1996). Psychology. New York: Harper Collins state in regards to IQ testing "the average is set arbitrarily at 100, and tests are constructed so that the standard deviation ... is always 15 or 16, depending on the test".


So, now to directly address your questions:


*

*Yes, zscores and tscores are both types of "Standard scores".  However, please note that your boss is right in calling the transformation you are doing a "standard score".

*I don't know of any standard abbreviation for standardized scores.

*As you can see above, I looked for a canonical source, but I was unable to find one. I think the best place to look for a citation people would believe is in the manual of the standardized test you are using.


Good luck.
A: Your question pertains to terminology used in the reporting of standardised psychometric tests.


*

*Charles Hale has notes on terminology in standardised testing.


My understanding:


*

*z-score: mean = 0; sd = 1

*t-score: mean = 50; sd = 10 (example test using t-scores) (interestingly, t-score means something different in the bone density literature)

*Typical IQ style scaling: mean = 100; sd = 15


All of the above are "standardised scores" in a general sense.
I have seen people use the term "standard score" exclusively for z-scores, and also for typical IQ style scaling (e.g., in this conversion table).
In terms of definitive sources of information, there might be something in The Standards for Educational and Psychological Testing from the American Psychological Association.
A: The Student's t test is used when you have a small sample and have to approximate the standard deviation (SD, $\sigma$). If you look at the distribution tables for the z-score and t-score you can see that they quickly approach similar values and that with more than 50 observations the difference is so small that it really doesn't matter which one you use. 
The term standard score indicates how many standard deviations away from the expected mean (the null hypothesis) your observations are and through the z-score you can then deduce the probability of that happening by chance, the p-value.
