# 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.

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:

A couple mentions of t-scores here supports my assertion regarding them:

A mention of standardized scores along my interpretation is here:

• 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.

• Great information, before I posted this question I had found some of the information that you posted (uconn.edu link for instance) and after I read these answers and then reread it, I realized I wasn't comprehending the t-score properly. Thanks! Commented Jul 25, 2011 at 16:15
• Great explanation, I hope this is taken by people to encourage them to describe the details of whatever standard scores they are using in their write-ups. Commented Apr 12, 2013 at 15:53

Your question pertains to terminology used in the reporting of standardised psychometric tests.

My understanding:

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.

• Funny to see we found the same table; I guess I was struggling in my efforts for revision fruitlessly. That is a much more clear and succinct answer. Commented Jul 24, 2011 at 6:51
• @drknexus. It's funny when the terminology of different fields collide. Commented Jul 24, 2011 at 7:10
• But, what seems ultra-funny here is that psychometrics isn't so far away from statistics/general psychology that the people working in the field aren't (generally) aware of the term collision. In this case there might be an interesting story about the time in which the T-score was developed versus when Student did his fine work. Commented Jul 24, 2011 at 14:15

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.

• Hmmm...I see what you're saying, but my favorite text does not make this distinction. Its definition of $z$ coincides with your $t$. Perhaps that's why someone downvoted this reply. (I wish they had been kind enough to indicate the reason.)
– whuber
Commented Jul 25, 2011 at 21:50
• @whuber I have heard that Gosset originally used $z$ for his statistics, what we now generally refer to as $t$, so your favorite text probably stays with that notation. The books that I have used and taught from (though a small subset of all available) all use $z$ for known $\sigma$ and $t$ for sample standard deviation. Commented Jul 26, 2011 at 16:34
• Yes, Gosset introduces $z$ in Section III of his 1908 ("Student") paper. I don't find a $t$ in there anywhere. Thanks for clarifying.
– whuber
Commented Jul 26, 2011 at 16:42
• Respectfully, I down voted this and Max's answers. Though the answers were well stated and made with good intentions, they were also misleading. Sorry guys. I should have made a statement to that effect earlier. I was focused on getting a more correct answer out there first and neglected to come back and do the house keeping. Commented Jul 27, 2011 at 5:37
• There is an extended discussion of the terminological issue @whuber raised above here: stats.stackexchange.com/questions/99717/… Commented Sep 17, 2014 at 21:34

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.