# Convert z-score to t-score?

#### unipsychologist

##### New Member
Hi, Is there a standard formula for converting z-scores to t-scores? I found something online that said t = 10z + 50, but I wasn't sure if that's universally true.

Thanks!

#### Dragan

##### Super Moderator
Hi, Is there a standard formula for converting z-scores to t-scores? I found something online that said t = 10z + 50, but I wasn't sure if that's universally true.

Thanks!
No, it is not universally true.

For example, many psychological and educational tests are purposely constructed to follow a normal distribution. The purpose of T-scores is primarily to get rid of the negative values and decimal numbers associated with Z-scores.

A common example would be the Wechsler Adult Intelligence Scale (WAIS) which uses

T = 100 + 10*Z

where the mean is 100 (not 50) and the values of T are rounded to whole numbers.

#### unipsychologist

##### New Member
Thanks. So in order to determine the proper conversion, I'd just need to identify the mean? Or do I also need to worry about the coefficient?

#### Dragan

##### Super Moderator
Thanks. So in order to determine the proper conversion, I'd just need to identify the mean? Or do I also need to worry about the coefficient?

Both. For example, GRE scores have a mean of 500 and a standard deviation of 100.

The main point is to turn standard normal scores (Z) into scores that the average person (who may have no background in statistics) can find meaningful.

For example, could you imagine telling the parents of a child that their son (or daughter) has an IQ of -1.562 (the Z score).

The response from the parents might be something like: "Oh my gosh my child has negative intelligence!"

#### amygdala9

##### New Member
No, it is not universally true.

For example, many psychological and educational tests are purposely constructed to follow a normal distribution. The purpose of T-scores is primarily to get rid of the negative values and decimal numbers associated with Z-scores.

A common example would be the Wechsler Adult Intelligence Scale (WAIS) which uses

T = 100 + 10*Z

where the mean is 100 (not 50) and the values of T are rounded to whole numbers.

Actually, the WAIS standard scores & scaled scores are not technically "t-scores." Therefore, the conversion is different, but it is also called a "Standard Score" (SS) for the full scale (mean of 100, SD of 10) or a "Scaled Score" (ss) for the indices (mean of 10, SD of 1).

Don't get confused, though, because t-scores are sometimes called "standardized" scores.

#### amygdala9

##### New Member
Actually, the WAIS standard scores & scaled scores are not technically "t-scores." Therefore, the conversion is different, but it is also called a "Standard Score" (SS) for the full scale (mean of 100, SD of 10) or a "Scaled Score" (ss) for the indices (mean of 10, SD of 1).

Don't get confused, though, because t-scores are sometimes called "standardized" scores.
Sorry - I got that wrong: Standard scores have a mean of 100, SD of 15, and scaled scores have a mean of 10, SD of 3.

#### Biff Spigler

##### New Member
it's a bit more complex that that. i was googling this and couldn't find a satisfactory explanation, so I dug it out of some computer code I wrote:

double PDF = (1.0 / (Math.Sqrt(2 * Math.PI * SD * SD))) * Math.Exp(-(X - Mean) / (2 * SD * SD));
double t = 1.0 / (1 + b0 * X);
double Percentile = 1 - PDF * ((b1 * t) + (b2 * t * t) + (b3 * t * t * t) + (b4 * t * t * t * t) + (b5 * t * t * t * t * t));
where:
double b0 = 0.2316419, b1 = 0.319381530, b2 = -0.356563782, b3 = 1.781477937, b4 = -1.821255978, b5 = 1.330274429;

You can drag it out to b6, b7, b8 . .. .. and I forgot how those values are calculated, but that's a pretty accurate solution where X in this case is a raw value. Oh, and Math.Exp(a) is e^a.