Probability and Statistics > Statistics Definitions

Contents:

T Scores in Statistics.

What is the T Score Formula?

T Score Formula Example

T Scores in Psychometrics

## T Scores in Statistics

Watch the video or read on below:

## What is the T Score Formula?

A t score is one form of a standardized test statistic (the other you’ll come across in elementary statistics is the z-score). The t score formula enables you to take an individual score and transform it into a standardized form — one which helps you to compare scores.

You’ll want to use the t score formula when you don’t know the population standard deviation and you have a small sample (under 30).

The t score formula is:

Where

x̄ = sample mean

μ_{0} = population mean

s = sample standard deviation

n = sample size

If you have only one item in your sample, the square root in the denominator becomes √1. This means the formula becomes:

In simple terms, the larger the t score, the larger the difference is between the groups you are testing. It’s influenced by many factors including:

- How many items are in your sample.
- The means of your sample.
- The mean of the population from which your sample is drawn.

The standard deviation of your sample.

## What is the T Score Formula used for?

You traditionally look up a t score in a t-table. The number of items in your sample, minus one, is your degrees of freedom. For example, if you have 20 items in your sample, then df = 19. You use the degrees of freedom along with the confidence level you are willing to accept, to decide whether to support or reject the null hypothesis.

The t score formula can also be used to solve probability questions. You won’t have an alpha level, but you can use the result from the formula, along with a calculator like the TI-83, to find probabilities.

## Example of the T Score Formula

**Sample question:**

A law school claims it’s graduates earn an average of $300 per hour. A sample of 15 graduates is selected and found to have a mean salary of $280 with a sample standard deviation of $50. Assuming the school’s claim is true, what is the probability that the mean salary of graduates will be no more than $280?

Step 1: Plug the information into the formula and solve:

x̄ = sample mean = 280

μ_{0} = population mean = 300

s = sample standard deviation = 50

n = sample size = 15

t = (280 – 300)/ (50/√15) = -20 / 12.909945 = -1.549.

Step 2: Subtract 1 from the sample size to get the degrees of freedom:

15 – 1 = 14. The degrees of freedom lets you know which form of the t distribution to use (there are many, but you can solve these problems without knowing that fact!).

Step 3: Use a calculator to find the probability using your degrees of freedom (8). You have several options, including the TI-83 (see the bottom part of How to find a t distribution on a TI 83) and this online calculator. Here’s the result from that calculator. Note that I selected the radio button under the left tail, as we’re looking for a result that’s no more than $280:

The probability is 0.0718, or 7.18%.

## T Scores in Psychometrics

A t score in psychometric (psychological) testing is a specialized term that is **not the same thing as a t score that you get from a t-test.**

T scores in t-tests can be positive or negative. **T scores in psychometric testing are always positive**, with a mean of 50.

A difference of 10 (positive or negative) from the mean is a difference of one standard deviation. For example, a score of 70 is two standard deviations above the mean, while a score of 0 is one standard deviations below the mean.

A t score is similar to a z score — it represents **the number of standard deviations from the mean**. While the z-score returns values from between -5 and 5 (most scores fall between -3 and 3) standard deviations from the mean, the t score has a greater value and returns results from between 0 to 100 (most scores will fall between 20 and 80). Many people prefer t scores because the lack of negative numbers means they are easier to work with and there is a larger range so decimals are almost eliminated. This table shows z-scores and their equivalent t scores.

## T Score Conversion in Psychometrics

Watch the video or read the article below:

Calculating a t score is really just a **conversion** from a z score to a t score, much like converting Celsius to Fahrenheit. The formula to convert a z score to a t score is:

T = (Z x 10) + 50.

**Sample question:** A candidate for a job takes a written test where the average score is 1026 and the standard deviation is 209. The candidate scores 1100. Calculate the t score for this candidate.

*Note: If you are given the z-score for a question, skip to Step 2.
*

Step 1: Calculate the z score. (See: How to calculate a z-score).. The z-score for the data in this sample question is .354.

Step 2: Multiply the z score from Step 1 by 10:

10 * .354 = 3.54.

Step 3: Add 50 to your result from Step 2:

3.54 + 50 = 53.54.

*That’s it!*

**Tips**:

- z-scores and t scores both represent standard deviations from the mean, but while “0” on a z-score is 0 standard deviations from the mean, a “50” on a t score represents the same thing. That’s because t scores use a mean of 50 and z-scores use a mean of 0.
- A t score of over 50 is above average; below 50 is below average. In general, a t score of above 60 means that the score is in the top one-sixth of the distribution; above 63, the top one-tenth. A t score

below 40 indicates a lowest one-sixth position; below 37, the bottom one-tenth.

Questions on t score conversion? Post a comment and I’ll do my best to help!

I am trying to assemble a table for converting raw scores in to T scores for a self-report measure, such that low T scores indicate poor ability and increasing T scores indicate increasing ability. However, there are two raw scores missing in the sequence of my sample scores (35 and 37, from a range of 32 to 80). Therefore my question is, is it acceptable to “skip” T scores in a table? If someone with a raw score of 35 (one of the missing raw score values) was to consult this table in order to find the corresponding T score (missing), how would they be able to work out their T score?

Help with this question would be greatly appreciated. Thank you.

“…is it acceptable to “skip” T scores in a table?”

Yes, if they don’t apply (i.e. no one will ever look for them).

No, if you expect that people will look for the missing raw scores.

“If someone with a raw score of 35 (one of the missing raw score values) was to consult this table in order to find the corresponding T score (missing), how would they be able to work out their T score?”

Not sure about this. How are you doing your conversions? With a formula?

Hi,

I appreciated your teaching very much. I find statistics really hard. I just chanced to see this here. I will visit again. My problem is that I do not have computer of my own, only when I go to the library, do I get a chance to use computer for my assignments, it draws me backwards especially on public holidays when libraries are not open. Thanks. Please, keep up the good work. Anne.

Best of luck with your class, Anne. Glad you found this helpful :)