**Contents:**

T Scores in Statistics.

What is the T Score Formula?

T Score Formula Example

T Scores in Psychometrics

## T Scores in Statistics

## 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:

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.

The following example shows how to calculate a t-score formula for a single sample. Paired samples and independent samples use different formulas.

- If you have paired samples, follow the instructions in the
*paired samples t-test*. - For independent samples, see:
*independent samples t-test*.

## Example of the T Score Formula

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**Example 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 How to find a t distribution on a TI 83). Here’s the result from an online calculator I found a few years back (now defunct). 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

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.

**Example 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 example 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.

## References

Levine, D. (2014). Even You Can Learn Statistics and Analytics: An Easy to Understand Guide to Statistics and Analytics 3rd Edition. Pearson FT Press