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?
Tech mistake A t score is one form of a standardized test statistic (the other you’ll come across in elementary statistics is the zscore). 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).
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 ttable. 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 TI83, to find probabilities.
The following example shows how to calculate a tscore 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 ttest.
 For independent samples, see: independent samples ttest.
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 TI83 (see 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 ttest.
T scores in ttests 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 zscore 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 zscores 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 zscore for a question, skip to Step 2.
Step 1: Calculate the z score. (See: How to calculate a zscore).. The zscore 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:
 zscores and t scores both represent standard deviations from the mean, but while “0” on a zscore 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 zscores 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 onesixth of the distribution; above 63, the top onetenth. A t score
below 40 indicates a lowest onesixth position; below 37, the bottom onetenth.
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