Z-Score & T-Score Calculator
Enter a score, the mean and the standard deviation to get the z-score and the T-score (50 + 10 × z). You can also enter a list of values and let the calculator find the mean and standard deviation, or work back from a z-score or T-score to a raw score.
z-score and T-score
A z-score tells how many standard deviations a score is above or below the mean:
z = (score − mean) ÷ standard deviation
A score equal to the mean has z = 0; a score one standard deviation above the mean has z = 1.
The T-score used here is the standardized score that is shifted and scaled to have a mean of 50 and a standard deviation of 10 (Wikipedia, "Standard score": "In educational assessment, T-score is a standard score Z shifted and scaled to have a mean of 50 and a standard deviation of 10"). It follows that
T = 50 + 10 × z
(the article read does not write this formula out; it follows from the definition). The same article says this score is known as hensachi in Japan, where it is widely used.
The calculation itself does not need the data to be normally distributed. It only rescales scores using the mean and the standard deviation. Turning a z-score or T-score into a rank or a percentage of people below a score does require assumptions about how the scores are distributed, and this calculator does not do that.
Other meanings of "T-score"
The name is used in other fields, so check that this is the meaning you need:
- In bone density measurement, the T-score is the standard score of a measurement compared with healthy 30-year-old adults, and it has the usual mean of 0 and standard deviation of 1. That is a different quantity from the one on this page.
- The t-statistic (from Student's t-distribution) is also different. Replacing the population mean and standard deviation with sample estimates in the z formula is related to the t-statistic, but this calculator does not calculate a t-statistic or run a t-test.
(Wikipedia's "Standard score" article notes both of these. Its list of other uses of "T-score" could not be read, and the pages on bone density and the t-statistic were not read.)
Population or sample standard deviation
- The z formula as usually defined uses the population mean and standard deviation (the standard deviation of the whole group, dividing by n).
- When the population values are not known, the sample mean and sample standard deviation can be used as estimates (dividing by n − 1).
In "Data list → z and T" you choose which one to use. The default is the population standard deviation (divide by n), which matches the definition of the standardized T-score for a whole group. Choose "A sample" when your values are a sample from a larger group. The two give slightly different results, and the difference shrinks as n grows. A sample standard deviation needs at least 2 values. To see both together, use the Standard Deviation Calculator.
In the other two calculations you enter the standard deviation yourself, so the choice is already made.
The three calculations
- Score → z and T: from a score, the mean and the standard deviation.
- Data list → z and T: enter the values; the calculator finds the mean and standard deviation (as chosen above), the number of values, the largest and the smallest, and the z-score and T-score of the score you pick. When the standard deviation is not an exact value, it is approximated with integer arithmetic and labeled a rounded value.
- z or T → score: the raw score is mean + standard deviation × z, or, from a T-score, mean + standard deviation × (T − 50) ÷ 10.
Examples
| Input | Result |
|---|---|
| Score 65, mean 50, standard deviation 10 | z = 1.5, T = 65 |
| Score 50, mean 50, standard deviation 10 | z = 0, T = 50 |
| Values 40, 50, 60, 70, 80; the score 70 (population) | Mean 60, standard deviation about 14.14, z about 0.71, T about 57.07 |
| The same values treated as a sample | Standard deviation about 15.81, z about 0.63, T about 56.32 |
| T-score 60, mean 50, standard deviation 10 | Score 60 (z = 1) |
Notes
- If the standard deviation is 0 (for example, all the values are the same, or there is only one value in the population case), the z-score and T-score cannot be calculated.
- A T-score can be below 0 or above 100 (when a score is far from the mean). A note is shown when that happens.
- Ranks, percentiles, "percent of people below" and grade or pass/fail judgments are not calculated. For quartiles and percentiles of your data, see the Percentile & Quartile Calculator.
- The results are exact up to the rounding you choose: z and T are rounded once for display.
Frequently asked questions
What range can a T-score take?
There is no fixed range; in principle it can be any real number. This calculator shows a note when it falls outside 0 to 100.
How is a T-score different from a z-score?
It is the same information on a different scale: the z-score has a mean of 0 and a standard deviation of 1, and the T-score has a mean of 50 and a standard deviation of 10.
Where do I get the mean and standard deviation?
From your data, or from the report of the test or study. If you have all the values, use "Data list → z and T". For only the mean or only the standard deviation, see the Mean & Median Calculator and the Standard Deviation Calculator.