tstatistic Calculator
Table of contents
What is tstatistic and Student's ttest?What is the tstatistic formula?How to use this tstatistic calculator?A tstatistic exampleFAQsUse the tstatistic calculator (tvalue calculator or t test statistic calculator) to compute the tvalue of a given dataset using its sample mean, population mean, standard deviation and sample size.
Read further where we answer to the following questions:
 What is the tstatistic?
 How do I calculate tstatistic?
 What is the difference between Tscore vs. Zscore?
If you are also interested in Ftest, check our Fstatistic calculator.
What is tstatistic and Student's ttest?
In statistics, the tstatistic, or tvalue, is a measure that describes the relationship between a sample and its population. The tstatistic is central to the Student's ttest, which is a test for evaluating hypotheses about the population mean.
More precisely, the tstatistic is used to determine whether to support or reject the null hypothesis. It is used in conjunction with the pvalue, or critical value, which indicates the probability that your results could have happened by chance. It is comparable to the zstatistic, with the difference being that the tstatistic is applied for small sample sizes or unknown population standard deviations.
What is the tstatistic formula?
You need to use the following tstatistic formula to calculate the tvalue:
Where:
 $\bar x$  Sample mean;
 $\mu$  Population mean;
 $n$  Sample size; and
 $s$  Standard deviation of the sample.
How to use this tstatistic calculator?
To compute the tstatistic, you need to provide the following four variables:
 Sample mean, $\bar x$;
 Population mean, $\mu$;
 Sample size, $n$; and
 Sample standard deviation, $s$.
Alternatively, you can use the tool in reverse; for example, you can recover the sample mean from the tstatistic, provided you input all other values.
A tstatistic example
Let's say you are a basketball player and your game score is 15 (x̄
) on average over 36 (n
) games, with a standard deviation of 6 (s
). You know that an average basketball player scores 10 (μ
). Should your performance be considered above average? Or are your scores due to luck? Finding the tstatistic and the probability value will give you some insight. More specifically, finding the tstatistic with the pvalue will let you know if there is a significant difference between your mean and the population mean of everyone else.
Applying the previously stated tstatistic formula, you can obtain the following equation.
$t = \dfrac{15  10}{6 / \sqrt{36}} = 5$
Now, we know that the tstatistic equals 5
, but what does it mean? To gain more knowledge, you should compare this value with a particular threshold (or significance level), let's say 5 percent (α = 5%
) of a Studentt distribution. Since the sample size is relatively large (n > 30
) we can use the critical value of the standard normal distribution. The critical value of a 5%
threshold in a standard normal distribution is 1.645
. Since our tstatistic is above the critical value, we can say that you play better than the average.
🙋 In fact, we have just performed a Student's ttest! Visit our dedicated ttest calculator to learn more.
What is the difference between tscore vs. Zscore?
Both tscore and Zscore aim to make comparisons and decide on the dissimilarity between the sample and the population mean. The main difference between Tscore vs. Zscore comes from the knowledge about the population standard deviation. For Zscore, we assume it is given, while for Tscore you need to estimate it. In addition, Tscore can be applied when you have a small sample size (less than 30 elements).
How do I calculate tstatistic?
To calculate tstatistic:

Determine the sample mean (
x̄
, x bar), which is the arithmetic mean of your data set. 
Find the population mean (
μ
, mu). 
Compute the sample standard deviation (
s
) by taking the square root of the variance.To find the variance, if it is not given, take each value in the sample, subtract it from the sample mean, square the difference, sum them up, and divide by the sample size minus one.

Calculate the tstatistic as
(x̄  μ) / (s / √n)
, wheren
denotes the sample size.
What is the origin of Student's tdistribution?
The student ttest was devised by Gosset, who developed the connected statistical theory in 1908. At the time, Gosset worked at the Guinness Brewery in Dublin, which had an internal policy of forbidding employees from publishing to preclude potential loss of trade secrets. Gosset, however, found a loophole: he was writing under the pseudonym of ‘Student’. As a consequence, the statistical student t distribution became known as student t rather than Gosset's t. So, next time you enjoying a pint of Guinness with your friend, you have a compelling story to share.