# t-statistic Calculator

Created by Tibor Pál, PhD candidate
Reviewed by Anna Szczepanek, PhD and Jack Bowater
Last updated: Aug 25, 2023

Use the t-statistic calculator (t-value calculator or t test statistic calculator) to compute the t-value of a given dataset using its sample mean, population mean, standard deviation and sample size.

• What is the t-statistic?
• How do I calculate t-statistic?
• What is the difference between T-score vs. Z-score?

If you are also interested in F-test, check our F-statistic calculator.

## What is t-statistic and Student's t-test?

In statistics, the t-statistic, or t-value, is a measure that describes the relationship between a sample and its population. The t-statistic is central to the Student's t-test, which is a test for evaluating hypotheses about the population mean.

More precisely, the t-statistic is used to determine whether to support or reject the null hypothesis. It is used in conjunction with the p-value, or critical value, which indicates the probability that your results could have happened by chance. It is comparable to the z-statistic, with the difference being that the t-statistic is applied for small sample sizes or unknown population standard deviations.

## What is the t-statistic formula?

You need to use the following t-statistic formula to calculate the t-value:

$t = \frac{\bar x - \mu}{s/\sqrt n}$

Where:

• $\bar x$ - Sample mean;
• $\mu$ - Population mean;
• $n$ - Sample size; and
• $s$ - Standard deviation of the sample.

## How to use this t-statistic calculator?

To compute the t-statistic, 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 t-statistic, provided you input all other values.

## A t-statistic 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 t-statistic and the probability value will give you some insight. More specifically, finding the t-statistic with the p-value will let you know if there is a significant difference between your mean and the population mean of everyone else.

Applying the previously stated t-statistic formula, you can obtain the following equation.

$t = \dfrac{15 - 10}{6 / \sqrt{36}} = 5$

Now, we know that the t-statistic 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 Student-t 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 t-statistic is above the critical value, we can say that you play better than the average.

## FAQ

### What is the difference between t-score vs. Z-score?

Both t-score and Z-score aim to make comparisons and decide on the dissimilarity between the sample and the population mean. The main difference between T-score vs. Z-score comes from the knowledge about the population standard deviation. For Z-score, we assume it is given, while for T-score you need to estimate it. In addition, T-score can be applied when you have a small sample size (less than 30 elements).

### How do I calculate t-statistic?

To calculate t-statistic:

1. Determine the sample mean (x̄, x bar), which is the arithmetic mean of your data set.

2. Find the population mean (μ, mu).

3. 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.

4. Calculate the t-statistic as (x̄ - μ) / (s / √n), where n denotes the sample size.

### What is the origin of Student's t-distribution?

The student t-test 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.

Tibor Pál, PhD candidate
Sample mean (x̄)
Population mean (μ)
Sample size
Sample standard deviation (s)
t-statistic
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