t-statistic calculator

This t-statistic calculator computes the t-statistic value based on user-provided inputs for sample mean, population mean, sample size, and sample standard deviation. The also calculator allows you to enter any four of these five values. Once four inputs are provided, it automatically calculates and displays the fifth missing value.

Please enter at least four values to compute the result.

Sample mean:

Population mean:

Sample size:

Sample standard deviation (%):

t-statistic:

Related calculators:

One-sample t-test Two-sample t-test P-value calculator Confidence interval

What is the t-statistic?

The t-statistic is a ratio that represents the difference between the sample mean and the population mean in units of standard error. It shows how many standard errors the sample mean is away from the population mean, which helps to decide if the difference between them is statistically significant.

Formula for the t-statistic

The formula for calculating the t-statistic depends on the context, but a common form for a one-sample t-test is:

t= \frac{\bar{X} - \mu}{\frac{s}{\sqrt{n}}}

Where:

$\bar{X}$ = Sample mean
$\mu$ = Population mean (or hypothesized mean in the test)
$s$ = Sample standard deviation
$n$ = Sample size

The denominator, $\frac{s}{\sqrt{n}}$ , is called the standard error of the sample mean. It represents the standard deviation of the sampling distribution of the mean.

Interpretation of the t-statistic

The value of the t-statistic indicates how far the sample mean is from the population mean in terms of standard error:

Large t-statistic: A large absolute t-value (either positive or negative) indicates a greater difference between the sample and population mean, suggesting it is unlikely to have occurred by random chance.
Small t-statistic: A t-value close to zero implies the sample mean is similar to the population mean, suggesting no significant difference.

To determine statistical significance, the t-statistic is compared to a critical value from the t-distribution, which depends on the desired significance level (e.g., 0.05) and degrees of freedom ( $n-1$ for a one-sample test). If the absolute value of the t-statistic exceeds the critical value, the difference is considered statistically significant.

Calculation and use cases

The t-statistic is calculated by:

Taking the difference between the sample mean and population mean.
Dividing this difference by the standard error (sample standard deviation divided by the square root of sample size).

The t-statistic is used in various types of t-tests:

One-sample t-test: To test if a sample mean significantly differs from a population mean.
Independent two-sample t-test: To compare the means of two independent groups.
Paired t-test: To compare means of two related groups (e.g., measurements before and after a treatment on the same subjects).

In general, the t-statistic is widely applied in fields like research, psychology, business, and medicine, where hypothesis testing is critical to making data-driven conclusions.