# Confidence Interval Using t Distribution Calculator

  

An online and easy to use calculator that calculates the confidence interval with a certain percentage, using the t distribution, is presented.
An online calculator that calculates the confidence interval using normal distribution calculator is included.

## Definition of Confidence Interval for the t Distribution

For a sample of size $n$ with standard deviation $s$, we define a $(1-\alpha)100\%$ confidence interval for $\mu$ as $\bar X \pm t_{\alpha/2} \dfrac{s}{\sqrt n}$ We say that we are $(1-\alpha)100\%$ confident that the mean $\mu$ of the population is within the interval $\left[\bar X - t_{\alpha/2} \dfrac{s}{\sqrt n} \quad , \quad \bar X + t_{\alpha/2} \dfrac{s}{\sqrt n} \right]$.
where $t_{\alpha/2}$ is the value of the t didtribution with $n - 1$ degrees of freedom such that the areas to the left and to the right are equal to $\alpha/2$ as shown in the graph below.
The graphical meaning of an interval of confidence is shown below.

The above definition is used when the standard deviation of the population $P$ is NOT known but the sample standard deviation $s$ is known and/or the sample size is not large $(n \lt 30)$.

## Confidence Interval Calculator

Enter the sample size $n$ as a positive integer, the sample mean $\bar X$, the sample standard deviation $s$ as a positive real number and the level of confidence (percentage) as a positive real number greater than $0$ and smaller than $100$.

Sample Size: $n$ =       Sample Mean: $\bar X$ =       Sample Standard Deviation: $s$ =       Confidence Level = $\%$
Decimal Places =