# 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 =

## Outputs