# Confidence Interval Using Normal Distribution Calculator

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

An online confidence interval using t distribution calculator is included.

## Definition of Confidence Interval for the Normal Distribution

For a sample of size \( n \) from a population that has a standard deviation \( \sigma \), we define a \( (1-\alpha)100\% \) confidence interval for \( \mu \) as
\[ \bar X \pm Z_{\alpha/2} \dfrac{\sigma}{\sqrt n} \]
We say that we are \( (1-\alpha)100\% \) confident that the mean \( \mu \) of the population is within the interval \[ \left[\bar X - Z_{\alpha/2} \dfrac{\sigma}{\sqrt n} \quad , \quad \bar X + Z_{\alpha/2} \dfrac{\sigma}{\sqrt n} \right] \].

The graphical meaning of an interval of confidence is shown below.

Note that: \( \quad \text{Area}_1 + \text{Area}_2 + \text{Area}_3 = 1 \)

The above definition is used when the standard deviation \( \sigma \) of the population \( P \) is known and

1) either the population \( P \) is normally distributed

2) or the population \( P \) is NOT normally distributed but the sample size \( n \) is greater than \( 30 \).

## Confidence Interval Calculator

Enter the sample size \( n \ge 30 \) as a positive integer, the sample mean \( \bar X \), the population standard deviation \( \sigma \) 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: \( \bar X \) =
Population Standard Deviation: \( \sigma \) =
Confidence Level = \( \% \)

Decimal Places =

## Outputs