If you don't know the population's standard deviation, **just leave it empty**

It is **not recommended** to exclude outliers unless you know the reasons

Shapiro Wilk test

Outliers count: ,based on the Tukey's fences method, k=1.5

**Information**

Calculates the **confidence interval** of the **mean** and the **standard deviation** using the **Normal distribution** or the **Student's t distribution** for the **mean** and the **Chi-Squared distribution** for the standard deviation.

When using the sample data we know the sample's statistics but we don't know the true value of the population's measures. Instead, we may treat the population's measures as random variables and calculate the confidence interval.

First, we need to define the **confidence level** which is the required certainty level that the true value will be in the **confidence interval**. Researchers commonly use a confidence level of **0.95**.**Mean confidence interval**

When we know the population's standard deviation (σ), we will use the normal distribution. The average's (X) distribution is Normal (Mean, SD/√n) Otherwise, we will use the sample size standard deviation with the t distribution with n-1 degrees of freedom. The (X-Mean)/(S/√n) distribution is T.**Standard deviation confidence interval**

(n-1)S^{2}/σ^{2} distributed Chi-squared with n-1 degrees of freedom.

n - sample size.

S - sample standard deviation.

SD - population standard deviation.

The following R code should produce the same results. unless you filled the population standard deviation as the R code use only the t distribution based on the sample standard deviation.

The sigma.test produce the confidence interval of the **variance** instead of the standard deviation