Confidence interval for a single sample (unknown variance)

Quick Guide: Confidence interval for a single sample - unknown variance

Spread The Word

Try the Live Calculator ⮕

The Confidence Interval Formula

When the population standard deviation is unknown, we use the following formula to calculate the interval for the mean:

Key Components for Calculation

• Sample Mean (x̄): The average value of the observations in your sample.
• Critical Value (t*): A multiplier found using a t-distribution table based on your confidence level (e.g., 95%) and degrees of freedom (df = n - 1).
• Sample Standard Deviation (s): The measure of variation calculated from your sample data to estimate the population's dispersion.
• Sample Size (n): The total number of observations in the sample.
• Standard Error (s / √n): The estimated standard deviation of the sample mean's distribution.

Why Use the t-Distribution?

We use the t-distribution instead of the Z-distribution because estimating the variance from a sample introduces extra uncertainty. The t-distribution has "heavier tails," meaning it accounts for this uncertainty by providing a wider interval. As the sample size (n) increases, the t-distribution becomes nearly identical to the standard normal distribution.