Visualization
The shaded regions represent the two tails with total area α. The vertical lines mark the critical values ±t*.
Paste sample data (or enter summary statistics) and click “Calculate”.
Estimation of a Population Mean σ Not Known the T Distribution
Statistics • Estimation of the Mean and Proportion
Frequently Asked Questions
How do you find a confidence interval for the mean when sigma is not known?
Use the t interval: xbar +/- t* x (s / sqrt(n)). The critical value t* depends on the confidence level and degrees of freedom df = n - 1.
What does the t critical value t* mean in this calculator?
t* is the cutoff on the t distribution that leaves total tail area alpha split evenly as alpha/2 in each tail. The calculator marks +/- t* on the curve and uses it to compute the margin of error.
Why do we use the t distribution instead of the normal distribution here?
When sigma is unknown, the sample standard deviation s is used, which introduces extra variability. The t distribution accounts for this, especially for smaller sample sizes.
How is the margin of error computed for a t confidence interval?
The margin of error is E = t* x (s / sqrt(n)). Increasing the confidence level increases t* and therefore increases E, while increasing n decreases E.
How should I paste my sample data into the tool?
Paste one value per line or paste CSV data using comma, semicolon, tab, or spaces as separators. If your pasted data has multiple columns, choose the correct column number so only that column is used.