About

This tool calculates the Confidence Interval (CI) for a population mean, a crucial metric in inferential statistics. Unlike a simple point estimate (like the average), a CI provides a range of values that likely contains the true population parameter. Using this calculator mitigates the risk of drawing incorrect conclusions from limited sample data by quantifying uncertainty.

The calculator automatically selects between the Z-distribution (Normal) and the t-distribution (Student's) based on your sample size and standard deviation settings. It adheres to standard statistical formulas used in academic research and quality control (ISO 2602).

Formulas

The general formula for a confidence interval is:

CI = x ± CV × s√n

Where:

ME = CV × SE

Variables:

x = Sample Mean
s = Sample Standard Deviation
n = Sample Size
CV = Critical Value (z^* or t^*)
SE = Standard Error (s / √n)
ME = Margin of Error

Reference Data

Confidence Level	Alpha (α)	Tail Area (α/2)	Z-Score (Critical Value)
80%	0.20	0.10	1.282
85%	0.15	0.075	1.440
90%	0.10	0.05	1.645
95%	0.05	0.025	1.960
98%	0.02	0.01	2.326
99%	0.01	0.005	2.576
99.5%	0.005	0.0025	2.807
99.9%	0.001	0.0005	3.291

Frequently Asked Questions

It means that if you were to take 100 random samples from the same population and calculate a confidence interval for each, approximately 95 of those intervals would contain the true population mean. It does NOT mean there is a 95% chance the true mean is inside this specific interval.

This tool handles it automatically. Generally, use the Z-distribution if the population standard deviation is known or the sample size is large (n > 30). Use the T-distribution if the population standard deviation is unknown and the sample size is small (n < 30).

As the sample size (n) increases, the Standard Error decreases. This makes the confidence interval narrower (more precise) because the denominator in the margin of error formula becomes larger.

For large sample sizes (n > 30), the Central Limit Theorem suggests that the sampling distribution of the mean approximates a normal distribution, making this standard CI valid. For very small, skewed samples, other methods (like bootstrapping) might be safer.