Calculadora de Intervalo de Confiança (média, σ conhecido)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
IC = x̄ ± z · σ / √n
About this calculator
This confidence interval calculator estimates the population mean when the population standard deviation (σ) is known. It uses the formula CI = x̄ ± z * σ / √n, where x̄ is the sample mean, z is the critical value from the normal distribution (based on the chosen confidence level), σ is the population standard deviation, and n is the sample size. The result provides an interval that, with a given confidence level, contains the true population mean.
The operation is simple: enter the sample mean, population standard deviation, sample size, and select the desired confidence level (90%, 95%, or 99%). The calculator automatically finds the corresponding z value (1.645, 1.96, or 2.576) and computes the margin of error. The interval is displayed as [lower limit, upper limit]. For levels not listed, consult a z-table or use another tool.
When to use? This calculator is ideal in situations where the population standard deviation is known, such as in controlled industrial processes or studies with historical data. Examples: quality control to check if a batch mean meets specifications, scientific research with large samples (n > 30) and known population variance, or estimating population parameters in well-established experiments.
Important precautions: the formula is valid only when σ is known and the sample is random and independent. If σ is unknown, use the Student's t-distribution. For small samples (n < 30), data normality should be verified. Correct interpretation: the confidence interval is not a probability that the mean lies within the interval; it is the frequency with which the method captures the population mean in repeated sampling.
Frequently asked questions
What if the population standard deviation is unknown?
If σ is unknown, use the Student's t-distribution with the confidence interval calculator for mean with unknown σ. In this case, use the sample standard deviation (s) and the critical t value.
What is the difference between a 95% and 99% confidence interval?
A 99% interval is wider, providing higher confidence that it contains the population mean, but with less precision. A 95% interval is narrower, but has a 5% chance of not containing the mean.
Can I use this calculator for small samples (n < 30)?
Yes, provided the population is normally distributed and σ is known. Otherwise, it is recommended to use the t-distribution, even with small n.
Does the confidence interval mean there is a 95% chance the population mean is inside the interval?
No. The 95% confidence level means that if we repeated the sampling process many times, 95% of the calculated intervals would contain the population mean. The specific interval either contains it or does not.
How do I choose the confidence level?
It depends on the acceptable risk. In scientific research, 95% is common. In critical areas like medicine, 99% may be used for higher reliability. Higher levels produce wider intervals.