1.825742
3.578454
7.1569
%
46.421546
53.578454
7.156908
1.825742
3.578454
7.1569
%
46.421546
53.578454
7.156908
The Confidence Interval Calculator computes the interval estimate around a sample mean that is likely to contain the true population mean at a specified confidence level. Confidence intervals are one of the most important tools in statistical inference, providing both a point estimate and a measure of uncertainty about that estimate.
Unlike a single point estimate, a confidence interval gives you a range of plausible values for the parameter of interest, along with a probability statement about how often such intervals would contain the true parameter if the study were repeated many times. This calculator supports 90%, 95%, and 99% confidence levels using z-critical values, making it appropriate for large samples or when the population standard deviation is known.
The confidence interval for a population mean is constructed using the following formulas:
Standard Error:
$$SE = \frac{\sigma}{\sqrt{n}}$$
where σ is the standard deviation and n is the sample size.
Margin of Error:
$$E = z_{\alpha/2} \cdot SE = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$
where $z_{\alpha/2}$ is the critical value from the standard normal distribution corresponding to the desired confidence level:
Confidence Interval:
$$CI = \bar{x} \pm E = \left(\bar{x} - z_{\alpha/2}\cdot\frac{\sigma}{\sqrt{n}},\;\bar{x} + z_{\alpha/2}\cdot\frac{\sigma}{\sqrt{n}}\right)$$
The width of the interval depends on three factors: the confidence level (higher → wider), the standard deviation (larger → wider), and the sample size (larger → narrower). This trade-off is fundamental to experimental design.
A 95% confidence interval means that if you were to repeat the sampling process many times and compute a confidence interval each time, approximately 95% of those intervals would contain the true population mean. It does not mean there is a 95% probability that the true mean lies within this particular interval.
The margin of error tells you how far the interval extends from the sample mean. A smaller margin of error indicates a more precise estimate. The CI width is simply twice the margin of error and represents the total span of the interval.
Inputs
Results
With a sample mean of 72.5, SD of 12, and n=50, the 95% CI is approximately (69.17, 75.83). We are 95% confident the true population mean lies within this range.
Inputs
Results
For systolic blood pressure with mean 120 mmHg, SD 15, n=100, the 99% CI is (116.14, 123.86). The wider interval reflects the higher confidence level.
A 95% CI means that if you repeated the sampling and interval construction process 100 times, about 95 of those intervals would contain the true population parameter. It is a statement about the procedure, not a probability statement about any single interval.
A higher confidence level requires you to cast a wider net to be more certain of capturing the true parameter. The z-critical value increases from 1.645 (90%) to 1.96 (95%) to 2.576 (99%), directly widening the margin of error.
Increasing the sample size reduces the standard error (SE = σ/√n), which narrows the confidence interval. Quadrupling the sample size halves the margin of error. This is why larger studies produce more precise estimates.
Use a z-interval when the population standard deviation is known or n > 30. Use a t-interval when σ is unknown and n is small. For large samples, the z and t intervals are virtually identical. This calculator uses z-critical values.
Yes. Two groups can have overlapping confidence intervals yet still show a statistically significant difference when tested directly. Confidence intervals for individual means are not the same as a confidence interval for the difference between means.
By the Central Limit Theorem, the sampling distribution of the mean is approximately normal for large samples (n ≥ 30), regardless of the population distribution. For small samples from non-normal populations, consider bootstrap confidence intervals or non-parametric methods.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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