Free 95% Confidence Interval Calculator
Choose what you want to estimate, enter your sample values, and click calculate.
Tip: For proportion data, this calculator shows both the normal approximation and the Wilson interval.
What is a 95% confidence interval?
A 95% confidence interval gives a range of plausible values for a population parameter based on your sample. Instead of returning only one number (like a mean or proportion), it returns an interval: lower bound to upper bound.
The “95%” does not mean there is a 95% chance your specific interval contains the true value. It means that if you repeated the same sampling process many times and built intervals the same way, about 95% of those intervals would capture the true parameter.
When to use each method
1) Mean with unknown population standard deviation
Use a t-interval when you have sample mean, sample standard deviation, and sample size, but do not know the population standard deviation. This is the most common case in real-world data analysis.
2) Mean with known population standard deviation
Use a z-interval when population standard deviation (σ) is known from prior reliable data. This is less common in practice, but common in textbook examples and quality-control settings.
3) Proportion confidence interval
Use this when your outcome is binary (success/failure), such as conversion rate, pass/fail, or yes/no outcomes. Enter successes and total sample size.
Formulas used in this calculator
Mean (unknown σ, t-interval)
95% CI = x̄ ± t* × (s / √n), where t* is the critical value from the t distribution with df = n - 1.
Mean (known σ, z-interval)
95% CI = x̄ ± 1.96 × (σ / √n)
Proportion (normal approximation)
95% CI = p̂ ± 1.96 × √(p̂(1 − p̂)/n), where p̂ = x/n
Proportion (Wilson interval)
Wilson is often more stable for smaller samples or proportions near 0 or 1. This tool reports Wilson as a recommended interval in those cases.
How to interpret your result correctly
- A wider interval means more uncertainty.
- A larger sample size typically makes the interval narrower.
- Higher variability (larger standard deviation) makes intervals wider.
- Confidence intervals describe uncertainty in estimation, not causality.
Worked example
Suppose your sample mean test score is 78, sample standard deviation is 12, and sample size is 64. The standard error is 12/√64 = 1.5. With a 95% critical value near 2, margin of error is roughly 3. So your confidence interval is approximately 75 to 81.
Interpretation: your data are consistent with a population mean score in that range under your model assumptions.
Common mistakes to avoid
- Confusing confidence level with probability for a single computed interval.
- Using mean-based intervals for highly skewed data without checking assumptions.
- Using tiny samples and overconfident interpretation.
- Ignoring data quality issues (measurement error, sampling bias, missingness).
FAQ
Does 95% confidence mean 95% of values lie in the interval?
No. It refers to the long-run performance of the interval procedure, not the spread of individual data points.
Should I always use 95%?
95% is a common default. Some contexts use 90% (narrower interval) or 99% (wider interval) depending on risk tolerance.
What if my data are not normal?
For large samples, mean intervals are often robust. For small or heavily skewed datasets, consider transformations, bootstrap intervals, or nonparametric approaches.
Bottom line
A 95 confidence interval calculator is one of the fastest ways to quantify uncertainty around your estimate. Use it with sound sampling practice and clear interpretation, and you will make more informed decisions from data.