Confidence Interval Calculator

Understanding Confidence Intervals

In statistics, a **confidence interval** is a type of estimate computed from the statistics of the observed data. This proposes a range of plausible values for an unknown parameter (for example, the average height of a population, or the true percentage of voters supporting a candidate).

A confidence interval is associated with a **confidence level** (e.g., 90%, 95%, or 99%). The confidence level represents the frequency (expressed as a percentage) with which the interval contains the parameter over repeated sampling.

Z-Interval vs. T-Interval

The decision of whether to calculate standard errors using Z-scores or Student's T-scores depends on whether the population variance is known:

• Z-Interval (σ known): Used when the population standard deviation ($\sigma$) is historically established or known.
• T-Interval (σ unknown): Used when the population standard deviation is unknown, which requires estimating it using the sample standard deviation ($s$). T-intervals adapt to smaller sample sizes by using degrees of freedom ($df = n-1$).

Wilson Score vs. Wald Intervals for Binomial Proportions

When analyzing survey or poll proportions, the standard method (the **Wald interval**) works by approximating a normal curve. However, this model suffers from severe bias when sample sizes are small or when the proportion is extremely close to 0% or 100%. The **Wilson Score interval** resolves this by centering around adjusted values, ensuring that confidence bounds stay realistically within 0% and 100% ranges.

Frequently Asked Questions