Statistics
Distribution, Sampling & Inference
About Statistics Calculators
Perform professional statistical analysis with our suite of calculators covering probability distributions, hypothesis testing, confidence intervals, and sample size determination. Whether you are a student learning inferential statistics, a researcher designing experiments, or a data analyst verifying results, our tools implement standard statistical methods with clear formula references and interpretation guidance.
- ✓Normal distribution calculator with z-scores, percentiles, and area-under-curve computation.
- ✓Confidence interval construction for means, proportions, and differences.
- ✓Sample size determination for achieving target statistical power and margin of error.
- ✓Binomial and probability calculators for discrete event analysis.
Distribution, Sampling & Inference
Statistics — Frequently Asked Questions
What is a confidence interval and how do I interpret it?
A 95% confidence interval means that if you repeated the same experiment 100 times, approximately 95 of the computed intervals would contain the true population parameter. It does NOT mean there is a 95% probability that the true value lies within your specific interval. Wider intervals indicate more uncertainty; narrower intervals indicate more precision. Our calculator computes CI using z-distribution (large samples) or t-distribution (small samples).
How do I determine the right sample size for my study?
Sample size depends on four factors: (1) desired confidence level (typically 95% or 99%), (2) acceptable margin of error, (3) expected population proportion or variability, and (4) statistical power (typically 80%). Larger samples reduce margin of error but increase cost. Our sample size calculator finds the minimum n needed to achieve your target precision.
What is the difference between a z-score and a t-score?
Z-scores use the standard normal distribution and are appropriate when the population standard deviation is known or the sample size is large (n > 30). T-scores use the Student's t-distribution with n−1 degrees of freedom and are used when the population standard deviation is unknown and estimated from the sample. As sample size increases, the t-distribution converges to the normal distribution.
When should I use a binomial vs. normal distribution?
Use the binomial distribution for discrete events with two outcomes (success/failure) over a fixed number of trials — like coin flips, defect rates, or survey responses. Use the normal distribution for continuous measurements (height, test scores, measurement errors). When n is large and p is not near 0 or 1, the normal distribution approximates the binomial (Central Limit Theorem): np > 5 and n(1−p) > 5.
What does a p-value actually mean?
A p-value is the probability of observing results as extreme as (or more extreme than) your data, assuming the null hypothesis is true. A p-value of 0.03 means there is a 3% chance of seeing your results if the null hypothesis were correct. It does NOT mean there is a 97% chance your alternative hypothesis is true. The conventional significance threshold of 0.05 is arbitrary and increasingly debated in the scientific community.