Statistics

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.

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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.