Critical Value Calculator

The critical value marks the cutoff for rejecting the null hypothesis.

How the Math Works

The z critical value is calculated using the inverse of the standard normal distribution. Given a confidence level (e.g., 95%), we first determine alpha by subtracting the confidence level from 100% (alpha = 1 - confidence level). For two-tailed tests, alpha is divided by 2, since the rejection region splits between both tails. The formula z = inverse_normal(1 - alpha/tails) finds the z-score that corresponds to the cumulative probability, marking the boundary beyond which we reject the null hypothesis in statistical testing.

Practical Applications

To apply this calculation, start by defining your desired confidence level (e.g., 90%, 95%, or 99%). Convert it to alpha by subtracting from 1 (e.g., 0.05 for 95% confidence). If performing a two-tailed test, divide alpha by 2. Use the inverse normal function (often labeled as 'qnorm' or 'invNorm' in calculators) to find the z-score that matches the cumulative probability (1 - alpha/2). This critical value is then used in hypothesis testing to compare against your test statistic or to construct confidence intervals by multiplying it with the standard error.

Day-to-Day Use

In everyday life, this concept helps interpret results from surveys, polls, or product quality assessments. For example, a political poll reporting a 95% confidence interval uses the critical z-value to show the margin of error around their results. Similarly, businesses might use it to ensure manufacturing processes meet specifications, knowing that 95% of products fall within a certain range. Understanding critical values also aids in evaluating claims like '90% of people prefer Brand X' by showing how reliable those results are likely to be.

Worked example

95% two-tailed gives z about 1.96.

FAQ