Critical Value Calculator Given Alpha and N

This calculator helps you find critical values for statistical tests when you know the significance level (alpha) and sample size (n). Critical values are essential for hypothesis testing in statistics.

What is a Critical Value?

A critical value is a threshold value from a statistical distribution that separates regions where the null hypothesis is accepted or rejected. In hypothesis testing, you compare your test statistic to the critical value to determine whether to reject the null hypothesis.

Critical values depend on:

The significance level (alpha) - typically 0.05 or 0.01
The sample size (n)
The type of test (one-tailed or two-tailed)
The distribution being used (usually t-distribution for small samples, normal distribution for large samples)

For large samples (n > 30), the normal distribution is often used. For small samples, the t-distribution is more appropriate.

How to Use This Calculator

Enter your significance level (alpha) - common values are 0.05 or 0.01
Enter your sample size (n)
Select whether you want a one-tailed or two-tailed test
Click "Calculate" to get the critical value

The calculator will display the critical value and show it on a chart for visualization.

Formula

The critical value depends on the distribution being used. For the t-distribution:

Critical value = t_{α/2, n-1} for two-tailed tests

Critical value = t_{α, n-1} for one-tailed tests

Where:

α = significance level
n = sample size
n-1 = degrees of freedom

For large samples (n > 30), the normal distribution is used instead of the t-distribution.

Worked Example

Let's find the critical value for a two-tailed test with α = 0.05 and n = 20.

Degrees of freedom = n - 1 = 19
For α = 0.05 and 19 degrees of freedom, the critical value from the t-distribution table is approximately ±2.093

This means you would reject the null hypothesis if your test statistic is less than -2.093 or greater than 2.093.

FAQ

What's the difference between a critical value and a p-value?

A critical value is a threshold from a distribution table, while a p-value is a probability calculated from your test statistic. Both are used to make decisions in hypothesis testing.

When should I use a one-tailed vs. two-tailed test?

Use a one-tailed test when you have a directional hypothesis (e.g., "greater than" or "less than"). Use a two-tailed test when you have a non-directional hypothesis.

What if my sample size is very large?

For large samples (n > 30), you can use the normal distribution instead of the t-distribution, as the t-distribution approaches the normal distribution.