Crit Value Calculator with N

Critical values are essential in statistical hypothesis testing. They help determine whether your sample data provides enough evidence to reject the null hypothesis. This calculator helps you find critical values for different distributions with n degrees of freedom.

What is a Critical Value?

A critical value is a threshold value from a statistical distribution that separates the region where the null hypothesis is rejected from the region where it is not rejected. In hypothesis testing, you compare your test statistic to the critical value to make a decision about the null hypothesis.

Critical values are often found in t-distribution tables, z-tables, or chi-square tables, depending on the type of test you're performing.

There are two types of critical values:

One-tailed test: The critical value is at one end of the distribution.
Two-tailed test: The critical values are at both ends of the distribution.

Critical values are typically denoted as zα, tα, or χ²α, where α is the significance level (usually 0.05 or 0.01).

How to Use This Calculator

Using this calculator is simple:

Select the distribution type (t, z, or chi-square).
Enter the degrees of freedom (n-1 for t-distribution).
Choose the significance level (α).
Select the test type (one-tailed or two-tailed).
Click "Calculate" to get the critical value.

Example Input

Distribution: t
Degrees of freedom: 10
Significance level: 0.05
Test type: Two-tailed

Formula

The formula for finding critical values depends on the distribution:

For t-distribution: tα = t_{n-1, α} For z-distribution: zα = Φ^-1(1-α) For chi-square distribution: χ²α = χ²_{n, α}

Where:

n = sample size
α = significance level
Φ^-1 = inverse of the standard normal cumulative distribution function

Worked Example

Let's find the critical value for a two-tailed t-test with n=11 and α=0.05.

Degrees of freedom = n-1 = 10
Look up t_{10, 0.025} in a t-distribution table (two-tailed test means we use α/2 = 0.025)
The critical value is approximately 2.228

Degrees of Freedom	Critical Value (α=0.05)
5	2.571
10	2.228
30	2.042

Interpreting Results

When you get a critical value:

If your test statistic is more extreme than the critical value, you reject the null hypothesis.
If your test statistic is less extreme, you fail to reject the null hypothesis.
For two-tailed tests, you compare the absolute value of your test statistic to the critical value.

Remember that failing to reject the null hypothesis doesn't mean the null hypothesis is true - it just means you don't have enough evidence to reject it.

Common Mistakes

Avoid these common errors when working with critical values:

Using the wrong degrees of freedom (should be n-1 for t-tests)
Confusing one-tailed and two-tailed tests
Using the wrong significance level (α)
Misinterpreting the results (remember, it's about evidence, not proof)

FAQ

What is the difference between a critical value and a p-value?

A critical value is a threshold from the distribution that you compare your test statistic to. A p-value is the probability of getting a result as extreme as yours if the null hypothesis is true. Both are used in hypothesis testing, but they work differently.

How do I know which distribution to use?

Use t-distribution for small samples (n < 30) with unknown population standard deviation. Use z-distribution for large samples (n ≥ 30) or when the population standard deviation is known. Use chi-square for goodness-of-fit tests.

What if my degrees of freedom aren't in the table?

For t-distribution, you can use interpolation between the closest available degrees of freedom. For chi-square, you can use the nearest available value or use statistical software for more precise values.