Probability Distribution Calculator

Calculates p-values and critical values for the normal, t, chi-square and F distributions and shades the region on the curve. It replaces a printed z table or t table.

How to use

1. Under "Distribution", choose Normal, t, Chi-square or F, and enter the mean and standard deviation or the degrees of freedom.
2. Under "Direction", choose "Value → probability" or "Probability → value".
3. Under "Region", choose a tail, "Two tails" or "Between two values" ("Central area" in probability → value mode), then enter the value or probability.
4. Read the result and the shaded curve. "Common critical values" and "z-score ↔ raw score" are below.

How it works

• z-score = (x − μ) ÷ σ. A normal probability is the same as the standard normal probability of this z-score.
• Chi-square and F distributions are not symmetric, so the two-tailed probability is taken as twice the smaller tail.
• "Common critical values" shows α = 0.10, 0.05 and 0.01 for the distribution and degrees of freedom selected, with α/2 in each end for two tails. Results can be shown with 4 to 6 decimal places.

Questions and answers

What is the probability for z = 1.96?

Under the standard normal distribution, the left tail P(Z ≤ 1.96) is about 0.9750 and the two tails together are about 0.05, so 1.96 is the two-tailed critical value at the 5% level.

How do I find a t critical value?

Choose t, enter the degrees of freedom, select "Probability → value" and "Two tails", and enter 0.05. With 10 degrees of freedom the result is about ±2.228.

How do I convert a raw score to a z-score?

Enter the mean, standard deviation and raw score in "z-score ↔ raw score". With a mean of 100 and a standard deviation of 15, a score of 130 is z = 2.

Can I get a p-value from a test statistic?

Choose the distribution and degrees of freedom for your test, select "Value → probability", enter the statistic, and pick "Two tails" or the matching single tail.