P-value Calculator

This p-value calculator converts between Z-scores and p-values using the standard normal distribution. Enter a Z-score to get the corresponding p-value for a left-, right-, or two-tailed test, or enter a p-value to get the Z-score. Set the significance level (α) and see immediately whether to reject or fail to reject the null hypothesis. The graph shades the tail area that equals the p-value and shows a Logic Trace with the exact formula, so you can find p-value from z-score (or the reverse) and understand how it’s derived.

Toggle the input mode to choose whether you are entering a Z-score (to find the p-value) or a p-value (to find the corresponding Z-score). Type your value into the input field. Set the significance level α, 0.05 is the most common threshold, to define the boundary for rejecting the null hypothesis. Select a tail type: Left-tailed for testing whether a value is significantly below a reference, Right-tailed for significantly above, or Two-tailed for significantly different in either direction. The Center (0 to Z) option shows the area between 0 and z, matching how many textbook z-tables are organized, but it is not used for hypothesis-test decisions. The result card displays the computed value, a Reject or Fail to Reject decision badge, and the shaded tail area on the interactive normal curve. The Logic Trace shows the exact CDF formula and substitution.

In hypothesis testing, the null hypothesis (H₀) is the default (e.g. no effect, no difference). The p-value is the probability, if H₀ were true, of getting a result as extreme as (or more extreme than) what you observed. A small p-value suggests the data are unlikely under H₀, so you reject the null hypothesis. A large p-value means you do not have enough evidence to reject H₀. The calculator gives you the p-value from your Z-score (or the Z from a p-value) and a clear decision: reject when p ≤ α, fail to reject when p > α. Understanding what it means to reject the null hypothesis is essential for interpreting significance in tests and reports.

The significance level α (often 0.05) is the threshold you choose for rejecting the null hypothesis. If p ≤ α, you reject H₀; if p > α, you fail to reject. So α is the largest p-value you will treat as “significant.” Choosing α = 0.05 means you accept up to a 5% chance of a Type I error (rejecting a true null). This calculator lets you set α from 0.01 to 0.10 and shows whether your result is significant at that level, giving a practical view of p-value and significance level together.

The standard normal CDF, Φ(z), gives P(X ≤ z). To find p-value from z-score (or convert z-score to p-value) you use the tail you care about. Left-tailed: Right-tailed: Two-tailed: The two-tailed p-value formula doubles the area in one tail beyond |z|, so you get the probability of a result that extreme in either direction. This tool uses the error function to compute Φ(z) to high precision and shades the matching region on the curve. The Logic Trace shows the formula with your numbers, and you can hover over the curve to see Φ(z) or the center area (0 to z) in Center mode, matching how values are often presented in standard normal distribution tables.

The interactive curve shows the standard normal distribution. The shaded area is the probability (p-value or center area) you are computing. For a left-tailed test, the shaded region is to the left of your Z; for a right-tailed test, to the right; for a two-tailed test, both tails beyond ±|z|. Choosing the correct tail is important: it depends on your alternative hypothesis (e.g. “greater than” → right-tailed; “different from” → two-tailed). The red dashed line(s) mark your Z (and −|z| in two-tailed mode), and the decision badge reflects whether p ≤ α. Use this p-value calculator to check homework, verify significance for a test statistic, or explore how p-values change with Z and α, all with a clear, visual link between the number and the area under the curve.