P Value From Z Score Calculator

Calculator Form

Test Label

Z Score

Tail Type

Alpha Level

Decimal Places

Observed Value Optional

Mean Optional

Standard Deviation Optional

Example Data Table

Z Score	Left Tailed P	Right Tailed P	Two Tailed P	Common Meaning
-2.58	0.0049	0.9951	0.0098	Very low left side result
-1.96	0.0250	0.9750	0.0500	Common two tailed cutoff
0.00	0.5000	0.5000	1.0000	Exactly at the mean
1.64	0.9495	0.0505	0.1010	Near a one tailed cutoff
2.33	0.9901	0.0099	0.0198	Strong right side result

Formula Used

The calculator uses the standard normal cumulative distribution function.

CDF formula: Φ(z) = 0.5 × [1 + erf(z ÷ √2)]

Left tailed p value: p = Φ(z)

Right tailed p value: p = 1 - Φ(z)

Two tailed p value: p = 2 × min[Φ(z), 1 - Φ(z)]

Raw value z score: z = (x - μ) ÷ σ

How To Use This Calculator

Enter a z score directly, or leave it blank.
If z score is blank, enter observed value, mean, and standard deviation.
Select left tailed, right tailed, or two tailed testing.
Enter the alpha level used for your decision.
Choose decimal places for the displayed result.
Press the calculate button.
Review the p value, confidence level, and decision.
Use CSV or PDF export for saving your result.

Understanding P Values From Z Scores

What The Result Means

A z score shows how far a value sits from the mean. It uses standard deviation units. A p value converts that distance into probability. The probability depends on the selected tail. A left tailed test checks unusually low results. A right tailed test checks unusually high results. A two tailed test checks both extremes. This calculator uses the standard normal curve. It first finds the cumulative probability up to your z score. Then it adjusts that probability for the chosen test direction.

Why Tail Choice Matters

Tail choice should match the hypothesis before calculation. Do not choose it after seeing the answer. For example, a right tailed test fits claims about increases. A left tailed test fits claims about decreases. A two tailed test fits claims about any difference. The same z score can create different p values. That is why the form shows left, right, and two tailed values.

Using Alpha And Confidence

Alpha is your cutoff for statistical significance. Common alpha levels are 0.10, 0.05, and 0.01. A smaller alpha needs stronger evidence. The calculator compares the final p value with alpha. If the p value is less than or equal to alpha, the result is marked significant. Confidence is shown as one minus alpha. It helps readers understand the strictness of your decision.

Practical Notes

A small p value does not prove a claim. It only suggests the observed z score is unusual under the null model. A large p value does not prove no effect. It simply means the test found limited evidence. Always report the z score, tail type, alpha, p value, and decision together. This makes your work easier to review. Use enough decimal places for clarity. Very small values may appear in scientific notation. The export tools help save the same result for worksheets, reports, and audit trails. The example table also shows how different z scores change probability. This can help students learn the curve.

When Raw Data Is Used

You may enter a direct z score. You may also enter an observed value, mean, and standard deviation. The page then builds the z score before finding probability.

FAQs

What is a z score?

A z score shows how many standard deviations a value is from the mean. Positive values are above the mean. Negative values are below the mean.

What does a p value mean?

A p value shows how unusual the result is under the null hypothesis. Smaller p values usually provide stronger evidence against that null model.

When should I use a two tailed test?

Use a two tailed test when your hypothesis checks for any difference. It works for results that may be unusually high or unusually low.

When should I use a left tailed test?

Use a left tailed test when your hypothesis expects a lower value. It checks the probability of results at or below the z score.

When should I use a right tailed test?

Use a right tailed test when your hypothesis expects a higher value. It checks the probability of results at or above the z score.

What alpha level should I choose?

Many studies use 0.05. Some use 0.01 for stricter evidence. Choose alpha before testing, based on your course, field, or project rule.

Can I calculate z score from raw data?

Yes. Leave the z score blank. Then enter the observed value, mean, and standard deviation. The calculator will compute z first.

Why does the calculator show scientific notation?

Scientific notation helps display very small p values clearly. It prevents tiny results from appearing as zero after rounding.