Normal Distribution, Z-Score and P-Value Calculator

Q: Does a p-value of exactly 0.05 reject H0?

By the p ≤ α rule, yes. In practice, the 0.05 threshold is a convention, and a p of 0.049 and one of 0.051 carry almost the same evidence. Report the p and the effect size, not only the decision.

Q: How do I convert a p-value into z?

Use the Inverse tab. For a two-tailed p of 0.05, ask for the two-tail option with 5%: z is ±1.96. For a one-tailed p of 0.01, ask for the right tail with 1%: z is 2.3263.

Q: What does it mean when a p-value shows as 2.5e-7?

It is scientific notation: 2.5 times 10 to the power of −7, or 0.00000025. It happens with distant statistics, such as z = 5, whose right tail is 2.9e-7. The tool shows it this way so it does not round everything to 0.000000.

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p value from z score

P-value from a z-score (and from t)

In the P-value tab, choose z or t, the test type, type the statistic (and the degrees of freedom for t) and α. The result brings the p, the decision and the shaded tail on the chart.

From z to the p-value, and from the p-value to the decision

The p-value is the probability, assuming the null hypothesis is true, of observing a statistic as extreme as the one obtained or more. In a right-tailed test, p = 1 − Φ(z); in a left-tailed test, p = Φ(z); in a two-tailed test, p = 2 · (1 − Φ(|z|)). For z = 1.96, the right tail is 0.0250 and the two-tailed p is 0.0500, which is why 1.96 is the classic 5% critical value. For z = 2.58 the two-tailed p is 0.0099, and for z = 3, 0.0027.
For a t test, the calculation is the same but uses the Student t tail, which depends on the degrees of freedom (df). The tail is heavier, so the same value yields a larger p: 2.5 gives a two-tailed p of 0.0124 for z, and 0.0245 for t with 15 df. With 5 df, t = 3 gives 0.0301, while z = 3 gives 0.0027. The more df, the closer t gets to z. The tool accepts fractional df, such as those of the Welch t test.
The decision compares p with the significance level α chosen before the test, usually 0.05: if p ≤ α, H0 is rejected. The critical value carries the same information on the statistic scale (±1.96 for a two-tailed α = 0.05). Cautions: p is not the probability that H0 is true, nor does it measure effect size; a p of 0.03 in a huge sample can be a practically irrelevant difference; and a one-tailed α of 0.05 corresponds to z = 1.6449, less demanding than the 1.96 of the two-tailed test, which is only justified if the direction was fixed before the data.

Checked examples

z = 1.96, two-tailed

Input: z = 1,96
Expected output: p = 0,0500

Rejects H0 at α = 0.05 (p = 0.04999).

z = 2.58, two-tailed

Input: z = 2,58
Expected output: p = 0,0099

Close to the 1% critical value.

z = 1.645, right-tailed

Input: z = 1,645
Expected output: p = 0,0500

One-tailed 5% critical value.

z = −2.326, left-tailed

Input: z = −2,326
Expected output: p = 0,0100

One-tailed 1% critical value.

t = 3 with 5 df, two-tailed

Input: t = 3, gl = 5
Expected output: p = 0,0301

Versus 0.0027 if it were z.

Full tool FAQ

How do you calculate a probability in the normal distribution?

Standardize the value with z = (x − μ) / σ and read the cumulative area Φ(z). For P(X < x) the answer is Φ(z); for P(X > x) it is 1 − Φ(z); for P(a < X < b) it is Φ(zb) − Φ(za). With μ = 100, σ = 15 and x = 115, z = 1 and P(X < 115) = 0.8413. In the calculator, pick the type, type the values and read the result with its step-by-step work.

What is a z-score and how do I interpret it?

How do I find the x value that matches a percentile?

What is the difference between a one-tailed and a two-tailed p-value?

What does the 68-95-99.7 empirical rule say?

How do I read the z table?

When should I use t instead of z for the p-value?

Why is P(X = x) zero?

How accurate are the calculations?

Can I use it for data that is not normal?

Frequently asked questions

Does a p-value of exactly 0.05 reject H0?

By the p ≤ α rule, yes. In practice, the 0.05 threshold is a convention, and a p of 0.049 and one of 0.051 carry almost the same evidence. Report the p and the effect size, not only the decision.

How do I convert a p-value into z?

Use the Inverse tab. For a two-tailed p of 0.05, ask for the two-tail option with 5%: z is ±1.96. For a one-tailed p of 0.01, ask for the right tail with 1%: z is 2.3263.

What does it mean when a p-value shows as 2.5e-7?

It is scientific notation: 2.5 times 10 to the power of −7, or 0.00000025. It happens with distant statistics, such as z = 5, whose right tail is 2.9e-7. The tool shows it this way so it does not round everything to 0.000000.

Next steps

Share of values within k standard deviations of the mean
Band	Range	Rule	Exact
μ ± 1σ	85 – 115	68%	68.2689%
μ ± 2σ	70 – 130	95%	95.45%
μ ± 3σ	55 – 145	99.7%	99.73%

Share of values within k standard deviations of the mean

Band

Range

Rule

Exact

μ ± 1σ

85 – 115

68%

68.2689%

μ ± 2σ

70 – 130

95%

95.45%

μ ± 3σ

55 – 145

99.7%

99.73%

From z to the p-value, and from the p-value to the decision

The p-value is the probability, assuming the null hypothesis is true, of observing a statistic as extreme as the one obtained or more. In a right-tailed test, p = 1 − Φ(z); in a left-tailed test, p = Φ(z); in a two-tailed test, p = 2 · (1 − Φ(|z|)). For z = 1.96, the right tail is 0.0250 and the two-tailed p is 0.0500, which is why 1.96 is the classic 5% critical value. For z = 2.58 the two-tailed p is 0.0099, and for z = 3, 0.0027.

For a t test, the calculation is the same but uses the Student t tail, which depends on the degrees of freedom (df). The tail is heavier, so the same value yields a larger p: 2.5 gives a two-tailed p of 0.0124 for z, and 0.0245 for t with 15 df. With 5 df, t = 3 gives 0.0301, while z = 3 gives 0.0027. The more df, the closer t gets to z. The tool accepts fractional df, such as those of the Welch t test.

The decision compares p with the significance level α chosen before the test, usually 0.05: if p ≤ α, H0 is rejected. The critical value carries the same information on the statistic scale (±1.96 for a two-tailed α = 0.05). Cautions: p is not the probability that H0 is true, nor does it measure effect size; a p of 0.03 in a huge sample can be a practically irrelevant difference; and a one-tailed α of 0.05 corresponds to z = 1.6449, less demanding than the 1.96 of the two-tailed test, which is only justified if the direction was fixed before the data.

Checked examples

z = 1.96, two-tailed

Input: z = 1,96
Expected output: p = 0,0500

Rejects H0 at α = 0.05 (p = 0.04999).

z = 2.58, two-tailed

Input: z = 2,58
Expected output: p = 0,0099

Close to the 1% critical value.

z = 1.645, right-tailed

Input: z = 1,645
Expected output: p = 0,0500

One-tailed 5% critical value.

z = −2.326, left-tailed

Input: z = −2,326
Expected output: p = 0,0100

One-tailed 1% critical value.

t = 3 with 5 df, two-tailed

Input: t = 3, gl = 5
Expected output: p = 0,0301

Versus 0.0027 if it were z.

Full tool FAQ

How do you calculate a probability in the normal distribution?

What is a z-score and how do I interpret it?

How do I find the x value that matches a percentile?

What is the difference between a one-tailed and a two-tailed p-value?

What does the 68-95-99.7 empirical rule say?

How do I read the z table?

When should I use t instead of z for the p-value?

Why is P(X = x) zero?

How accurate are the calculations?

Can I use it for data that is not normal?

Frequently asked questions

Does a p-value of exactly 0.05 reject H0?

By the p ≤ α rule, yes. In practice, the 0.05 threshold is a convention, and a p of 0.049 and one of 0.051 carry almost the same evidence. Report the p and the effect size, not only the decision.

How do I convert a p-value into z?

Use the Inverse tab. For a two-tailed p of 0.05, ask for the two-tail option with 5%: z is ±1.96. For a one-tailed p of 0.01, ask for the right tail with 1%: z is 2.3263.