2026-08-07 10 min

P-value Calculator

Convert z-scores to p-values (or invert p to z) on the standard normal distribution, with tail-shaded curve and significance verdict.

Calculator

From Z-score From P-value

Z-score Help

Standard deviations from the mean

Results update live as you type

Status:Two-tailed p = 0.049996Significant at α = 0.05

P-values for Z = 1.9600

P(x < Z) left0.975002

P(x > Z) right0.024998

P(0 to Z)0.475002

Two-tailed0.049996

Results

Critical

Two-Tailed p

0.049996

P(x<−Z or x>Z)

Left-Tail p

0.975002

P(x < Z)

Right-Tail p

0.024998

P(x > Z)

P(−Z < x < Z)

0.950004

Central mass

Significant at α=0.05?

Yes — reject H₀

Two-tailed vs 0.05

Standard-normal model with ~1e−7 CDF accuracy. Two-tailed is the default; p is continuous evidence — pair with effect size and intervals.

Visualizations

Tail Probabilities

1Left

0Right

0Two-tail

Tail vs Central Mass

Tails (p)5%

Center95%

Evidence Against H₀

1 − p (two-tail)95%

Two-Tailed p vs α = 0.05

p-value0

Formula

The CDF Φ accumulates the bell from −∞ to z; tails are its complements; doubling covers both directions. Inversion runs the Acklam rational approximation backwards.

p₂ = 2·min(Φ(z), 1 − Φ(z))

Alternative Form

z = Φ⁻¹(1−p) one-tailed · Φ⁻¹(1−p/2) two-tailed

Variables

Symbol	Name	Unit	Description
z	Z-score	σ	Standardized distance from H₀
Φ	Normal CDF	probability	½(1+erf(z/√2))
p	P-value	probability	Tail mass at least as extreme
α	Significance level	probability	Pre-set rejection line

Explanation

Further-out z leaves less tail mass: the curve area beyond ±z is exactly the two-tailed p. Small p = surprising under H₀.

Derivation

Φ via erf (A&S 7.1.26, ~1e−7); tails 1−Φ; double for two-sided; invert with Acklam central/tail rationals.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

State H₀ and pick α before seeing data (commonly 0.05).

Reject H₀ ⟺ p ≤ α

α:0.05

Tails:two

Intermediate: Direction decides one vs two tails

Result: Two-tailed test at α = 0.05

Step 2

Standardize to z = (estimate − H₀ value) / SE.

z ∼ N(0,1)

z:1.96

Intermediate: 68-95-99.7 anchors: 1, 1.96, 2.58

Result: Working z = 1.96

Step 3

Map z through the standard normal CDF.

Φ(z) = ½(1 + erf(z/√2))

P(x<Z):0.975002

P(x>Z):0.024998

Intermediate: Abramowitz–Stegun erf approximation

Result: Tails split at 0.975002 / 0.024998

Step 4

Double the outer tail for the two-sided question.

p₂ = 2·min(Φ, 1−Φ)

Two-tailed p:0.049996

Intermediate: One-tailed is half of two-tailed

Result: p = 0.049996

Step 5

Compare against α and state the verdict in words.

p ≤ 0.05 → reject H₀

Verdict:Reject H₀

Intermediate: Non-rejection never proves H₀

Result: Statistically significant

Step 6

Read benchmarks: 1.96/2.58/3.29 ↔ .05/.01/.001.

Acklam inverse normal

Critical z:422.3070

Intermediate: Exact p preserves more information than bands

Result: p = 0.049996

Final Answer

0.049996

Worked Examples

See how the calculator works with different scenarios.

Scenario

Textbook 95% boundary.

Inputs

z:1.96

Calculation

Φ(1.96) ≈ 0.975 → right tail 0.025 → double

Result

Two-tailed p ≈ 0.05

Interpretation

Exactly the α = 0.05 line: borderline significance.

Understanding Your Result

Your Result: Two-Tailed p: 0.049996

Typical range: p < 0.001 very strong · < 0.01 strong · < 0.05 standard · ≥ 0.05 insufficient.

What This Means

p is compatibility of data with H₀, not P(H₀ true). 0.05 = 1-in-20 surprise level; 0.001 = 1-in-1000.

Why It Matters

Publication, regulation and product calls hinge on this number — misread it and noise ships as discovery.

Professional Interpretation

Pair exact p with effect size and CI; correct for multiplicity; prefer two-tailed unless pre-registered otherwise.

Recommendations

•Fix α before collecting data
•Report exact p, not bands
•Add effect size + CI always
•Correct multiple tests
•Default to two-tailed

Educational Content

Introduction

NHST asks: if nothing were happening, how surprising would this result be? The p-value is that surprise meter on the normal curve.

Concept Explanation

Standardize → accumulate tails via Φ → double for two-sided → compare to α. Inversion reverses the accumulation to recover critical z.

Applications

▸Clinical and A/B testing

▸Regression coefficient tests

▸ANOVA and chi-square follow-ups

▸GWAS and physics thresholds

▸QC control-chart signals

▸Meta-analytic pooling

Benefits

•Universal evidence dialect
•Exact continuous reading
•Bidirectional z↔p travel
•Benchmark-anchored intuition
•Pairs with CIs naturally

Limitations

•Not P(H₀ true)
•Sample-size gamed easily
•Hides effect magnitude
•Forking-paths fragility
•Dichotomania at 0.05

Best Practices

•Preregister tails and α
•Show CIs beside p
•Adjust for many looks
•Replicate before claiming
•Treat p as continuous evidence

Common Misconceptions

Myth: p is the chance H₀ is true.

Truth: p assumes H₀ to compute data surprise; posteriors need Bayes.

Myth: p > 0.05 proves no effect.

Truth: It may be underpowered noise — absence of evidence ≠ evidence of absence.

Myth: Smaller p always means bigger effect.

Truth: Huge n makes trivia "significant" — check effect size.

Myth: One-tailed doubles your power fairly.

Truth: Only if direction was fixed in advance; otherwise it is hacking.

Practical Tips

•Set α and tails before peeking
•Report exact p-values
•Pair every p with effect + CI
•Correct for multiple comparisons
•Recheck outliers first
•Prefer two-tailed by default
•Replicate surprising hits

Common Mistakes

Reading p as P(H₀ true)

Translate as: surprise of data given H₀ — use Bayes for hypothesis odds.

P-hacking to 0.049

Preregister analyses; report all looks with corrections.

Ignoring multiplicity

Apply Bonferroni/Holm/FDR when testing many hypotheses.

Confusing significance with importance

Require meaningful effect size alongside small p.

Hiding exact values as stars

Print p = 0.032 style exactness for re-analysis.

Assumptions

This calculator makes the following assumptions to provide estimates:

Normal Reference: Standard normal N(0,1) sampling distribution.

Approximation Error: erf/inverse accurate to ~1e−7.

Pre-set α: Threshold chosen before data.

Single Look: No uncorrected peeking or multiplicity.

Local Compute: All CDF math runs in-browser.

Limitations

Be aware of these limitations when using this calculator:

Z-Only Model: No t/df, chi-square or F distributions.

No Effect Sizes: Magnitude must be computed separately.

Approximation: Rational fits, not arbitrary precision.

No Sequential Monitoring: Interim-look spending not modeled.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a p-value?

The probability of a test statistic at least as extreme as observed, assuming the null is true. Small p (typically ≤ 0.05) signals incompatibility with the null — not the probability the null is true.

How do you calculate p-value from z-score?

Via the standard normal CDF Φ: left-tail Φ(z), right-tail 1−Φ(z), two-tailed 2·min(Φ(z), 1−Φ(z)). This calculator uses the Abramowitz–Stegun erf approximation.

What does a p-value of 0.05 mean?

A 5% chance of so-extreme data under the null — the conventional significance line. p ≤ 0.05 rejects H₀ at α = 0.05.

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

One-tailed tests one direction; two-tailed tests either (exactly double for symmetric laws). Two-tailed is the conservative default.

What is the relationship between z-score and p-value?

Inverse: bigger |z| → smaller p. |z| = 1.96 ↔ p ≈ 0.05, 2.58 ↔ 0.01, 3.29 ↔ 0.001 (two-tailed).

What is the null hypothesis and significance level?

H₀ claims no effect; α (often 0.05, 0.01, 0.10) is the pre-set rejection line. Reject H₀ when p ≤ α; otherwise fail to reject (never "accept").

What are Type I and Type II errors?

Type I = false positive (reject true H₀; rate α). Type II = false negative (miss real effect; rate β). Power = 1−β.

How do you convert a p-value back to a z-score?

Inverse normal: one-tailed z = Φ⁻¹(1−p); two-tailed z = Φ⁻¹(1−p/2). This calculator inverts via the Acklam approximation.

Can a p-value exceed 1 or prove H₀?

No to both: p ∈ [0,1], and p > α is absence of evidence, not evidence of absence — possibly just low power.

What is p-hacking and multiple testing?

Trying analyses until p < 0.05 inflates false positives; multiple tests need Bonferroni/Holm/FDR correction. Preregistration and exact-p reporting mitigate this.

Statistical vs practical significance?

Huge samples make trivial effects "significant". Always pair p with effect size (Cohen’s d, r, odds ratio) and confidence intervals.

How do you report a p-value?

Exact values (p = 0.032) or bounded small ones (p < 0.001), with test statistic, df and effect size — never bare "NS".

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Reading p without fooling yourself

ASA warnings in plain language.

Effect sizes that matter

Beyond stars and thresholds.

Multiple-testing survival

Bonferroni to FDR in practice.

Glossary

Technical terms used in mortgage calculations explained in plain language.

P-value: Tail probability at least as extreme under H₀.
Z-score: Sigmas from the null mean.
Null hypothesis: No-effect baseline being tested.
Alpha: Pre-set rejection threshold.
One-tailed: Directional single-side test.
Two-tailed: Either-direction test (2× one-tail).
Critical value: z boundary at chosen α (±1.96).
Type I / II error: False positive / false negative.
Power: 1−β: chance of catching real effects.
P-hacking: Gaming analyses to breach 0.05.

Authoritative References

Trusted sources and standards referenced by this calculator.

Wikipedia — P-value Wikipedia — Statistical hypothesis testing Wikipedia — Normal distribution Khan Academy — Significance tests Stat Trek — P-values Scribbr — P-values

Applicable Standards

ASA Statement on Statistical Significance and P-Values (2016)APA reporting standards for exact p-values

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Page Information

Last Updated:

2026-08-07

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.