P Value Calculator: From z, t, Chi-Square, F & r (Free)

Quick answer: A p-value is the probability of getting a result at least as extreme as yours if the null hypothesis were true. Small p = your data would be surprising under "no effect". Convention: p < 0.05 is statistically significant. Key landmarks: z = 1.96 → p = 0.05, z = 2.576 → p = 0.01, z = 3.29 → p = 0.001. A one-tailed p-value is exactly half the two-tailed one.

🔑 Key Takeaways

1p = P(data this extreme | null is true) — not the probability the null is true, and not the probability your finding is a fluke. That reversal is the single most common error in statistics.
2p < 0.05 is a convention, not a law of nature. Fisher picked it as a rough rule; physics demands ~0.0000003, and many fields now argue for 0.005.
3The p-value says nothing about effect size. With n = 10,000 a meaningless difference gets p < .001. Always report an effect size (Cohen's d, η², r) beside it.
4One-tailed p = two-tailed p ÷ 2. Only use one-tailed if you predicted the direction before collecting data — switching afterwards is p-hacking.
5Chi-square and F tests are naturally one-tailed (upper tail only), because only large values indicate departure from the null.
6"Not significant" does not mean "no effect." It means insufficient evidence — often just a small sample. Check the confidence interval before concluding anything.
7Report exact p-values (p = .032, not "p < .05"), with no leading zero, and use p < .001 only for very small values — that is the APA 7 standard.

📋 On This Page

Calculator What is a p-value How to use P-value in SPSS, R, Excel, Python P-value table Formulas Interpretation APA reporting Worked examples Cheat sheet FAQ References

🧮 Calculate Your P-Value

z-score From a z-test or standardized value

t-score

Degrees of freedom (df) n − 1 for one sample; n₁+n₂−2 for two groups

Chi-square (χ²)

Degrees of freedom (df) (rows−1)(cols−1) for contingency tables

Chi-square tests use the upper tail only — the tail setting below is ignored.

F value

df between (df₁)

df within (df₂)

F tests use the upper tail only — the tail setting below is ignored.

Correlation r

Sample size (n)

Converted to t = r√(n−2) / √(1−r²) with df = n − 2.

Choose your test Compares the means of two independent groups assuming equal variances.

Comma-separated input is the default. Group names are editable. Paired tests and correlation require both columns to have the same number of values, entered in matching order.

Tail: Significance level (α):

📊 Results

Full Results Table

Visualizations

Distribution with P-Value Shaded

Your Statistic vs Critical Value

P-Value on the Significance Scale

How P Changes with the Statistic

❓ What Is the P-Value?

The p-value is the probability of seeing a result at least as extreme as the one you got, assuming the null hypothesis is true. It is a measure of surprise. A small p-value means your data would be unusual in a world where nothing is going on, so the "nothing is going on" story starts to look like a poor explanation.

Written formally, for an observed test statistic T_obs:

p = P( |T| ≥ |T_obs| | H₀ true )

The idea in plain English

Suppose a coin is fair (that is the null hypothesis). You flip it 20 times and get 18 heads. The p-value answers one question only: if the coin really were fair, how often would I see something this lopsided or worse? The answer is about 0.0004. That does not prove the coin is loaded. It says that "fair coin" is a very awkward explanation for what you just saw.

The two hypotheses behind every p-value

A p-value only makes sense once you have written down two competing statements about the world. You always test the first one and, if the data are surprising enough, you side with the second.

Null hypothesis (H₀) — "nothing is happening"

The boring, default statement: no difference, no effect, no relationship. It is the assumption the maths runs on. You never prove H₀ true; you either reject it or fail to reject it. It always contains an equals sign.

Medicine: The new drug and the placebo lower blood pressure by the same amount. (μ_drug = μ_placebo)
Ecology: Tiger density is the same inside and outside the protected reserve. (μ_inside = μ_outside)
Education: There is no correlation between hours studied and exam score. (ρ = 0)

Alternative hypothesis (H₁) — "something is happening"

The claim you are actually interested in: there is a difference, effect, or relationship. It is what you accept when the p-value falls below α. Write it as two-sided (≠) unless you predicted the direction before collecting data, in which case it is one-sided (> or <).

Medicine: The drug and the placebo differ in how much they lower blood pressure. (μ_drug ≠ μ_placebo) — one-sided version: the drug lowers it more.
Ecology: Tiger density differs between reserve and non-reserve areas. (μ_inside ≠ μ_outside) — one-sided version: density is higher inside.
Education: Hours studied and exam score are correlated. (ρ ≠ 0) — one-sided version: the correlation is positive.

	Null hypothesis (H₀)	Alternative hypothesis (H₁)
Says	No effect, no difference	There is an effect or difference
Symbol	=	≠, >, or <
Role	Assumed true while computing p	What you argue for if p ≤ α
If p ≤ α	Reject H₀	Supported by the data
If p > α	Fail to reject H₀ (not "accept")	Not supported — this time

Quick rule: H₀ is the statement you try to knock down, H₁ is the statement you hope survives. The p-value measures only how awkward your data look for H₀.

How the p-value is produced

State the null hypothesis (H₀). Usually "no difference", "no effect", or "no association".
Compute a test statistic. A z, t, chi-square, F, or r that measures how far your data sit from what H₀ predicts, in units of noise.
Place that statistic on its null distribution. This is the curve you would get from repeated sampling if H₀ were true.
Measure the tail area beyond it. That shaded area is the p-value. Two-tailed tests count both tails; one-tailed tests count one.

What the p-value is NOT

Common claim	Why it is wrong
"p = .03 means there is a 3% chance the null is true."	Reversed conditional. p is P(data \| H₀), not P(H₀ \| data). Getting the posterior needs Bayes and a prior.
"p = .03 means a 97% chance the effect is real."	Same reversal. The p-value says nothing about the probability of your alternative hypothesis.
"p < .05 means the effect is large or important."	Size and significance are different things. Large samples turn trivial differences into tiny p-values.
"p = .20 proves there is no effect."	Absence of evidence is not evidence of absence. Usually it means low power or a small sample.
"p = .049 and p = .051 are different findings."	The 0.05 line is a convention, not a cliff. Treat p as a continuous measure of evidence.
"p is the chance of a false positive in my study."	The long-run Type I error rate is α, set in advance. p is a property of the data you actually collected.

Reading the value you get

P-value	Evidence against H₀	Typical wording
p > 0.10	None worth mentioning	Not statistically significant
0.05 < p ≤ 0.10	Weak, suggestive only	Marginal; do not overclaim
0.01 < p ≤ 0.05	Moderate	Statistically significant
0.001 < p ≤ 0.01	Strong	Highly significant
p ≤ 0.001	Very strong	Report as p < .001

Where alpha fits in

α (alpha) is the threshold you commit to before looking at the data, the false positive rate you are willing to tolerate. The p-value is what the data give you after. The decision rule is simply: reject H₀ when p ≤ α. Choosing α after seeing p is p-hacking, and it destroys the guarantee that makes the whole procedure work.

Always report it with company

A p-value on its own is a thin result. Pair it with an effect size (Cohen's d, η², r, odds ratio) and a confidence interval. The p-value tells you whether an effect is distinguishable from noise; the effect size tells you whether it matters; the interval tells you how precisely you have pinned it down. APA 7 asks for all three, and reviewers increasingly insist on it.

🛠 How to Calculate a P-Value with This Tool

Pick your test statistic. z-score, t-score, chi-square, F, correlation r, or paste raw data for an automatic t-test — or load one of the ten sample scenarios.
Enter the value and degrees of freedom. z needs no df; t needs df; chi-square needs df; F needs both df₁ and df₂; r needs the sample size.
Choose the tail. Two-tailed is the default and correct choice unless you predicted the direction in advance. Chi-square and F ignore this — they are upper-tail tests by nature.
Set α. 0.05 is conventional; the tool compares your p-value against it and states the decision explicitly.
Read, check, report. Read the exact p-value and verdict, confirm the shaded tail area on the curve, then copy an APA sentence — remembering to add an effect size.

💻 How to Find the P-Value in SPSS, R, Excel, Python and JASP

Most people arrive at a p-value through software, and every package labels it differently. Here is where to look, and the exact command if you only have a test statistic.

SPSS

The p-value is the column headed Sig. or Sig. (2-tailed). For a one-tailed test, halve it. A displayed value of .000 never means zero — report it as p < .001.

R

Every test object prints p-value directly (t.test(), chisq.test(), aov(), cor.test()). From a statistic alone:

2 * pnorm(-abs(z))
2 * pt(-abs(t), df)
pchisq(x, df, lower.tail = FALSE)
pf(F, df1, df2, lower.tail = FALSE)

Excel / Google Sheets

Use the distribution functions, or the built-in test functions for raw data:

=2*(1-NORM.S.DIST(ABS(z),TRUE))
=T.DIST.2T(ABS(t),df)
=CHISQ.DIST.RT(x,df)
=F.DIST.RT(F,df1,df2)
=T.TEST(range1,range2,tails,type)

Python (SciPy)

from scipy import stats
2 * stats.norm.sf(abs(z))
2 * stats.t.sf(abs(t), df)
stats.chi2.sf(x, df)
stats.f.sf(F, df1, df2)

JASP, Jamovi, Minitab, Stata

JASP and Jamovi print the column simply as p. Minitab uses P-Value. Stata uses Prob > |t|, Prob > chi2, or Prob > F — all are the same quantity, the tail area beyond your statistic.

Software	What the p-value is called	Default tail
SPSS	Sig. / Sig. (2-tailed)	Two-tailed
R	p-value	Two-tailed
Excel	P(T<=t) two-tail	Both offered
Python (SciPy)	pvalue	Two-tailed
Stata	Prob > \|t\| / chi2 / F	Two-tailed (chi2 and F upper)
JASP / Jamovi	p	Two-tailed
Minitab	P-Value	Two-tailed

📑 P-Value Table: Critical Values & Landmarks

These are the critical values people look up most. A statistic larger than the critical value means p is smaller than that α.

Standard normal (z) — no degrees of freedom needed

α (two-tailed)	Critical z	α (one-tailed)	Critical z
0.10	1.645	0.10	1.282
0.05	1.960	0.05	1.645
0.01	2.576	0.01	2.326
0.001	3.291	0.001	3.090

Student's t — critical values by degrees of freedom (two-tailed)

df	α = 0.10	α = 0.05	α = 0.01	α = 0.001
5	2.015	2.571	4.032	6.869
10	1.812	2.228	3.169	4.587
20	1.725	2.086	2.845	3.850
30	1.697	2.042	2.750	3.646
60	1.671	2.000	2.660	3.460
∞ (= z)	1.645	1.960	2.576	3.291

Chi-square — critical values (upper tail)

df	α = 0.10	α = 0.05	α = 0.01
1	2.706	3.841	6.635
2	4.605	5.991	9.210
3	6.251	7.815	11.345
4	7.779	9.488	13.277
5	9.236	11.070	15.086

ƒ P-Value Formulas Used

①

Definition of the P-Value

p = P( test statistic at least as extreme | H₀ true )

H₀The null hypothesis — usually "no effect" or "no difference"

NoteIt is a tail area under the null distribution, never a probability about H₀ itself

②

P-Value from a z-Score

two-tailed: p = 2[1 − Φ(|z|)] · one-tailed: p = 1 − Φ(|z|)

ΦStandard normal cumulative distribution function

Checkz = 1.96 → p = .050 · z = 2.576 → p = .010

③

P-Value from a t-Score

p = 2[1 − F_t(|t|, df)] where F_t uses the incomplete beta function

dfn − 1 (one sample) or n₁ + n₂ − 2 (two independent groups)

Notet converges to z as df grows beyond about 100

④

P-Value from Chi-Square

p = 1 − P(df/2, χ²/2) (regularized lower incomplete gamma)

df(rows − 1)(columns − 1) for independence tests

TailUpper tail only — large χ² means poor fit to the null

⑤

P-Value from an F Statistic

p = 1 − I_x(df₁/2, df₂/2), x = df₁F / (df₁F + df₂)

I_xRegularized incomplete beta function

UseANOVA, regression overall fit, variance ratio tests

⑥

P-Value for a Correlation r

t = r√(n − 2) ÷ √(1 − r²), df = n − 2

ThenFeed t into the t-distribution formula above

⑦

Independent-Samples t (Raw Data Mode)

t = (M₁ − M₂) ÷ [ s_p√(1/n₁ + 1/n₂) ], df = n₁+n₂−2

s_pPooled SD = √[((n₁−1)s₁²+(n₂−1)s₂²)/(n₁+n₂−2)]

🔍 P-Value Interpretation — In Detail

The p-value answers one narrow question: if there were truly no effect, how often would I see data at least this extreme? A p of .03 means results like yours would arise 3% of the time under a true null. That is evidence against the null — but it is not a measure of how large, important, or reproducible your effect is.

What the number does and does not mean

A p-value is not the probability that the null hypothesis is true; that reversal (confusing P(data|H₀) with P(H₀|data)) is the single most common statistical error in published research. Nor is p = .04 "twice as significant" as p = .08 — p-values are noisy, and in replications they bounce around dramatically. Nor does p > .05 prove there is no effect; absence of evidence is not evidence of absence, and small samples routinely produce non-significant results for real effects. Finally, the p-value says nothing about whether your study design, measures, or sampling were sound — it assumes all of that was correct.

Reading the magnitude

Treat p as a continuous measure of evidence rather than a switch. Values near .05 are weak evidence; below .01 is moderately strong; below .001 is strong — always conditional on the model being right. Because the 0.05 line is arbitrary, never write "approaching significance" for p = .06 or treat .049 and .051 as categorically different. The American Statistical Association's 2016 statement makes this explicit: scientific conclusions should not be based only on whether a p-value crosses a threshold.

Always pair p with an effect size and interval

Sample size drives the p-value: with n = 20 a large effect can miss significance, while with n = 20,000 a trivial effect achieves p < .001. So the complete report is statistic + df + exact p + effect size + confidence interval. The effect size tells you whether the finding matters; the interval tells you how precisely you measured it; the p-value only tells you how surprising the data would be under the null. Our effect size calculators and confidence interval calculator produce the companions this number needs.

One-tailed versus two-tailed

A two-tailed test asks "is there a difference in either direction?" and is the default. A one-tailed test asks "is group 1 greater?" and halves the p-value, which makes it tempting — and makes switching to it after seeing the data a form of p-hacking. Choose the tail before collecting data, state the choice in your methods, and stick to it. Note chi-square and F tests are inherently upper-tail: only large values signal departure from the null, so tail selection does not apply.

✍ How to Report P-Values in APA Style

APA 7 wants the exact p-value, no leading zero, two or three decimals, italicised, and never travelling alone: the test statistic, its degrees of freedom, and an effect size must sit beside it. Below are publication-ready sentences for every common test. Click Copy, paste into your manuscript, and swap the highlighted placeholders for your own numbers.

The five rules that cover 95% of cases

✓ Do this

Exact value, no leading zero: p = .032
Italicise the statistic: t, F, r, p
df in parentheses: t(28)
Effect size always: d = 0.66
Very small values: p < .001

✗ Not this

Leading zero: p = 0.032
Vague banding: p < .05 when you know the exact value
Impossible zero: p = .000
Naked p with no effect size
"Significance" used to mean "importance"

Ready-to-paste sentences by test

✅ Significant t-test

The treatment group scored significantly higher than the control group, t(30) = 7.76, p < .001, d = 2.74, 95% CI [1.78, 3.71].

Use p < .001 only when the value truly is below .001, never as a lazy substitute for an exact p.

⛔ Non-significant result

The groups did not differ significantly, t(28) = 1.80, p = .083, d = 0.66, 95% CI [−0.09, 1.40]. Because the interval extends to a moderate effect, the study was inconclusive rather than evidence of no difference.

Never write "there was no effect". Write "we did not detect a difference" and let the interval speak.

📊 One-way ANOVA (F test)

There was a significant effect of condition on recall score, F(2, 27) = 5.20, p = .012, η² = .28, indicating a large effect.

Both df go in the parentheses: numerator first, denominator second. Partial η² is labelled η_p².

🏁 Chi-square test

The association between treatment and recovery was significant, χ²(1, N = 120) = 3.84, p = .050, Cramér's V = .18.

Chi-square is the one test where APA asks for the sample size inside the parentheses alongside df.

🔗 Correlation

Study hours correlated significantly with exam score, r(28) = .42, p = .021, 95% CI [.07, .68].

df for a correlation is n − 2, not n. Drop the leading zero from r as well as p.

📐 Paired-samples t-test

Scores increased significantly from pre-test (M = 14.2, SD = 3.1) to post-test (M = 17.8, SD = 2.9), t(24) = 4.11, p < .001, d_z = 0.82.

Give M and SD for both time points; the paired effect size is d_z.

📈 Regression coefficient

Rainfall significantly predicted species richness, b = 0.34, SE = 0.11, t(47) = 3.09, p = .003, β = .41. The model explained a significant proportion of variance, R² = .28, F(1, 47) = 9.55, p = .003.

Report the unstandardised b with its SE, then the standardised β, then the model-level R² and F.

📏 Non-parametric (Mann-Whitney)

Group A (Mdn = 12) scored significantly higher than Group B (Mdn = 9), U = 142.5, z = −2.31, p = .021, r = .33.

Report medians rather than means, and the rank-biserial or r = z/√N effect size.

Decimal and formatting rules at a glance

Value	Correct APA form	Why
p = 0.032	p = .032	No leading zero for anything that cannot exceed 1
p = 0.0001	p < .001	APA caps reporting precision at three decimals
p = .000 (SPSS)	p < .001	A p-value is never exactly zero
p = .05000	p = .050	Two or three decimals, no more
r = 0.42	r = .42	Correlations also drop the leading zero
d = .79	d = 0.79	Cohen's d can exceed 1, so it keeps the zero
t(28)=1.8	t(28) = 1.80	Two decimals for test statistics, spaces around =
P value, p-Value	p	Lower case, italic, no hyphen inside a statistic string

Reporting many p-values at once

When a study reports more than a handful of tests, APA prefers a table to a wall of parentheses. Put the statistic, df, exact p, and effect size in adjacent columns; add a note stating the correction method. A single line such as "All p-values are Benjamini-Hochberg corrected for 18 comparisons" earns far more reviewer goodwill than silence.

Note. Reported p-values are two-tailed and corrected for multiple comparisons using the Benjamini-Hochberg procedure (18 tests, FDR = .05). Effect sizes are Hedges' g with 95% confidence intervals.

Final checklist before submission

Exact p to two or three decimals, or p < .001 — never p = .000 and never a bare p < .05.
Leading zero removed from p, r, β, and η²; kept for d, t, F, and M.
Test statistic and degrees of freedom stated in parentheses immediately before p.
An effect size accompanies every p-value, with a confidence interval where available.
Tail stated explicitly if the test was one-tailed, along with the pre-registered justification.
Correction method named whenever multiple comparisons were run.
Statistics italicised (p, t, F, r, M, SD, N) but Greek letters (χ², η², β) left upright.
Wording avoids "proves", "approaching significance", and "trend toward significance".

🔬 Example Results

Six worked examples matching the sample scenarios in the dropdown. Load any of them and click Calculate to reproduce the numbers exactly.

Example 1 — The 0.05 Landmark: z = 1.96

z = 1.96 · two-tailed
p = 2 × [1 − Φ(1.96)] = 0.0500
Verdict: significant at α = 0.05 (exactly on the line)

The most famous number in statistics. Each red tail holds 2.5% of the area; together they make the 5% that defines the conventional significance threshold.

Example 2 — Typical t-Test: t = 2.50, df = 20

t = 2.50 · df = 20 · two-tailed
critical t = 2.086 · p = 0.0212
2.50 > 2.086 → reject H₀ at α = 0.05

A routine significant result. Report it as t(20) = 2.50, p = .021 — and add Cohen's d, because this p-value alone says nothing about how large the difference actually is.

Example 3 — Borderline: t = 1.80, df = 28 (p = 0.083)

t = 1.80 · df = 28 · two-tailed
critical t = 2.048 · p = 0.0827
1.80 < 2.048 → fail to reject H₀

Do not call this "approaching significance." The honest reading: the data are compatible with the null, but also with a moderate real effect — report the confidence interval and treat the study as inconclusive.

Example 4 — Chi-Square: χ² = 3.841, df = 1

χ² = 3.841 · df = 1
p = 0.0500 (upper tail only)
The classic df = 1 critical value

Chi-square is right-skewed and one-tailed by nature: only large values indicate the observed frequencies depart from expectation. Note 3.841 = 1.96² — the same 5% threshold seen through a different distribution.

Example 5 — ANOVA F: F = 5.20, df = 2, 27

F = 5.20 · df₁ = 2, df₂ = 27
critical F = 3.354 · p = 0.0123
Report: F(2, 27) = 5.20, p = .012, η² = .28

A significant one-way ANOVA. The F distribution is also upper-tail only. Pair it with η² or ω² — our effect size calculators compute both from exactly these inputs.

Example 6 — How the P-Value Falls as the Statistic Grows

z = 1.00 → p = .317
z = 1.65 → p = .099
z = 1.96 → p = .050
z = 2.58 → p = .010
z = 3.29 → p = .001

The relationship is steep and non-linear. Small changes in the statistic near the threshold swing the p-value a lot, which is exactly why treating .049 and .051 as categorically different is indefensible.

⚡ P-Value Cheat Sheet

The numbers people look up most, in one place.

Question	Answer
z = 1.96 → p	0.050 (two-tailed)
z = 2.576 → p	0.010 (two-tailed)
z = 3.29 → p	0.001 (two-tailed)
χ² = 3.841, df = 1 → p	0.050 (upper tail)
One-tailed p	Two-tailed p ÷ 2
Two-tailed p	One-tailed p × 2
p = .000 in output	Report as p < .001
Decision rule	Reject H₀ if p ≤ α
Correlation r → p	t = r√(n−2)/√(1−r²), df = n − 2
APA format	t(20) = 2.50, p = .021, d = 0.79
Bonferroni correction	Compare p to α ÷ number of tests
Chi-square and F tails	Upper tail only, never two-tailed

🧭 When to Use This Tool

Use this p-value calculator whenever you have a test statistic and need the exact probability — reading a paper that reports only t or F, checking software output, doing homework or exam prep, converting between one- and two-tailed values, teaching hypothesis testing, or running a quick t-test from raw data. It suits students, researchers, reviewers, analysts, and anyone who needs a p-value without opening SPSS or R.

⚠ Assumptions & Limitations

Every p-value assumes its test's conditions hold: random sampling, independent observations, and the relevant distributional assumptions (normality for t and z tests, expected cell counts of at least 5 for chi-square, homogeneity of variance for the pooled t-test and ANOVA). If those fail, the p-value is not merely imprecise — it is answering a question about a model that does not describe your data. The raw-data mode here uses Student's pooled t-test; with clearly unequal variances Welch's version is preferable. P-values also assume a single pre-specified analysis: running many tests, peeking at data as it accumulates, or choosing the tail after seeing the result all inflate false positives (multiplicity and p-hacking). Correct for multiple comparisons when appropriate. Finally, p-values are famously unstable across replications, so a single p near .05 is weak evidence — effect sizes, intervals, and replication carry more information.

🏁 Conclusion

A p-value is a tail area: the probability of data at least as extreme as yours, computed under the assumption that the null hypothesis is true. Mechanically it is simple — find where your statistic falls on the null distribution and measure the area beyond it. For z that is the normal curve, for t the Student distribution with its degrees of freedom, for chi-square and F the upper tail only, and for a correlation you convert r to t first.

The difficulty is never the arithmetic; it is the interpretation. The p-value does not give the probability that the null is true, does not measure effect size, does not indicate importance, and does not become meaningful simply by crossing 0.05. It is one piece of evidence about compatibility between data and a model. Treat the threshold as a convention, report the exact value, and resist the urge to bin results into "significant" and "not significant."

The habit that fixes most p-value misuse is simple: never report one alone. Statistic, degrees of freedom, exact p, effect size, and confidence interval together tell a complete story — how surprising the data are, how big the effect is, and how precisely you measured it. This calculator gives you the exact p-value from any common test statistic, with plots that show the tail area you are actually computing, plus APA-ready sentences. Pair it with our effect size and confidence interval calculators, and your results section will satisfy any reviewer.

❓ Frequently Asked Questions

What is a p-value in simple terms?

The probability of getting results at least as extreme as yours if the null hypothesis (no real effect) were true. A small p-value means your data would be surprising in a world with no effect.

What does p < 0.05 mean?

If the null were true, data this extreme would occur less than 5% of the time. By convention that counts as statistically significant and the null is rejected at the .05 level — but it does not prove the effect is real or important.

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

Two-tailed: p = 2[1 − Φ(|z|)], where Φ is the standard normal CDF. One-tailed: p = 1 − Φ(|z|). For example z = 1.96 gives p = .050 and z = 2.576 gives p = .010.

How do you calculate a p-value from a t-score?

Use the t distribution with your degrees of freedom rather than the normal curve: p = 2[1 − F_t(|t|, df)]. Because t has heavier tails, its p-value is always slightly larger than the z-based one for the same statistic.

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

Two-tailed tests look for a difference in either direction; one-tailed tests look in one direction only and give exactly half the p-value. Choose the tail before seeing the data — switching afterwards is p-hacking.

Is the p-value the probability the null hypothesis is true?

No — this is the most common misinterpretation. It is P(data | null true), not P(null true | data). Answering the second question requires Bayesian methods and a prior.

Does a smaller p-value mean a bigger effect?

No. The p-value depends on both effect size and sample size. A tiny effect measured in 50,000 people yields a minuscule p; a large effect in 12 people may not reach significance. Only an effect size measures magnitude.

What does a non-significant p-value mean?

Insufficient evidence to reject the null — not proof the null is true. Absence of evidence is not evidence of absence. Check the confidence interval: if it spans from trivial to large effects, the study was simply inconclusive.

Why is 0.05 the standard threshold?

Ronald Fisher suggested it in the 1920s as a convenient rule of thumb, and convention hardened around it. There is nothing special about it mathematically — particle physics uses ~0.0000003, and many methodologists now advocate 0.005.

Can a p-value be exactly zero?

Not mathematically — the tail area is always positive. Software showing "p = .000" means the value is smaller than the displayed precision, which should be reported as p < .001.

How do I report p-values in APA style?

Exact values to two or three decimals with no leading zero: p = .032. Use p < .001 for very small values. Italicize p, include the test statistic and df, and always add an effect size.

What is p-hacking?

Manipulating analysis until p drops below .05 — trying different tests, dropping outliers selectively, adding participants until it works, or switching to one-tailed after the fact. It dramatically inflates false positives and is a major driver of the replication crisis.

What p-value do chi-square and F tests use?

The upper tail only. Both distributions are right-skewed and only large values indicate departure from the null, so there is no two-tailed option for them.

How do degrees of freedom affect the p-value?

Lower df means heavier tails and a larger p-value for the same statistic. t = 2.5 gives p = .067 at df = 5 but p = .021 at df = 20 and p = .012 at df = 100 — small samples are penalized appropriately.

How do I get a p-value for a correlation?

Convert r to t: t = r√(n−2)/√(1−r²) with df = n − 2, then read the t-distribution p-value. This tool does it automatically in the correlation tab.

Should I correct p-values for multiple comparisons?

Yes, when running many tests on the same data. Twenty independent tests at α = .05 give about a 64% chance of at least one false positive. Bonferroni (α/k) is simplest; Benjamini-Hochberg FDR is less conservative.

What is the difference between p-value and alpha?

Alpha is the threshold you set before the study (usually .05) — your tolerated false-positive rate. The p-value is what you compute from the data. You compare p to alpha to make a decision.

Can I compare two p-values?

Not directly. p = .04 is not "twice as significant" as p = .08, and a significant result in one study plus a non-significant one in another does not mean the studies disagree — compare effect sizes and their intervals instead.

What does the ASA say about p-values?

The American Statistical Association's 2016 statement warns that p-values do not measure effect size or the probability the hypothesis is true, and that scientific conclusions should not rest on whether p crosses a threshold.

What should I report alongside the p-value?

The test statistic, degrees of freedom, exact p, an effect size (Cohen's d, η², r), and a confidence interval. That combination answers how surprising, how large, and how precise — a p-value alone answers only the first.

Where is the p-value in SPSS output?

It is the column labelled Sig. or Sig. (2-tailed). Halve it for a one-tailed test. SPSS printing .000 means p < .001, never exactly zero.

How do I calculate a p-value in Excel?

Use =T.DIST.2T(ABS(t),df) for t, =2*(1-NORM.S.DIST(ABS(z),TRUE)) for z, =CHISQ.DIST.RT(x,df) for chi-square, and =F.DIST.RT(F,df1,df2) for F. For raw data use =T.TEST(range1,range2,tails,type).

How do I get a p-value in R from a test statistic?

2*pnorm(-abs(z)) for z, 2*pt(-abs(t),df) for t, pchisq(x,df,lower.tail=FALSE) for chi-square, and pf(F,df1,df2,lower.tail=FALSE) for F.

How do I get a p-value in Python?

With SciPy: 2*stats.norm.sf(abs(z)), 2*stats.t.sf(abs(t),df), stats.chi2.sf(x,df), or stats.f.sf(F,df1,df2). Test functions such as stats.ttest_ind return the p-value directly as .pvalue.

What does "Sig. (2-tailed)" mean?

It is simply SPSS's name for the two-tailed p-value. It is the same number this calculator gives for a two-tailed test, and half of it is the one-tailed value.

🔗 Related Calculators & Guides

A p-value is only one part of a complete result. These tools cover the rest:

T-Test Calculator — run paired, and independent t-tests from raw data.
Chi-Square Calculator — goodness of fit and tests of independence from a contingency table.
One-Way ANOVA Calculator — F statistic, p-value, and post-hoc comparisons.
Effect Size Calculator — Cohen's d, Hedges' g, and eta squared to report beside your p.
Confidence Interval Calculator — the precision estimate every p-value should travel with.
Sample Size & Power Calculator — plan the study so a non-significant p is actually informative.
Correlation Calculator — Pearson and Spearman r with significance testing.
Z-Score Calculator — standardize a value before converting it to a p-value.

📚 References

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Wasserstein, R. L., Schirm, A. L., & Lazar, N. A. (2019). Moving to a world beyond "p < 0.05". The American Statistician, 73(sup1), 1–19. https://doi.org/10.1080/00031305.2019.1583913
Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337–350. https://doi.org/10.1007/s10654-016-0149-3
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