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For learning and exploration only. Not a substitute for professional statistical review or clinical-trial analysis.
Reference
The tool above converts a test statistic into a p-value in your browser. The reference below is what you need when computing the same p-values in code or pulling critical values from a table: tail formulas, language-by-language snippets for Z/T/chi-square/F, the standard critical values students still pull on paper, and the common pitfalls when a pasted number does not match what the textbook says.
These are the calculations the tool runs. Same formulas across every library and language.
| Tail | Formula | Notes |
|---|---|---|
| Left-tailed | p = CDF(t) | One-sided, predicted lower direction |
| Right-tailed | p = 1 - CDF(t) (or SF(t) if available) | One-sided, predicted upper direction |
| Two-tailed (symmetric: Z, T) | p = 2 * min(CDF(t), 1 - CDF(t)) | Equivalent to 2 * (1 - CDF(abs(t))) |
| Two-tailed (asymmetric: chi-square, F) | p = 1 - CDF(t) | Test is intrinsically upper-tail |
A common mistake is applying the symmetric two-tailed formula to chi-square or F. Both are bounded at zero, so the “extreme” side is the upper tail only. Calling them two-tailed in textbook language usually still means computing 1 - CDF(t).
from scipy import stats # Z (standard normal) p_two = 2 * (1 - stats.norm.cdf(abs(z))) p_left = stats.norm.cdf(z) p_right = stats.norm.sf(z) # 1 - CDF, more numerically stable # T (Student's T) p_two = 2 * (1 - stats.t.cdf(abs(t), df)) p_left = stats.t.cdf(t, df) p_right = stats.t.sf(t, df) # Chi-square (always upper tail for goodness-of-fit, contingency) p = stats.chi2.sf(x, df) # F (always upper tail for ANOVA, variance ratio) p = stats.f.sf(F, df1, df2)
stats.t.sf(t, df) is preferred over 1 - stats.t.cdf(t, df) when the statistic is far in the tail. The subtraction loses precision near 1.
| Distribution | Lower tail | Upper tail | Two-tailed (symmetric) |
|---|---|---|---|
| Z | pnorm(z) | pnorm(z, lower.tail = FALSE) | 2 * pnorm(-abs(z)) |
| T | pt(t, df) | pt(t, df, lower.tail = FALSE) | 2 * pt(-abs(t), df) |
| Chi-square | pchisq(x, df) | pchisq(x, df, lower.tail = FALSE) | (use upper) |
| F | pf(F, df1, df2) | pf(F, df1, df2, lower.tail = FALSE) | (use upper) |
The lower.tail = FALSE flag is R’s equivalent of sf() in scipy. It computes the right-tail probability directly and avoids precision loss for extreme statistics.
import jStat from 'jstat'; // Z const pZRight = 1 - jStat.normal.cdf(z, 0, 1); const pZTwo = 2 * (1 - jStat.normal.cdf(Math.abs(z), 0, 1)); // T const pTTwo = 2 * (1 - jStat.studentt.cdf(Math.abs(t), df)); // Chi-square const pChi = 1 - jStat.chisquare.cdf(x, df); // F const pF = 1 - jStat.centralF.cdf(F, df1, df2);
jStat does not expose a survival function, so subtracting from 1 is the standard idiom. For very small p values, switch to stdlib.js, which has dedicated upper-tail routines.
| Test | Excel / Google Sheets |
|---|---|
| Z, two-tailed | =2*(1-NORM.S.DIST(ABS(z), TRUE)) |
| Z, right-tailed | =1-NORM.S.DIST(z, TRUE) |
| T, two-tailed | =T.DIST.2T(ABS(t), df) |
| T, right-tailed | =T.DIST.RT(t, df) |
| Chi-square, upper | =CHISQ.DIST.RT(x, df) |
| F, upper | =F.DIST.RT(F, df1, df2) |
LibreOffice mirrors the Excel names. Older Excel (pre-2010) used legacy names like TDIST, CHIDIST, and FDIST. These return upper-tail probabilities and are still supported in compatibility mode.
| Platform | Right-tail T-test p |
|---|---|
| Stata | display ttail(df, t) |
| SAS | p = 1 - probt(t, df); |
| SPSS syntax | COMPUTE p = 1 - CDF.T(t, df). |
| MATLAB | 1 - tcdf(t, df) |
| Julia | ccdf(TDist(df), t) (Distributions.jl) |
| PostgreSQL (with pl/R) | pt(t, df, lower.tail = FALSE) |
Pure SQL has no built-in statistical CDFs. Production setups call out to R (via pl/R), Python (via pl/Python), or a separate stats microservice.
For when a paper reports a z-score and you need the cutoff on hand.
| Alpha | One-tailed z* | Two-tailed z* |
|---|---|---|
| 0.10 | 1.282 | 1.645 |
| 0.05 | 1.645 | 1.960 |
| 0.025 | 1.960 | 2.241 |
| 0.01 | 2.326 | 2.576 |
| 0.005 | 2.576 | 2.807 |
| 0.001 | 3.090 | 3.291 |
| 5-sigma (particle physics) | N/A | 5.000 |
Each cell is the |t| at which 2 * (1 - pt(|t|, df)) equals the column alpha. For one-tailed, halve the column alpha.
| df | α = 0.10 | α = 0.05 | α = 0.01 | α = 0.001 |
|---|---|---|---|---|
| 1 | 6.314 | 12.706 | 63.657 | 636.619 |
| 5 | 2.015 | 2.571 | 4.032 | 6.869 |
| 10 | 1.812 | 2.228 | 3.169 | 4.587 |
| 15 | 1.753 | 2.131 | 2.947 | 4.073 |
| 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 |
| 120 | 1.658 | 1.980 | 2.617 | 3.373 |
| ∞ (Z) | 1.645 | 1.960 | 2.576 | 3.291 |
As df grows, the T critical values converge on the corresponding Z values. By df = 120, the difference is in the third decimal.
| df | α = 0.10 | α = 0.05 | α = 0.01 | α = 0.001 |
|---|---|---|---|---|
| 1 | 2.706 | 3.841 | 6.635 | 10.828 |
| 2 | 4.605 | 5.991 | 9.210 | 13.816 |
| 5 | 9.236 | 11.070 | 15.086 | 20.515 |
| 10 | 15.987 | 18.307 | 23.209 | 29.588 |
| 20 | 28.412 | 31.410 | 37.566 | 45.315 |
| 30 | 40.256 | 43.773 | 50.892 | 59.703 |
A 2-by-2 contingency table has df = 1. A 3-by-3 has df = 4. A goodness-of-fit on k categories has df = k - 1, minus any parameters estimated from the data.
ANOVA’s most common cutoff. df1 across, df2 down.
| df2 \ df1 | 1 | 2 | 3 | 5 | 10 | 20 |
|---|---|---|---|---|---|---|
| 5 | 6.61 | 5.79 | 5.41 | 5.05 | 4.74 | 4.56 |
| 10 | 4.96 | 4.10 | 3.71 | 3.33 | 2.98 | 2.77 |
| 20 | 4.35 | 3.49 | 3.10 | 2.71 | 2.35 | 2.12 |
| 30 | 4.17 | 3.32 | 2.92 | 2.53 | 2.16 | 1.93 |
| 60 | 4.00 | 3.15 | 2.76 | 2.37 | 1.99 | 1.75 |
| ∞ | 3.84 | 3.00 | 2.60 | 2.21 | 1.83 | 1.57 |
For α = 0.01, multiply roughly by 1.6 to 2.4 depending on df1, or compute qf(0.99, df1, df2) in R for the exact value.
| Symptom | Cause | Fix |
|---|---|---|
| Excel returns #NUM! for T.DIST.2T(-2, 10) | T.DIST.2T requires a non-negative argument | Wrap in ABS(): =T.DIST.2T(ABS(t), df) |
| Pasted p does not match scipy / R | Mixed one-tailed and two-tailed conventions | Re-check the tail; halve or double as needed |
| Chi-square test reported as “two-tailed” | Chi-square is intrinsically upper-tail | Compute 1 - CDF, ignore the two-tailed prompt |
| Welch's T p slightly different from a pooled-T calculator | Welch uses Welch-Satterthwaite df (often non-integer) | Use the non-integer df your stats package reported |
| Reported p = 0.0000 in a table | Rounded display, real value below the precision floor | Switch to < 0.0001 or report exp(...) from raw |
| p = 1.000 returned for a tiny statistic | Statistic in the wrong tail for the test you specified | Flip to the opposite one-tailed test, or use two-tailed |
| df = 0 returns NaN / inf | Distribution undefined at df = 0 | Recompute df; one-sample T needs n - 1, not n |
| Bonferroni-corrected p > 1 | Manual p * k exceeded 1 | Cap at 1: min(p * k, 1) |
| Convention | Format | Where used |
|---|---|---|
| Exact value, three significant figures | p = 0.0312 | APA 7th edition, most journals |
| Below display floor | p < 0.0001 | When the real value is smaller than reportable precision |
| Star notation (avoid in modern reports) | *p < 0.05, **p < 0.01, ***p < 0.001 | Older social-science tables only |
| Cohen's d alongside p | t(18) = 2.05, p = 0.055, d = 0.96 | Required by most psychology journals |
| Effect size with confidence interval | r = 0.42, 95% CI [0.18, 0.61], p = 0.003 | Preferred over star notation in clinical research |
Most journals dropped star formats between 2015 and 2020 in favor of exact values plus effect sizes. APA 7 explicitly requires the exact value to three significant figures when feasible.
1 - CDF, computed directly for tail precision), and percent-point function (the inverse CDF that returns the critical value for a given alpha).statsmodels.stats.weightstats) or R (equivalence package); a generic p-value calculator does not compute these directly.