# P-value Calculator - From z, t, or Chi-Square

Free online tool to turn a z-score, t-statistic, or χ² statistic into a left-, right-, or two-tailed p-value. Runs in your browser.

### Input Values

Distribution

z-score _1.96_

Tail

Clear All

The p-value updates as you type. df is required only for t and chi-square and must be > 0. Chi-square statistics must be non-negative.

### P-value Result

The p-value will appear here

### What Is a P-value Calculator and Why Do You Need One?

A **p-value** is the probability, under a stated null hypothesis H0, of seeing a test statistic at least as extreme as the one you observed. It is not the probability that H0 is true, and it is not the probability that you made a mistake. It is a tail probability on a reference distribution - usually the standard normal (z), Student’s t, or chi-square.

This **free online p-value calculator** takes that statistic and reports a left-tailed, right-tailed, or two-tailed probability. Many homework keys and papers still compare the result to α = 0.05. That cutoff is a convention used in a lot of fields, not a law of nature. A result just below 0.05 is not magically “true,” and a result just above it is not magically “false.”

Everything runs in your browser. Nothing is uploaded. Pair it with the [average calculator](https://www.toolcomb.com/average-calculator) when you are summarizing a sample, or with the [scientific notation calculator](https://www.toolcomb.com/scientific-notation-calculator) when a tiny p-value is easier to read as a × 10n.

### How to Use This Free Online P-value Calculator

Using this **z, t, and chi-square tail tool** is straightforward:

1. **Choose the distribution**: normal (z-score), Student’s t (t-statistic plus df), or chi-square (χ² plus df).
2. **Enter the test statistic**. Chi-square values must be ≥ 0. Degrees of freedom, when shown, must be a finite number greater than 0.
3. Select the **tail**: left P(X ≤ statistic), right P(X ≥ statistic), or two-tailed.
4. Read the **p-value**. Very small results (below 10−6) are shown in scientific notation. Copy the number, or clear the fields to start over. The last inputs are saved locally in your browser for up to 30 days.

**Note:** these tails are numerical approximations (erf, incomplete beta, incomplete gamma). They are not a substitute for R, SPSS, or a published statistical library.

### H0, Tails, and the α = 0.05 Convention

p = P(as extreme | H0)

Compare to α only if your protocol named α in advance

Under H0 the test statistic is assumed to follow the distribution you picked. A **one-tailed** p-value (left or right) matches a directional alternative: you only count probability in one direction. A **two-tailed** p-value on a symmetric null (z or t) is twice the more extreme one-sided tail, so both large positive and large negative statistics count as evidence against H0.

Chi-square tests in practice are usually **right-tailed**(Pearson goodness-of-fit, independence in a table). A two-tailed χ² option is included for completeness; it is not the usual Pearson setup because the chi-square density is not symmetric.

Many labs still call p < 0.05 “statistically significant.” That line is a **conventional cutoff**, not a scientific law. Effect size, design, and whether the hypothesis was pre-registered matter more than whether the third decimal happened to land at 0.049 or 0.051.

### z vs. t vs. Chi-Square

| Family | Statistic | Needs df? | Typical use |
| --- | --- | --- | --- |
| **Normal (z)** | z-score | No | Known σ, large-n Wald tests, proportions |
| **Student’s t** | t | Yes, df > 0 | Unknown σ, one-sample or two-sample means |
| **Chi-square** | χ² ≥ 0 | Yes, df > 0 | Goodness-of-fit, independence, variance tests |

The z tail uses the error function (erf) to approximate the standard normal CDF Φ. The t tail uses a regularized incomplete beta. The chi-square survival function is a regularized incomplete gamma Q(df/2, χ²/2). All three are approximations implemented in this page - honest enough for a homework check, not a claim of library-grade precision.

### Worked Examples

#### 1. z = 1.96, two-tailed

Φ(1.96) is about 0.975. The right tail 1 − Φ(1.96) is about 0.025, so the two-tailed p-value is 2 × (1 − Φ(|1.96|)) ≈ **0.05**. That is why 1.96 is the textbook two-sided normal critical value at α = 0.05. This calculator should land very close to 0.05; leftover digits are from the erf approximation, not from a different definition of Φ.

#### 2. t = 2.0, df = 10

Student’s t with 10 degrees of freedom has heavier tails than z. For t = 2.0 the two-tailed p-value is about **0.073** - larger than the normal two-tailed value for z = 2 (about 0.046). Same statistic, extra uncertainty in the variance, weaker evidence against H0 if you insist on the 0.05 convention. The one-sided complementary probability is about 0.037.

#### 3. χ² = 3.84, df = 1, right tail

For one degree of freedom, χ² is a squared standard normal, so P(χ²1 ≥ 3.84) matches the two-tailed normal p-value at |z| ≈ 1.96, again about **0.05**. Use the right tail for a Pearson statistic; a left tail would ask how surprisingly _small_ the chi-square was.

### This Is an Approximation

Tail probabilities here are evaluated with classical numerical recipes: an Abramowitz–Stegun erf for the normal, Lanczos log-gamma plus a continued-fraction incomplete beta for t, and series/continued-fraction incomplete gamma for chi-square. Double-precision JavaScript will underflow extremely far tails toward 0, and the approximations are weaker than a well-tested C library behind R or SPSS.

Use this page to check a homework step, to see whether a published z or t is in the same ballpark as p = 0.05, or to convert a statistic you already computed. Do not treat the last displayed digit as a published p-value, and do not skip the design, assumptions, and effect size that the number cannot capture.

### Frequently Asked Questions (FAQ) - P-value

#### How do I find a p-value from a z-score?

Treat the z-score as a standard normal deviate. A left-tailed p-value is Φ(z), a right-tailed p-value is 1 − Φ(z), and a two-tailed p-value is 2 × (1 − Φ(|z|)). This calculator approximates Φ with the error function (erf).

#### How do I calculate a p-value from a t-statistic and degrees of freedom?

Use the Student’s t distribution with the given df (df must be positive). Left, right, and two-tailed probabilities follow the same pattern as z, but the tails are heavier for small df. The CDF is evaluated with an incomplete-beta approximation.

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

One-tailed (left or right) measures probability in a single direction, matching a directional alternative hypothesis. Two-tailed doubles the more extreme one-sided tail for a symmetric null (z or t). Chi-square tests are usually right-tailed; a two-tailed χ² option is not the usual Pearson goodness-of-fit setup.

#### How do I get a p-value from a chi-square statistic?

Enter the χ² statistic and positive degrees of freedom. The usual p-value is the right-tail probability P(χ²_df ≥ statistic), approximated with the incomplete gamma function. A left tail is P(χ²_df ≤ statistic).

#### Is this p-value calculator as accurate as R or SPSS?

No. Tail probabilities here are numerical approximations (erf, incomplete beta, incomplete gamma) in the browser. They are fine for homework checks and rough significance calls, not a substitute for a statistics package when you need published-precision p-values.

#### Why does the p-value calculator require positive degrees of freedom?

The t and chi-square families are defined only for df > 0. Zero or negative df would make the density undefined. Enter the df from your test (n − 1 for a one-sample t, (r−1)(c−1) for a contingency table, and so on).

### Why Choose Our P-value Calculator?

- **Free, no account.** Unlimited z, t, and χ² tail conversions.
- **Private.** All arithmetic stays in the browser.
- **Left, right, and two tails** so the number matches the alternative you actually used.
- **Honest about α = 0.05** - a convention, not a law - and honest that the math is an approximation.
- **Scientific notation** for tiny p-values, with a complementary one-sided probability shown beside the result.
- **Works offline** after the first page load. Last inputs are stored locally for 30 days.

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### Related guides

Read the how-to, then come back to this tool when you are ready to run it locally.

- [Confidence Interval vs P-Value: Which Calculator? Interval estimates vs hypothesis-test evidence, z and t tables, and a comparison table for picking the right statistics tool.](https://www.toolcomb.com/blog/confidence-interval-vs-p-value-which-calculator)

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