P-Value Calculator from Z-Score

Test Type

p = 2 · P(Z ≥ |z|) — for tests where direction is unspecified.

Solution

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Worked Examples

Two-Tailed

What is the p-value for z = 1.96 (two-tailed)?

The most cited example in introductory statistics — the borderline case near α = 0.05.

  1. Identify the test type: two-tailed.
  2. Take the absolute z-score: |1.96| = 1.96.
  3. Find the right-tail area: 1 − Φ(1.96) ≈ 0.025.
  4. Double it for two tails: 2 × 0.025 = 0.05.
  5. So the two-tailed p-value rounds to about 0.050.
  6. The unrounded p-value sits just inside the rejection region, so under the strict rule p < α the result is just barely significant at α = 0.05.

This is why ±1.96 is taught as the universal two-tailed 5% boundary.

Right-Tailed

What is the p-value for z = 2.33 (right-tailed)?

A classic one-tailed 1% significance example — used when the alternative hypothesis specifies the upper direction.

  1. Identify the test type: right-tailed (upper-tail alternative).
  2. Find the right-tail area: 1 − Φ(2.33) ≈ 0.0099.
  3. So the one-tailed p-value is approximately 0.0099.
  4. Compare to α = 0.01: 0.0099 < 0.01, so the result is significant at the 1% level.

Z = 2.33 is the standard one-tailed 1% critical value. The corresponding p-value sits just under 0.01.

Left-Tailed

What is the p-value for z = -1.65 (left-tailed)?

A common left-tailed example near the standard 5% one-tailed cutoff.

  1. Identify the test type: left-tailed (lower-tail alternative).
  2. Find the left-tail area: Φ(-1.65) ≈ 0.0495.
  3. So the one-tailed p-value is approximately 0.0495.
  4. Compare to α = 0.05: 0.0495 < 0.05, so the result is just barely significant at the 5% level.

The standard one-tailed 5% boundary is z = ±1.6449. Test statistics near this cutoff produce p-values that hover near 0.05.

P-Value from Z-Score

The two-tailed p-value is twice the right-tail probability of the absolute z-score. For a one-tailed test, the p-value equals the tail area in the direction of the alternative hypothesis. Φ is the standard normal CDF.

p = 2 · (1 − Φ(|z|)) (two-tailed)

How It Works

A p-value tells you how likely your observed test statistic would be — or one even more extreme — if the null hypothesis were true. To compute a p-value from a z-score, you locate the z on the standard normal distribution and read off the tail area. The exact tail depends on the kind of test. A left-tailed test reports the area to the left of the z-score. A right-tailed test reports the area to the right. A two-tailed test reports twice the right-tail area of the absolute z-score, since extremes in either direction count as evidence against the null. Compare the resulting p to your chosen significance threshold (often 0.05 or 0.01) to decide whether to reject the null hypothesis.

Example Problem

A two-tailed z-test produces z = 1.96. What is the p-value, and is the result statistically significant at α = 0.05?

  1. Identify the z-score: z = 1.96 (a two-tailed test).
  2. Take the absolute value: |z| = 1.96.
  3. Find the right-tail area: 1 − Φ(1.96) ≈ 0.025.
  4. Double it for two tails: 2 × 0.025 = 0.050.
  5. So the p-value rounds to about 0.050.
  6. The unrounded p-value sits just under 0.05, satisfying the strict rule p < α — the textbook borderline case for two-tailed 5% significance.

This is why ±1.96 is taught as the universal two-tailed 5% boundary — its corresponding p-value rounds to exactly 0.05 even though the unrounded value lies just inside the rejection region.

Key Concepts

A p-value is the probability of observing a test statistic at least as extreme as the one you got, assuming the null hypothesis is true. It is not the probability that the null hypothesis is true. Smaller p-values indicate stronger evidence against the null. Common thresholds: p < 0.05 (significant), p < 0.01 (very significant), p < 0.001 (highly significant). The choice between one-tailed and two-tailed depends on whether the alternative hypothesis specifies a direction. Always pre-specify the test direction; running both and choosing the smaller p-value inflates the false-positive rate.

Applications

  • Hypothesis testing in medical, social, and scientific research
  • A/B testing for product, marketing, and UX experimentation
  • Quality control where deviations from a specification are evaluated for significance
  • Financial analysis to test whether a return differs from a benchmark
  • Clinical trials to compare treatment groups against a control
  • Public-policy analyses comparing program outcomes

Common Mistakes

  • Reporting a one-tailed p-value when the test is actually two-tailed (or vice versa)
  • Treating the p-value as the probability that H₀ is true — it is not
  • Comparing p-values across studies with different sample sizes without adjustment
  • Using a one-tailed test after seeing the data instead of pre-specifying direction
  • Confusing 'not significant' with 'evidence the effect is zero'
  • Reporting only the p-value without effect size or confidence interval

Frequently Asked Questions

What is a p-value in plain language?

A p-value is the probability of seeing data at least as extreme as what you observed if the null hypothesis is true. Smaller p-values mean your data is harder to explain by chance alone.

What p-value is considered statistically significant?

A p-value below 0.05 is the conventional threshold for statistical significance. Stricter thresholds like 0.01 or 0.001 are used in fields where false-positive control is more important.

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

A one-tailed p-value measures the probability of an extreme result in one specified direction. A two-tailed p-value measures the probability of an extreme result in either direction. Two-tailed tests are more conservative because they account for both directions.

How do I get a two-tailed p-value from a one-tailed one?

If your one-tailed p-value matches the direction of the test statistic, double it to get the two-tailed p-value: p_two = 2 × p_one. This works because of the symmetry of the standard normal distribution.

What p-value does z = 1.96 give?

A z-score of 1.96 produces a two-tailed p-value of approximately 0.050 — the standard threshold for 5% significance. The one-tailed p-value (right-tailed) for z = 1.96 is approximately 0.025.

What p-value does z = 2.58 give?

A z-score of 2.58 produces a two-tailed p-value of approximately 0.010 — the standard threshold for 1% significance.

Can a p-value be exactly zero?

Mathematically no — the normal distribution has infinite tails — but for very large z-scores the p-value can be smaller than what most computers can represent. Calculators may report it as 0 or in scientific notation.

Should I use a z-test or a t-test?

Use a z-test when the population standard deviation is known or the sample is large (typically n > 30). Use a t-test when the population standard deviation is unknown and the sample is small. Both produce p-values, but the t-test uses the t-distribution instead of the standard normal.

Reference: P-values are computed from the standard normal cumulative distribution function using the Abramowitz and Stegun error-function approximation, with double-tailing applied for two-tailed tests.

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