P-Value Calculator: Z-Test, T-Test, Chi-Square Test for Statistical Significance

How to use the tool

1. Select the test Example: Z-test for a known population σ, T-test for small unknown-σ samples, or Chi-Square for frequency data.
2. Fill the fields accurately
     • Sample Mean (𝑥̄): 112.0 or 97.6
     • Population Mean (μ₀): 100 or 105
     • Population SD (σ): 14.8 or 18.2
     • Sample SD (s): 4.1 or 6.3
     • Sample Size (n): 49 or 22
     • Observed (Oᵢ): 45, 35, 20 or 18, 22, 30, 30
     • Expected (Eᵢ): 40, 40, 20 or 25, 25, 25, 25
3. Choose test direction: one-tailed (specific direction) or two-tailed (any difference).
4. Optionally set α: type 0.01 or 0.10; default is 0.05.
5. Press “Calculate”: review Test Statistic, Degrees of Freedom, P-value, and automated conclusion.

Formulas used

• Z-test: $$Z = rac{\,\bar x – μ_0\,}{σ/√n}$$
• T-test: $$t = rac{\,\bar x – μ_0\,}{s/√n}, \quad df = n-1$$
• Chi-Square: $$χ^2 = Σ rac{(O_i – E_i)^2}{E_i}, \; df = k-1$$
• Two-tailed p-value: $$p = 2\bigl(1 – Φ(|Z|)\bigr)$$   with Φ the standard-normal CDF.
• Student-t and χ² p-values follow the respective cumulative distributions (NIST e-Handbook).

Worked examples

• Z-test: 𝑥̄ = 112, μ₀ = 100, σ = 15, n = 49 →   Z = 5.60, p ≈ 0.000 000 01 (reject H₀).
• T-test: 𝑥̄ = 23, μ₀ = 20, s = 4.5, n = 10 →   t = 2.11, df = 9, p ≈ 0.063 (fail to reject at α = 0.05).
• Chi-Square: O = 45,35,20; E = 40,40,20 →   χ² = 1.25, df = 2, p ≈ 0.53.

Quick-Facts

• α = 0.05 remains the predominant significance level in health research (Curran-Everett, 2017).
• The Z-test assumes n ≥ 30 and known σ (Rossi, StatHandbook).
• T-tests are robust to moderate departures from normality up to |skewness| ≈ 1 (Lumley et al., 2002).
• Chi-Square cells should each expect ≥ 5 counts for validity (McHugh, 2013).

FAQ

What is a p-value?

A p-value gauges how incompatible your data are with the null hypothesis: smaller values signal stronger evidence against H₀ (Wasserstein & Lazar, 2016).

When should you use a Z-test instead of a T-test?

Use a Z-test when the population standard deviation is known and sample size is large; otherwise choose a T-test (NIST e-Handbook).

How does test direction affect results?

One-tailed tests halve the critical region, cutting the p-value in two for the hypothesized direction; two-tailed tests split it across both tails (UCLA ATS).

Why check degrees of freedom?

Degrees of freedom shape the T and χ² distributions; fewer df widen tails and raise p-values, demanding stronger effects to reach significance (Motulsky, Intuitive Biostatistics).

Can you change the α level?

Yes. Lower α (e.g., 0.01) reduces false positives but increases false negatives; regulatory drug trials often use 0.025 one-sided (FDA Guidance 2017).

What happens if expected counts are < 5 in a χ² test?

The approximation breaks down; switch to Fisher’s exact test or merge categories for reliable inference (McDonald, Handbook of Biological Statistics).