Hypothesis Test Calculators (Z, t, Chi-Square)

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Hypothesis Test Resources

A centralized guide for selecting tests, entering data, interpreting results, and understanding assumptions.

A hypothesis test is a formal procedure for investigating our ideas about the world. It evaluates whether sample evidence is strong enough to reject a null hypothesis ($H_0$) about a population parameter in favor of an alternative hypothesis ($H_1$ or $H_a$).

Think of it like a courtroom trial: the null hypothesis ($H_0$) is the defendant's presumption of innocence. We only reject this presumption if the evidence from our sample data (the p-value) is overwhelmingly strong.

Typical questions addressed by a hypothesis test include:

“Is the population mean $\mu$ equal to a specific value $\mu_0$?”
“Are two population proportions different ($p_1 \neq p_2$)?”
“Is there a relationship between two categorical variables (a $\chi^2$ test)?”

Multiple test families: Z, t, and $\chi^2$ (chi-square) — with more planned.
Modes: Standard Results and Learning Mode (step-by-step derivations).
Input options: Summary data or Raw data (paste or upload CSV).
Outputs: Test statistic, p-value, CI, decision summary, optional chart, effect size, and assumptions checklist.

Select a test family (Z, t, $\chi^2$) and a subtype.
Choose data mode (Summary or Raw) and enter your data.
Set the alternative hypothesis (two-sided, left-tailed, right-tailed) and significance level $\alpha$ (e.g., $0.05$).
Click Run Test to view the test statistic, p-value, CI, and decision summary.
Toggle Show Steps (Learning Mode) to see the step-by-step formulas and intermediate values.

Z-test — One-Sample Mean: known population $\sigma$, tests $\mu$ vs $\mu_0$.
Z-test — One-Sample Proportion: tests $p$ vs $p_0$.
Z-test — Two-Sample Proportions: tests $p_1 - p_2$.
t-test — One-Sample Mean: unknown $\sigma$, tests $\mu$ vs $\mu_0$.
t-test — Two-Sample Mean (Independent): pooled or Welch, tests $\mu_1 - \mu_2$.
t-test — Paired: tests mean of differences $\mu_d$.
Chi-Square — Test of Independence: tests association between two categorical variables.

Z-test — Two-Sample Mean (known $\sigma$’s)
Chi-Square — Goodness of Fit
ANOVA (one-way, possibly two-way later)
Linear Regression (inference on slope, $R^2$, residual diagnostics)

Match your data type and goal to a test:

Single mean ($\mu$): use one-sample t (unknown $\sigma$) or one-sample Z (known $\sigma$).
Difference in means ($\mu_1 - \mu_2$): independent samples → two-sample t; paired data → paired t.
Single proportion ($p$): one-sample proportion Z.
Difference in proportions ($p_1 - p_2$): two-sample proportion Z.
Association of categories: $\chi^2$ independence.

Tip: If you’re unsure, start from the test family selector in the calculator, then pick the subtype that matches your data layout and question.

Use Summary when you already have aggregated values from a study (e.g., $\bar{x}, s, n$ or $x, n$). Use Raw to paste/import your individual observations and let the calculator compute the summaries for you.

Summary Data

Means (Z/t): sample mean $\bar{x}$, sample std dev $s$ (or population $\sigma$ for Z), sample size $n$, and hypothesized mean $\mu_0$.
Proportions: number of successes $x$ and sample size $n$ (or sample proportion $\hat{p}$ and $n$), and hypothesized proportion $p_0$.
Chi-Square: contingency table counts.

Example: "I surveyed 50 people ($n=50$) and found the average height was 175cm ($\bar{x}=175$) with a standard deviation of 10cm ($s=10$). The null hypothesis is that the population mean is 170cm ($\mu_0=170$). Use summary data."

Raw Data

Means (Z/t): one numeric column (one-sample), two numeric columns or one numeric + group label (two-sample), paired columns (paired t).
Proportions: one categorical column (one-sample) or two columns (group + outcome) / two categorical columns (two-sample).
Chi-Square: two categorical columns (variables) or a pre-made table.

Example: "I have a list of 50 student GPAs. I need to test if the mean is different from 3.0. Paste the raw data."

Test statistic: a standardized score ($z$, $t$, or $\chi^2$) that measures how far your sample result is from the null hypothesis value.
p-value: the probability of obtaining a result at least as extreme as your sample, assuming the null hypothesis ($H_0$) is true.
Confidence interval (CI): an estimated range for the population parameter or difference. For example, a $95\%$ CI means we are $95\%$ confident the true population parameter lies within this range.
Decision summary: a clear conclusion to either 'Reject $H_0$' or 'Fail to Reject $H_0$' at the chosen significance level $\alpha$.
Chart: a visual representation of your test statistic and the critical/rejection regions.
Effect size: a measure of the magnitude of the observed effect, independent of sample size.

Interpretation tip: A p-value of $0.03$ means there is a $3\%$ chance of seeing your result (or something more extreme) if the null hypothesis were true. Remember, statistical significance (a small p-value) does not guarantee practical significance—always check the effect size and confidence interval.

Means (t/Z): Cohen’s $d$. $$ d = \frac{\bar{x} - \mu_0}{s} \quad \text{or} \quad \frac{\bar{x}_1 - \bar{x}_2}{s_{\text{pooled}}} $$ Common interpretation:
- Small effect: $d \approx 0.2$
- Medium effect: $d \approx 0.5$
- Large effect: $d \approx 0.8$
Proportions: Cohen’s $h$. This measures the difference between two proportions after an arcsine transformation. $$ h = 2\arcsin\sqrt{p_1} - 2\arcsin\sqrt{p_2} $$
Chi-Square Independence: Cramér’s $V$ or $\phi$ (for $2\times2$ tables). $$ V = \sqrt{\frac{\chi^2}{n \cdot \min(r-1, c-1)}} \quad \phi = \sqrt{\frac{\chi^2}{n}} $$

Use effect sizes alongside p-values and CIs to get a complete picture of your results. Effect sizes capture the magnitude of the effect, while p-values tell you if the effect is statistically significant.

t-tests (means)

Independence: Observations within and between samples are independent.
Normality: The data is approximately normally distributed, or the sample size ($n$) is large enough for the Central Limit Theorem (CLT) to apply ($n \ge 30$).
Two-sample independent: Assumes equal variances (pooled) or unequal variances (Welch).

Z-tests

For mean Z: Population standard deviation $\sigma$ is known; independence; normal data or large $n$ (CLT).
For proportions: Independence; success–failure condition ($np \ge 10$ and $n(1-p) \ge 10$ for the null, or $np_1, n_1(1-p_1)$, etc. for two-sample).

Chi-Square Independence

Independence: Observations are independent.
Count data: Data are counts within categories.
Expected Counts: All expected counts should be at least 5.

The calculator surfaces assumption notes and warnings contextually (e.g., small expected counts, extreme proportions, etc.) to help you check these conditions.

What's the difference between a p-value and a significance level ($\alpha$)? The p-value is the probability of your result under the null hypothesis. The significance level ($\alpha$) is a predetermined threshold that you set before the test to decide if the p-value is small enough to reject the null.
When should I use a two-sided vs. one-sided test? Use a two-sided test if you are testing for any difference (e.g., $\mu \neq \mu_0$). Use a one-sided test if you have a specific directional hypothesis (e.g., $\mu > \mu_0$ or $\mu < \mu_0$). One-sided tests have more power, but should only be used if you can justify the direction *before* seeing the data.
When to use Welch vs. pooled t-test? The Welch t-test is more robust and should be used by default, especially if your sample sizes or variances are unequal. The pooled t-test should only be used when you have strong evidence that the population variances are equal.
What if my data is not normal? For large sample sizes ($n \ge 30$), the Central Limit Theorem allows you to proceed with Z or t-tests. For small samples, consider data transformations or use a non-parametric test (e.g., Wilcoxon, Mann-Whitney U, future roadmap).

Agresti, A. Categorical Data Analysis.
Casella, G., & Berger, R. Statistical Inference.
Wasserman, L. All of Statistics.
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences.
Newcombe, R. G. (1998). Interval estimation for the difference between independent proportions.