Probability and Statistics · t-test and chi-square

Which test fits which question:

question	test	scipy
is the mean equal to some value?	one-sample t	ttest_1samp
do two independent groups differ?	Welch two-sample t	ttest_ind(equal_var=False)
before/after on the same subjects?	paired t	ttest_rel
are two categorical variables related?	χ² independence	chi2_contingency

Welch’s version doesn’t assume equal variances, which makes it a safer default. A paired test removes the between-subject variation, so it’s more powerful when the design really is paired. Chi-square compares the observed table with the counts expected under independence.

Alongside the p-value I also report the effect size (here the mean difference), because a tiny effect can be “significant” with enough data.

import numpy as np
from scipy import stats

rng = np.random.default_rng(7)
group_a = rng.normal(72, 8, 40)       # e.g. scores with method A
group_b = rng.normal(77, 12, 35)      # method B
t, p = stats.ttest_ind(group_a, group_b, equal_var=False)
print(f"Welch t={t:.3f}, p={p:.4f}, mean diff={group_b.mean() - group_a.mean():.2f}")

before = rng.normal(60, 10, 25)
after = before + rng.normal(3, 4, 25)  # same students, small improvement
print("independent test p =", round(stats.ttest_ind(after, before).pvalue, 4),
      "| paired test p =", round(stats.ttest_rel(after, before).pvalue, 6))

table = np.array([[30, 10],    # rows: branch ICT, CSE ; cols: uses Linux yes/no
                  [18, 22]])
chi2, p, dof, expected = stats.chi2_contingency(table)
print(f"chi2={chi2:.3f}, dof={dof}, p={p:.4f}")
print("expected under independence:\n", expected.round(1))

Output:

Welch t=-3.853, p=0.0003, mean diff=8.31
independent test p = 0.3794 | paired test p = 0.001915
chi2=6.302, dof=1, p=0.0121
expected under independence:
 [[24. 16.]
 [24. 16.]]