Parametric Tests

Parametric tests assume (approximately) normal data and continuous measurements.
They are among the most commonly used statistical tests in practice.

One-Sample t-Test

Question: Does the mean of a sample differ from a specific value?

Used for:

• Comparing a sample to a known benchmark
• Checking whether a measurement meets a target
• Testing changes from a historical or expected mean

Example

from scipy import stats​sample = [5.2, 5.5, 4.9, 5.1, 5.0]​t_stat, p_value = stats.ttest_1samp(sample, popmean=5.0)print(t_stat, p_value)

Interpretation

• p < 0.05 → sample mean is significantly different from 5.0
• p ≥ 0.05 → no significant difference

Independent Two-Sample t-Test

Question: Do two independent groups have different means?

Used for:

• A/B tests
• Male vs female comparisons
• Treatment vs control groups

Example

group1 = [4.9, 5.1, 5.0, 5.3]group2 = [5.6, 5.7, 5.4, 5.8]​t_stat, p_value = stats.ttest_ind(group1, group2)print(t_stat, p_value)

Welch’s t-test (if variances differ)

t_stat, p_value = stats.ttest_ind(group1, group2, equal_var=False)

Interpretation

• p < 0.05 → groups differ significantly
• p ≥ 0.05 → no evidence of difference

Paired t-Test

Question: Do two related measurements differ?

Used when:

• Measurements are on the same subjects
• Before/after experiments
• Paired samples (e.g. left hand vs right hand)

Example

before = [100, 102, 98, 105]after  = [103, 107, 101, 110]​t_stat, p_value = stats.ttest_rel(before, after)print(t_stat, p_value)

Interpretation

• p < 0.05 → the before/after difference is significant
• Good for clinical improvements, performance upgrades, etc.

One-Way ANOVA (3+ groups)

Question: Do 3 or more groups differ?

Used for:

• Comparing multiple product versions
• Multiple study groups
• Any ≥3 independent samples

Example

group_a = [5.1, 5.3, 5.2]group_b = [6.2, 6.1, 6.3]group_c = [4.9, 5.0, 4.8]​f_stat, p_value = stats.f_oneway(group_a, group_b, group_c)print(f_stat, p_value)

Interpretation

• p < 0.05 → at least one group differs
• Does not tell which groups differ (requires post-hoc tests)

Post-hoc example using Statsmodels

(Not SciPy, but commonly paired.)

pip install statsmodels

import statsmodels.stats.multicomp as mcimport pandas as pd​data = pd.DataFrame({    'value': group_a + group_b + group_c,    'group': ['A']*3 + ['B']*3 + ['C']*3})​comp = mc.MultiComparison(data['value'], data['group'])post_hoc = comp.tukeyhsd()print(post_hoc)

wo-Way ANOVA (Not in SciPy)

SciPy only supports one-way ANOVA directly.

For factorial designs (e.g., "treatment × gender"), use Statsmodels:

import statsmodels.api as smfrom statsmodels.formula.api import ols​df = pd.DataFrame(...)​model = ols("score ~ C(treatment) * C(gender)", data=df).fit()anova_table = sm.stats.anova_lm(model, typ=2)print(anova_table)

Z-Tests (Statsmodels, not SciPy)

SciPy does not include z-tests, but Statsmodels does.

Two-sample z-test

from statsmodels.stats.weightstats import ztest​z_stat, p_value = ztest(group1, group2)print(z_stat, p_value)

Use for:

• Large samples (n > 30)
• Known population variance (rare)

One-sample z-test

z_stat, p_value = ztest(group1, value=5.0)print(z_stat, p_value)

Assumptions Checklist (Practical)

Parametric tests work well if:

• Data are roughly normal
• Measurements are continuous
• Groups have similar variances (for t-tests & ANOVA)
• Sample size is reasonably large (CLT helps)

If not:

• Use non-parametric tests

Quick Summary Table (Parametric Tests)