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name: statistical-analysis description: Probability, distributions, hypothesis testing, and statistical inference. Use for A/B testing, experimental design, or statistical validation. sasmp_version: "1.3.0" bonded_agent: 02-mathematics-statistics bond_type: PRIMARY_BOND

Statistical Analysis

Apply statistical methods to understand data and validate findings.

Quick Start

from scipy import stats
import numpy as np

# Descriptive statistics
data = np.array([1, 2, 3, 4, 5])
print(f"Mean: {np.mean(data)}")
print(f"Std: {np.std(data)}")

# Hypothesis testing
group1 = [23, 25, 27, 29, 31]
group2 = [20, 22, 24, 26, 28]
t_stat, p_value = stats.ttest_ind(group1, group2)
print(f"P-value: {p_value}")

Core Tests

T-Test (Compare Means)

# One-sample: Compare to population mean
stats.ttest_1samp(data, 100)

# Two-sample: Compare two groups
stats.ttest_ind(group1, group2)

# Paired: Before/after comparison
stats.ttest_rel(before, after)

Chi-Square (Categorical Data)

from scipy.stats import chi2_contingency

observed = np.array([[10, 20], [15, 25]])
chi2, p_value, dof, expected = chi2_contingency(observed)

ANOVA (Multiple Groups)

f_stat, p_value = stats.f_oneway(group1, group2, group3)

Confidence Intervals

from scipy import stats

confidence_level = 0.95
mean = np.mean(data)
se = stats.sem(data)
ci = stats.t.interval(confidence_level, len(data)-1, mean, se)

print(f"95% CI: [{ci[0]:.2f}, {ci[1]:.2f}]")

Correlation

# Pearson (linear)
r, p_value = stats.pearsonr(x, y)

# Spearman (rank-based)
rho, p_value = stats.spearmanr(x, y)

Distributions

# Normal
x = np.linspace(-3, 3, 100)
pdf = stats.norm.pdf(x, loc=0, scale=1)

# Sampling
samples = np.random.normal(0, 1, 1000)

# Test normality
stat, p_value = stats.shapiro(data)

A/B Testing Framework

def ab_test(control, treatment, alpha=0.05):
    """
    Run A/B test with statistical significance

    Returns: significant (bool), p_value (float)
    """
    t_stat, p_value = stats.ttest_ind(control, treatment)

    significant = p_value < alpha
    improvement = (np.mean(treatment) - np.mean(control)) / np.mean(control) * 100

    return {
        'significant': significant,
        'p_value': p_value,
        'improvement': f"{improvement:.2f}%"
    }

Interpretation

P-value < 0.05: Reject null hypothesis (statistically significant)

P-value >= 0.05: Fail to reject null (not significant)

Common Pitfalls

• Multiple testing without correction
• Small sample sizes
• Ignoring assumptions (normality, independence)
• Confusing correlation with causation
• p-hacking (searching for significance)

Troubleshooting

Common Issues

Problem: Non-normal data for t-test

# Check normality first
stat, p = stats.shapiro(data)
if p < 0.05:
    # Use non-parametric alternative
    stat, p = stats.mannwhitneyu(group1, group2)  # Instead of ttest_ind

Problem: Multiple comparisons inflating false positives

from statsmodels.stats.multitest import multipletests

# Apply Bonferroni correction
p_values = [0.01, 0.03, 0.04, 0.02, 0.06]
rejected, p_adjusted, _, _ = multipletests(p_values, method='bonferroni')

Problem: Underpowered study (sample too small)

from statsmodels.stats.power import TTestIndPower

# Calculate required sample size
power_analysis = TTestIndPower()
sample_size = power_analysis.solve_power(
    effect_size=0.5,  # Medium effect (Cohen's d)
    power=0.8,        # 80% power
    alpha=0.05        # 5% significance
)
print(f"Required n per group: {sample_size:.0f}")

Problem: Heterogeneous variances

# Check with Levene's test
stat, p = stats.levene(group1, group2)
if p < 0.05:
    # Use Welch's t-test (default in scipy)
    t, p = stats.ttest_ind(group1, group2, equal_var=False)

Problem: Outliers affecting results

from scipy.stats import zscore

# Detect outliers (|z| > 3)
z_scores = np.abs(zscore(data))
clean_data = data[z_scores < 3]

# Or use robust statistics
median = np.median(data)
mad = np.median(np.abs(data - median))  # Median Absolute Deviation

Debug Checklist

• Check sample size adequacy (power analysis)
• Test normality assumption (Shapiro-Wilk)
• Test homogeneity of variance (Levene's)
• Check for outliers (z-scores, IQR)
• Apply multiple testing correction if needed
• Report effect sizes, not just p-values