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Statistical analysis fundamentals including hypothesis testing, regression analysis, ANOVA, and probability distributions for data analysis.

ffsshhttiikk By ffsshhttiikk schedule Updated 2/28/2026
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name: statistics-math description: Statistical analysis fundamentals including hypothesis testing, regression analysis, ANOVA, and probability distributions for data analysis. category: mathematics tags: - mathematics - statistics - hypothesis-testing - regression - anova - data-analysis - statistical-inference difficulty: intermediate author: neuralblitz

Statistics

What I do

I provide comprehensive expertise in statistical analysis, the science of collecting, analyzing, interpreting, and presenting data. I enable you to perform descriptive statistics, conduct hypothesis tests, build regression models, analyze variance, estimate confidence intervals, and make data-driven decisions. My knowledge spans from basic statistical measures to advanced techniques like multiple regression and ANOVA, essential for scientific research, data science, business analytics, and evidence-based decision making.

When to use me

Use statistics when you need to: analyze experimental results and determine significance, build predictive models from observational data, compare groups or treatments for differences, estimate population parameters from samples, detect patterns and relationships in datasets, perform A/B testing for product decisions, conduct quality control analysis, or communicate uncertainty in findings to stakeholders.

Core Concepts

  • Descriptive Statistics: Summarizing data through measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation, range).
  • Sampling and Estimation: Methods for selecting representative samples and estimating population parameters from sample statistics.
  • Confidence Intervals: Ranges providing plausible values for unknown parameters with specified confidence levels.
  • Hypothesis Testing: Formal procedures for making decisions about population parameters using sample data.
  • Regression Analysis: Modeling relationships between dependent and independent variables for prediction and inference.
  • Analysis of Variance (ANOVA): Statistical methods for comparing means across multiple groups simultaneously.
  • Correlation and Association: Measures quantifying the strength and direction of relationships between variables.
  • Statistical Power: The probability of detecting an effect when it truly exists, influencing experimental design.
  • Residual Analysis: Examining model residuals to validate regression assumptions and detect violations.
  • Multiple Comparisons: Adjusting significance levels when conducting multiple statistical tests to control error rates.

Code Examples

Descriptive Statistics and Data Summaries

import numpy as np
import pandas as pd
from scipy import stats

# Generate sample data
np.random.seed(42)
data = np.random.normal(loc=50, scale=10, size=1000)
data_with_outliers = np.concatenate([data, [100, 110, 120]])

# Central tendency measures
mean_val = np.mean(data)
median_val = np.median(data)
mode_result = stats.mode(data, keepdims=True)

print(f"Mean: {mean_val:.2f}")
print(f"Median: {median_val:.2f}")
print(f"Mode: {mode_result.mode[0]:.2f}")

# Dispersion measures
variance_val = np.var(data, ddof=1)
std_val = np.std(data, ddof=1)
iqr_val = np.percentile(data, 75) - np.percentile(data, 25)
range_val = np.max(data) - np.min(data)

print(f"Variance: {variance_val:.2f}")
print(f"Standard Deviation: {std_val:.2f}")
print(f"IQR: {iqr_val:.2f}")
print(f"Range: {range_val:.2f}")

# Shape measures
skewness = stats.skew(data)
kurtosis = stats.kurtosis(data)
print(f"Skewness: {skewness:.4f} (0 = symmetric)")
print(f"Kurtosis: {kurtosis:.4f} (0 = normal)")

# Five-number summary
five_num = np.percentile(data, [0, 25, 50, 75, 100])
print(f"Five-number summary: {five_num}")

# Using pandas for comprehensive summary
df = pd.DataFrame({'values': data})
print(df.describe())

Hypothesis Testing

import numpy as np
from scipy import stats

# One-sample t-test
np.random.seed(42)
sample = np.random.normal(loc=102, scale=15, size=30)  # Population mean = 100
pop_mean = 100

# Two-sided t-test
t_stat, p_value = stats.ttest_1samp(sample, pop_mean)
print(f"One-sample t-test:")
print(f"  t-statistic: {t_stat:.4f}")
print(f"  p-value: {p_value:.4f}")
print(f"  Reject H0 at α=0.05: {p_value < 0.05}")

# Two-sample t-test (independent samples)
np.random.seed(42)
group1 = np.random.normal(loc=50, scale=10, size=30)
group2 = np.random.normal(loc=55, scale=10, size=30)

t_stat2, p_value2, dof = stats.ttest_ind(group1, group2, equal_var=True)
print(f"\nTwo-sample t-test:")
print(f"  t-statistic: {t_stat2:.4f}")
print(f"  p-value: {p_value2:.4f}")
print(f"  Degrees of freedom: {dof}")

# Welch's t-test (unequal variances)
t_stat_welch, p_value_welch = stats.ttest_ind(group1, group2, equal_var=False)
print(f"\nWelch's t-test:")
print(f"  t-statistic: {t_stat_welch:.4f}")
print(f"  p-value: {p_value_welch:.4f}")

# Paired t-test
pre_scores = [85, 90, 78, 92, 88, 76, 95, 89]
post_scores = [89, 92, 82, 96, 90, 80, 98, 93]
t_paired, p_paired = stats.ttest_rel(pre_scores, post_scores)
print(f"\nPaired t-test:")
print(f"  t-statistic: {t_paired:.4f}")
print(f"  p-value: {p_paired:.4f}")

# Chi-square test for independence
contingency_table = np.array([[30, 20], [20, 30]])
chi2, p_chi, dof_chi, expected = stats.chi2_contingency(contingency_table)
print(f"\nChi-square test:")
print(f"  χ² statistic: {chi2:.4f}")
print(f"  p-value: {p_chi:.4f}")

# Effect size (Cohen's d)
pooled_std = np.sqrt(((len(group1)-1)*np.var(group1, ddof=1) + 
                       (len(group2)-1)*np.var(group2, ddof=1)) / 
                      (len(group1) + len(group2) - 2))
cohens_d = (np.mean(group1) - np.mean(group2)) / pooled_std
print(f"\nEffect size (Cohen's d): {cohens_d:.4f}")

Confidence Intervals

import numpy as np
from scipy import stats
from statsmodels.stats.proportion import proportion_confint

# One-sample confidence interval for mean
np.random.seed(42)
sample = np.random.normal(loc=100, scale=15, size=50)
sample_mean = np.mean(sample)
sample_std = np.std(sample, ddof=1)
n = len(sample)
confidence_level = 0.95

t_critical = stats.t.ppf((1 + confidence_level) / 2, df=n-1)
margin_of_error = t_critical * sample_std / np.sqrt(n)
ci_lower = sample_mean - margin_of_error
ci_upper = sample_mean + margin_of_error

print(f"95% Confidence Interval for mean:")
print(f"  Sample mean: {sample_mean:.2f}")
print(f"  CI: [{ci_lower:.2f}, {ci_upper:.2f}]")

# Using bootstrap for confidence intervals
def bootstrap_ci(data, statistic=np.mean, n_bootstrap=10000, confidence_level=0.95):
    np.random.seed(42)
    bootstrap_stats = []
    n = len(data)
    
    for _ in range(n_bootstrap):
        sample_boot = np.random.choice(data, size=n, replace=True)
        bootstrap_stats.append(statistic(sample_boot))
    
    alpha = (1 - confidence_level) / 2
    return np.percentile(bootstrap_stats, [alpha*100, (1-alpha)*100])

ci_bootstrap = bootstrap_ci(sample)
print(f"Bootstrap 95% CI: [{ci_bootstrap[0]:.2f}, {ci_bootstrap[1]:.2f}]")

# Confidence interval for proportion
n_successes = 45
n_trials = 100
ci_prop = proportion_confint(n_successes, n_trials, method='wilson')
print(f"\n95% CI for proportion (45/100): [{ci_prop[0]:.4f}, {ci_prop[1]:.4f}]")

# Difference of means confidence interval
np.random.seed(42)
sample1 = np.random.normal(loc=100, scale=15, size=30)
sample2 = np.random.normal(loc=105, scale=15, size=35)

mean_diff = np.mean(sample1) - np.mean(sample2)
se_diff = np.sqrt(np.var(sample1, ddof=1)/len(sample1) + np.var(sample2, ddof=1)/len(sample2))
dof = len(sample1) + len(sample2) - 2
t_crit_diff = stats.t.ppf((1 + confidence_level) / 2, df=dof)
ci_diff_lower = mean_diff - t_crit_diff * se_diff
ci_diff_upper = mean_diff + t_crit_diff * se_diff

print(f"\n95% CI for difference of means: [{ci_diff_lower:.2f}, {ci_diff_upper:.2f}]")

Regression Analysis

import numpy as np
import pandas as pd
from scipy import stats
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score, mean_squared_error

# Simple linear regression
np.random.seed(42)
X = np.linspace(0, 10, 100)
true_slope = 2.5
true_intercept = 1.0
noise = np.random.normal(0, 2, 100)
y = true_slope * X + true_intercept + noise

# Using scipy for simple regression
slope, intercept, r_value, p_value, std_err = stats.linregress(X, y)

print("Simple Linear Regression (scipy):")
print(f"  Slope: {slope:.4f} (true: {true_slope})")
print(f"  Intercept: {intercept:.4f} (true: {true_intercept})")
print(f"  R-squared: {r_value**2:.4f}")
print(f"  p-value: {p_value:.2e}")
print(f"  Standard error: {std_err:.4f}")

# Multiple linear regression with sklearn
np.random.seed(42)
n_samples = 100
X1 = np.random.normal(0, 1, n_samples)
X2 = np.random.normal(0, 1, n_samples)
X3 = np.random.normal(0, 1, n_samples)
noise = np.random.normal(0, 0.5, n_samples)
y = 1 + 2*X1 + 3*X2 + 0.5*X3 + noise

X_multi = np.column_stack([X1, X2, X3])
model = LinearRegression()
model.fit(X_multi, y)

print("\nMultiple Linear Regression (sklearn):")
print(f"  Intercept: {model.intercept_:.4f}")
print(f"  Coefficients: {model.coef_}")
print(f"  R-squared: {model.score(X_multi, y):.4f}")

# Predictions and residuals
y_pred = model.predict(X_multi)
residuals = y - y_pred

# Residual analysis
print(f"\nResidual Statistics:")
print(f"  Mean of residuals: {np.mean(residuals):.6f} (should be ~0)")
print(f"  Std of residuals: {np.std(residuals):.4f}")

# Polynomial regression
X_poly = np.linspace(-3, 3, 50)
y_poly = 0.5 * X_poly**3 - 2 * X_poly**2 + X_poly + np.random.normal(0, 2, 50)

from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline

poly_model = make_pipeline(PolynomialFeatures(degree=3), LinearRegression())
poly_model.fit(X_poly.reshape(-1, 1), y_poly)
y_poly_pred = poly_model.predict(X_poly.reshape(-1, 1))

print(f"\nPolynomial Regression (degree=3):")
print(f"  R-squared: {r2_score(y_poly, y_poly_pred):.4f}")

ANOVA and Group Comparisons

import numpy as np
from scipy import stats
from scipy.stats import f_oneway, kruskal

# One-way ANOVA
np.random.seed(42)
group1 = np.random.normal(loc=70, scale=10, size=30)
group2 = np.random.normal(loc=75, scale=10, size=30)
group3 = np.random.normal(loc=80, scale=10, size=30)

f_stat, p_anova = f_oneway(group1, group2, group3)

print("One-way ANOVA:")
print(f"  F-statistic: {f_stat:.4f}")
print(f"  p-value: {p_anova:.4f}")
print(f"  Significant difference at α=0.05: {p_anova < 0.05}")

# Effect size (eta-squared)
ss_between = sum(len(g) * (np.mean(g) - np.mean(np.concatenate([group1, group2, group3])))**2 
                  for g in [group1, group2, group3])
ss_total = sum((np.concatenate([group1, group2, group3]) - np.mean(np.concatenate([group1, group2, group3])))**2)
eta_squared = ss_between / ss_total
print(f"  Effect size (η²): {eta_squared:.4f}")

# Post-hoc test (Tukey's HSD)
from scipy.stats import tukey_hsd

result = tukey_hsd(group1, group2, group3)
print(f"\nTukey's HSD results:")
print(f"  group1 vs group2: p={result.pvalue[0,1]:.4f}")
print(f"  group1 vs group3: p={result.pvalue[0,2]:.4f}")
print(f"  group2 vs group3: p={result.pvalue[1,2]:.4f}")

# Kruskal-Wallis test (non-parametric alternative)
h_stat, p_kruskal = kruskal(group1, group2, group3)
print(f"\nKruskal-Wallis test:")
print(f"  H-statistic: {h_stat:.4f}")
print(f"  p-value: {p_kruskal:.4f}")

# Two-way ANOVA (simplified using regression)
np.random.seed(42)
factor_a = np.repeat([0, 1], 50)
factor_b = np.tile(np.repeat([0, 1], 25), 2)
interaction = factor_a * factor_b
y_two_way = 50 + 5*factor_a + 3*factor_b + 2*interaction + np.random.normal(0, 5, 100)

# Using statsmodels for proper ANOVA
import statsmodels.api as sm
from statsmodels.formula.api import ols

data_anova = pd.DataFrame({
    'y': y_two_way,
    'A': factor_a,
    'B': factor_b
})
model_anova = ols('y ~ C(A) + C(B) + C(A):C(B)', data=data_anova).fit()
anova_table = sm.stats.anova_lm(model_anova, typ=2)
print("\nTwo-way ANOVA table:")
print(anova_table)

Best Practices

  • Always check assumptions (normality, equal variances, independence) before applying statistical tests and use robust alternatives when violations are detected.
  • Report effect sizes alongside p-values to communicate practical significance, not just statistical significance.
  • Use confidence intervals instead of or alongside hypothesis tests to convey uncertainty in estimates.
  • Adjust for multiple comparisons using Bonferroni, Holm, or FDR methods to control family-wise error rate.
  • Distinguish between correlation and causation; correlation does not imply causal relationships.
  • Validate regression models through residual analysis, checking for heteroscedasticity and non-linearity.
  • Consider statistical power during experimental design to ensure adequate sample sizes for detecting meaningful effects.
  • Use bootstrapping for confidence intervals when distributional assumptions cannot be met.
  • Report data preprocessing steps and any data exclusions that might affect generalizability.
  • Use appropriate visualizations (box plots, QQ plots, residual plots) to complement statistical analyses.
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