Understanding Descriptive vs Inferential Statistics: A Complete Guide for Beginners

📗 Chapter 3: Inferential Statistics – Making Predictions and Testing Hypotheses

Draw Conclusions from Data, Test Assumptions, and Power Your Decisions with Confidence

🧠 Introduction

While descriptive statistics help you summarize what’s already in the data, inferential statistics help you do something much more powerful: make predictions, draw conclusions, and test theories about a larger population — even when you only have a small sample.

Inferential statistics bridges the gap between what we know and what we want to know.

It’s the backbone of:

• Political polling
• A/B testing in marketing
• Clinical trial decisions
• Social science experiments
• Machine learning model validation

In this chapter, we’ll cover:

• The core concepts of sampling and population inference
• Confidence intervals and standard error
• Hypothesis testing (null, alternative, p-value)
• Common statistical tests (t-test, chi-square, ANOVA)
• Regression and correlation basics

Let’s start making sense of uncertainty — statistically.

📘 Section 1: Population vs. Sample

🧩 Definitions

📌 Example

• You want to know the average height of adults in a country (population).
• You survey 1,000 adults (sample).
• The sample mean becomes your estimate of the population mean.

📘 Section 2: Confidence Intervals

A confidence interval is a range of values we believe, with a certain degree of confidence, contains the true population parameter.

✅ Formula (for mean):

CI = x̄ ± z * (σ/√n)

💻 Code Example:

python

import numpy as np

import scipy.stats as stats

data = np.random.normal(loc=70, scale=10, size=100)

mean = np.mean(data)

sem = stats.sem(data)

confidence = 0.95

interval = stats.t.interval(confidence, len(data)-1, loc=mean, scale=sem)

print(f"95% Confidence Interval: {interval}")

📘 Section 3: Hypothesis Testing Basics

🔍 Goal:

To test an assumption (hypothesis) about a population parameter.

🧪 Steps in Hypothesis Testing:

✅ Definitions

📘 Section 4: t-Tests – Comparing Means

📍 Use when:

• You’re comparing the means of two groups
• Sample size is small or population SD is unknown

💻 Code Example:

python

group1 = np.random.normal(75, 8, 50)

group2 = np.random.normal(70, 10, 50)

t_stat, p_val = stats.ttest_ind(group1, group2)

print("t-statistic:", t_stat)

print("p-value:", p_val)

📊 Interpretation:

If p-value < 0.05 → Reject H₀ → Groups are significantly different.

📘 Section 5: Chi-Square Test – Categorical Data

📍 Use when:

• You want to test the association between two categorical variables

💻 Code Example:

python

from scipy.stats import chi2_contingency

import pandas as pd

# Contingency Table

data = [[20, 30],

        [25, 25]]

chi2, p, dof, expected = chi2_contingency(data)

print("Chi-Square Statistic:", chi2)

print("p-value:", p)

📘 Section 6: ANOVA – Comparing Multiple Means

📍 Use when:

• Comparing means across 3 or more groups

💻 Code Example:

python

group1 = np.random.normal(72, 6, 50)

group2 = np.random.normal(75, 7, 50)

group3 = np.random.normal(78, 6, 50)

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

print("F-statistic:", f_stat)

print("p-value:", p_val)

📘 Section 7: Correlation & Linear Regression (Basics)

✅ Correlation

Measures strength and direction of linear relationship (Pearson's r)

python

import seaborn as sns

tips = sns.load_dataset("tips")

corr = tips['total_bill'].corr(tips['tip'])

print("Correlation:", corr)

✅ Simple Linear Regression

python

from sklearn.linear_model import LinearRegression

X = tips[['total_bill']]

y = tips['tip']

model = LinearRegression()

model.fit(X, y)

print("Slope:", model.coef_[0])

print("Intercept:", model.intercept_)

📋 Section 8: Summary Table