Inferential Statistics

The Taste Tester & The Election Pollster

A soup chef is preparing a 50-litre pot of dal for a wedding feast of 500 guests. Does she pour every drop into bowls and serve one to each guest just to check the salt? Of course not. She picks up a single ladle — maybe 100 ml — stirs well, tastes it, adjusts the seasoning, and confidently serves the entire pot.

One ladle. 500 guests. That is inferential statistics in its purest form.

Every day, decisions worth billions of rupees — drug approvals, crop yield forecasts, product quality checks, credit-risk models — are made not by measuring everyone and everything, but by measuring a few and reasoning about the rest. That is the job of inferential statistics.

What Is Inferential Statistics?

Statistics splits into two broad branches. Descriptive statistics summarises and describes the data you already have — means, medians, charts. Inferential statistics goes further: it uses a sample to draw conclusions — inferences — about a larger population that you could not fully measure.

Population vs Sample

Before you can infer anything, you must be crystal clear about what you are measuring and what you are trying to say. This is the population-vs-sample distinction — one of the most important conceptual divides in all of statistics.

Sampling Techniques — How to Pick a Sample

Choosing who goes into your sample is one of the most consequential decisions in any study. The wrong technique introduces sampling bias — a systematic distortion that no amount of clever statistics can later fix. There are five major techniques, each suited to different situations.

1. Simple Random Sampling

Every member of the population has an equal and independent probability of being selected. Think of a lottery draw — each ticket has the same chance. In Python, this is random.sample() or df.sample(n=100). It is the gold standard for small, homogeneous populations but becomes impractical when the population is geographically spread or has important sub-groups.

2. Stratified Sampling

The population is first divided into non-overlapping sub-groups (strata) based on a key characteristic (income level, age group, state), and random samples are drawn from each stratum proportionally. An NSSO survey of Indian household income uses stratified sampling — it ensures rural, semi-urban, and urban households are all represented, not accidentally dominated by one group.

3. Cluster Sampling

The population is divided into natural clusters (villages, schools, city blocks), a random subset of clusters is selected, and every member of the chosen clusters is surveyed. This is far cheaper and faster than travelling to survey individuals scattered across the country. A government health survey might randomly pick 50 villages and survey all households in those villages. The trade-off: if the selected clusters happen to be unusually similar, the sample may not be representative.

4. Systematic Sampling

Choose a random starting point, then select every k-th element from an ordered list, where k = N / n (the sampling interval). For example, to sample 200 patients from a hospital register of 4,000: k = 4000 / 200 = 20. Pick a random start between 1 and 20, then take every 20th patient. Simple, fast, and nearly as good as random — unless there is a periodic pattern in the list (e.g. every 20th patient is always a Monday morning emergency case).

5. Convenience Sampling

Select whoever is easiest to reach — students in the researcher's classroom, followers on social media, shoppers at a nearby mall. It is fast and cheap but almost always biased. It cannot support valid statistical inference about any broader population. Use it only for early exploratory pilots or qualitative insights, never for drawing quantitative conclusions.

Sampling Distribution — The Heart of Inference

Here is where the magic happens. Imagine you want to estimate the average monthly salary of all 500 million workers in India. You take a random sample of 100 workers and compute the sample mean x̄. You get ₹32,500.

But what if someone else took a different random sample of 100 workers? They would get a slightly different x̄ — maybe ₹31,800. A third researcher gets ₹33,100. Every possible sample of size 100 produces its own x̄. The distribution of all these sample means is called the Sampling Distribution of the Mean.

The Central Limit Theorem in Action

The CLT is arguably the most powerful theorem in all of statistics. It says that no matter how strange, skewed, or non-normal your population distribution looks, the sampling distribution of the mean will become approximately normal as your sample size grows — typically by n ≥ 30.

Python Implementation

Sampling Techniques with Pandas & NumPy

import pandas as pd
import numpy as np

np.random.seed(42)

# Simulated population: 10,000 workers with income data
population = pd.DataFrame({
    'worker_id': range(1, 10001),
    'income':    np.random.lognormal(mean=10.5, sigma=0.5, size=10000),
    'region':    np.random.choice(['Rural', 'Urban', 'Metro'], size=10000,
                                   p=[0.60, 0.30, 0.10]),
    'sector':    np.random.choice(['Agriculture', 'Manufacturing', 'Services'],
                                   size=10000, p=[0.45, 0.25, 0.30])
})

print(f"Population mean income: ₹{population['income'].mean():,.0f}")
print(f"Population size N: {len(population)}")

# ── 1. Simple Random Sampling ──────────────────────────────────
srs = population.sample(n=200, random_state=42)
print(f"\nSRS Sample mean:  ₹{srs['income'].mean():,.0f}")
print(f"SRS Sample size n = {len(srs)}")

# ── 2. Stratified Sampling (by region, proportional) ───────────
strata_sizes = {'Rural': 120, 'Urban': 60, 'Metro': 20}   # 60/30/10 split

stratified = pd.concat([
    population[population['region'] == region].sample(n=n, random_state=42)
    for region, n in strata_sizes.items()
])
print(f"\nStratified Sample mean:  ₹{stratified['income'].mean():,.0f}")
print(f"Region distribution:\n{stratified['region'].value_counts()}")

# ── 3. Systematic Sampling ──────────────────────────────────────
n_sample  = 200
k         = len(population) // n_sample    # sampling interval = 50
start     = np.random.randint(0, k)        # random start between 0 and 49

systematic_idx = range(start, len(population), k)
systematic = population.iloc[systematic_idx]
print(f"\nSystematic Sample mean:  ₹{systematic['income'].mean():,.0f}")
print(f"Sampling interval k = {k}")

Demonstrating the Sampling Distribution & CLT

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(0)

# Population: exponential (heavily right-skewed)
population_data = np.random.exponential(scale=5, size=100_000)
print(f"Population mean: {population_data.mean():.4f}")    # ~5.0
print(f"Population std:  {population_data.std():.4f}")     # ~5.0

# Draw 10,000 samples at different sizes and record sample means
for n in [2, 10, 30, 100]:
    sample_means = [
        np.random.choice(population_data, size=n, replace=False).mean()
        for _ in range(10_000)
    ]
    se_theoretical = population_data.std() / np.sqrt(n)
    se_observed    = np.std(sample_means)

    print(f"\nn = {n:>3d}  |  "
          f"Mean of x̄ = {np.mean(sample_means):.4f}  |  "
          f"SE theoretical = {se_theoretical:.4f}  |  "
          f"SE observed = {se_observed:.4f}")

# Output (approximate):
# n =   2  |  Mean of x̄ = 4.9982  |  SE theoretical = 3.5355  |  SE observed = 3.5101
# n =  10  |  Mean of x̄ = 4.9998  |  SE theoretical = 1.5811  |  SE observed = 1.5693
# n =  30  |  Mean of x̄ = 5.0003  |  SE theoretical = 0.9129  |  SE observed = 0.9041
# n = 100  |  Mean of x̄ = 5.0001  |  SE theoretical = 0.5000  |  SE observed = 0.4987

Calculating Standard Error and Confidence Interval

import numpy as np
from scipy import stats

# Sample data: monthly income of 100 workers (₹)
sample = np.random.lognormal(mean=10.5, sigma=0.5, size=100)

x_bar = np.mean(sample)
s     = np.std(sample, ddof=1)
n     = len(sample)
se    = s / np.sqrt(n)                         # Standard Error

# 95% Confidence Interval using t-distribution (n < 1000 rule of thumb)
ci = stats.t.interval(confidence=0.95, df=n-1, loc=x_bar, scale=se)

print(f"Sample mean (x̄):    ₹{x_bar:,.2f}")
print(f"Standard deviation:  ₹{s:,.2f}")
print(f"Standard Error (SE): ₹{se:,.2f}")
print(f"95% CI: (₹{ci[0]:,.2f},  ₹{ci[1]:,.2f})")
# Interpretation: We are 95% confident the true population mean income
# lies between these two values.

Key Concepts — Comparison Table

Golden Rules