Sampling & the central limit theorem in practice

You almost never measure a whole population. A bank studies 2,000 loan files, not 20 lakh; a survey firm calls 1,500 voters, not a whole state. Statistics is the art of learning about the population from a well-chosen sample - and knowing how wrong you might be.

Population, sample, statistic

Term	Meaning	Example
Population	Everyone you care about	All 3.2 lakh customers of a sweets chain
Parameter	A true population value (usually unknown)	True mean bill value, mu
Sample	The subset you actually observe	500 randomly chosen bills
Statistic	A value computed from the sample	Sample mean, x-bar = ₹412

Sampling methods

• Simple random: every unit has an equal chance - df.sample(n=500, random_state=42).
• Stratified: sample within groups (e.g. each city) so small groups are not missed - df.groupby("city").sample(frac=0.05).
• Cluster: pick whole groups (e.g. 20 random stores) and survey everyone inside - cheaper, but less precise.
• Systematic: every k-th invoice - fine unless the list has a repeating pattern.

Sampling bias beats sample size

A huge biased sample is worse than a small random one. Classic traps:

• Selection bias: an online survey misses customers who never use the app.
• Survivorship bias: studying only shops that are still open hides why others failed.
• Non-response bias: unhappy customers skip feedback forms.

The central limit theorem (CLT)

The CLT says: if you take many random samples of size n and compute each sample's mean, those means form an approximately normal distribution centred on the population mean, with spread equal to the standard error:

SE = sigma / sqrt(n)

This holds even when the underlying data is skewed, as long as n is reasonably large (30+ is a common rule of thumb; heavier skew needs more).

See it with a simulation

Bill values are strongly right-skewed - most bills are small, a few wedding orders are huge.

import numpy as np

rng = np.random.default_rng(0)
population = rng.exponential(scale=400, size=320_000)    # skewed bill values in INR
print(f"Population mean: {population.mean():.0f}, sd: {population.std():.0f}")

for n in (5, 30, 200):
    means = np.array([rng.choice(population, n).mean() for _ in range(5_000)])
    print(f"n={n:>3}  mean of means={means.mean():.0f}  "
          f"sd of means={means.std():.1f}  theory={population.std() / np.sqrt(n):.1f}")

Plot a histogram of means for each n: at n=5 it is still skewed, at n=30 it looks bell-shaped, and at n=200 it is narrow and symmetric. The observed spread matches sigma/sqrt(n) closely.

Why this matters

• It justifies confidence intervals and t-tests on non-normal business data.
• Precision grows with the square root of n: to halve the standard error you need four times the sample.
• It applies to means (and proportions), not to individual values - a single bill can still be ₹25,000.

Common mistakes

• Believing a large sample automatically fixes bias.
• Applying the CLT to tiny samples of extremely skewed or outlier-heavy data.
• Confusing standard deviation (spread of data) with standard error (spread of an estimate).