Samples, the central limit theorem and confidence intervals

Why samples?

An election survey cannot ask 97 crore voters. A factory cannot open every biscuit packet to check its weight. So we take a sample, a smaller group, and use it to estimate a number about the whole population. The question is: how far off might our estimate be?

Good and bad samples

A random sample gives every member the same chance of being picked. A biased sample does not. An online poll about internet speeds only reaches people who are online. A survey done only in Mumbai cannot speak for all of Maharashtra. No formula can fix a biased sample, so how you collect data matters more than any calculation.

Standard error: how much the average wobbles

Take many samples of the same size, and each one gives a slightly different average. The standard error (SE) is the standard deviation of those averages. You can estimate it from one sample: SE = s ÷ √n, where s is the sample standard deviation and n is the sample size. Bigger samples give smaller SE, but slowly: to halve the SE you need four times the data.

import math
from statistics import mean, stdev

times = [10, 12, 14, 16, 18]          # delivery times in minutes
se = stdev(times) / math.sqrt(len(times))
print(mean(times), round(se, 3))

The central limit theorem (CLT)

The central limit theorem says: if you take many samples of a decent size and compute each sample's average, those averages form a roughly normal (bell-shaped) distribution around the true mean, even if the original data is not bell-shaped. Delivery times may be skewed by a few very late orders, but the average of 100 orders behaves like a bell curve. This is why the normal distribution appears everywhere in statistics.

import random
from statistics import mean

random.seed(7)
population = [random.expovariate(1 / 30) for _ in range(100000)]   # skewed times
sample_means = [mean(random.sample(population, 50)) for _ in range(2000)]
print(round(mean(population), 1), round(mean(sample_means), 1))

Confidence intervals

A 95% confidence interval (CI) for a mean is roughly mean ± 1.96 × SE. The 1.96 comes from the normal curve: 95% of the area lies within 1.96 standard deviations of the centre. For the delivery times: 14 ± 1.96 × 1.414, so about 11.23 to 16.77 minutes.

What a 95% CI means

It means: if we repeated the whole sampling process many times and built an interval each time, about 95% of those intervals would contain the true mean. It does NOT mean "there is a 95% chance the true mean is in this particular interval" in the strict classic sense, and it does not mean 95% of the data lies inside it. The interval is about the average, not about single values.

Common mistakes

• Thinking a huge biased sample beats a small random one. Fix: check how the data was collected first.
• Reading a CI as the range of individual values. Fix: a CI of the mean is much narrower than the spread of the data.
• Comparing two groups by checking whether their CIs overlap. Overlap does not prove "no difference". Fix: test the difference directly (next topic).
• Forgetting that SE shrinks with √n, not n. Four times the data only halves the error.