A 1936 poll with 2.4 million replies got it wrong — a tiny sample got it right.
To taste a whole pot of soup, must you drink it all — or does one spoonful of a well-stirred pot suffice?
A well-stirred spoonful is a miniature of the whole.
Experiment
Hands-on experiment
Predict first — to know 50 million voters' support within ±3 points, how many must you survey?
Draw samples from the hidden population (the truth stays secret)
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
National approval ratings, defect rates, TV ratings — a full census is impossible or ruinously expensive. Most of what humanity knows came from samples.
Two key questions: how must you draw so the sample resembles the whole (randomness), and how many are enough (the √n law)?
Draw samples over and over and their averages pile up into a bell shape. That is the hill you watched grow in Stage 2. The name for this fact is the central limit theorem, and every calculation here stands on that hill.
Insight
Insights from the video
“A sample's power comes from randomness, not size.”
A biased million (calls at one time of day, say) is garbage; a random thousand is treasure. In 1936 the Literary Digest's 2.4-million-person poll failed while Gallup's 50,000 succeeded — the classic proof.
“Ten times the accuracy costs a hundred times the sample.”
Error shrinks as 1/√n: 1,000 → ±3.1 points; 10,000 → ±1. This diminishing return is the economics behind '1,000 respondents' becoming the world standard.
Misconception
Common misconceptions
Knowing 50 million people surely takes a sample of hundreds of thousands.
The margin of error depends on sample size, not population size. A well-drawn 1,000 gives ±3 points — however big the pot, one spoonful suffices if it's stirred.
A ±3-point margin means the poll is inaccurate.
±3 points is a precise statement: 'at 95% confidence, the truth lies within this band'. Not a confession of inaccuracy but a specification of accuracy. Two candidates 1 point apart? The correct reading is 'a dead heat'.
Formula
Writing it as math
The size of the wobble you watched in repeated draws, as formulas:
Standard error of a sample proportion
The wobble of the sample rate. Note the √n in the denominator — bigger samples calm the wobble at √n speed.
The 95% margin of error
The normal distribution's ±2σ (95%) rule, applied. n=1,000 gives about ±3.1 points — the number from the news.
What the confidence level means
Not 'the truth is here with 95% probability' but 'this METHOD catches the truth 95 times out of 100' — a statement about the method.
In Real Life
Where you meet it in real life
Election polls
'A 45%, B 44%, margin ±3.1' — a 1-point gap is inside the margin: statistically a dead heat. Declaring a leader is a misreading.
TV ratings
A few thousand panel households estimate national ratings. Even 0.1-point competition rests on √n arithmetic.
Factory inspection
Destructive testing of everything is impossible — defect rates are estimated and controlled from n samples per lot.
Clinical trials
A few thousand participants estimate effects for all humanity. Sample design IS the quality of the science.
Try Yourself
Test yourself
Q1To halve the margin (±3 → ±1.5 points), how much bigger must the sample be?Show answer ▾
4x (1,000 → 4,000). Error goes as 1/√n, so halving it takes 4x the n. Ten times the accuracy takes 100x — diminishing returns.
Q2'Margin ±3, candidate A at 47%' — is it impossible for the truth to lie outside 44–50%?Show answer ▾
It's possible — about 5% of the time. A 95% confidence level includes 'about 1 in 20 misses the band'. That's why averages of many polls beat any single poll.
Q3You want the average height of your school's 500 students. Measure your 30 basketball-team friends?Show answer ▾
No — the problem isn't the size (30) but the bias. The basketball team isn't a miniature of the population. A random 30 beats a biased 300.
Try answering yourself before revealing it — getting it wrong is where learning starts.
Connection
Concepts connect
Previous concept
Normal Distribution
The sample mean's distribution being bell-shaped is what justifies the ±3-point computation — meet the bell first.
← Normal Distribution labLeads to next
Correlation
Having estimated single values from samples, it's time to read relationships between two variables — and meet that art's greatest trap.
Go to the Correlation lab →Related
Labs worth exploring together
Related lab
Probability
The law of large numbers — experience rates approaching truth as samples grow, first.
Go to the Probability lab →