🎯 By the end of this lesson you will…
📏 Mean vs Median — The Most Abused Distinction in Statistics
Consider a company with 9 employees earning £25,000 each and one CEO earning £1,000,000. The mean salary is £107,500. The median salary is £25,000. Both are technically correct. Which one would you use if you were the CEO claiming the company pays well? Which would you use if you were a union representative?
Use the mean when data is roughly symmetric with no extreme outliers. Use the median when there are outliers or skewed distributions — house prices, salaries, wealth. The median is almost always more representative of a "typical" person's experience. Politicians and companies overwhelmingly quote the mean when it flatters them.
UK government statements about "average earnings" almost always use the mean. The Office for National Statistics publishes both. In 2023, mean full-time weekly earnings were £682. Median full-time weekly earnings were £612 — £70 lower. The gap exists because high earners pull the mean up. Most workers earn below the mean average.
📉 Seven Ways Graphs Mislead
You don't need to lie with numbers if you can choose how to display them. These are the most common techniques — used in newspapers, political campaigns, and advertising every day.
Starting the y-axis at something other than zero makes small changes look enormous. A 2% rise in crime looks like a 400% rise if the axis runs from 98 to 102.
Choosing a start or end date that makes your trend look better. Stock prices always look impressive if you start from the right trough.
Using two different y-axes on the same graph can make completely unrelated trends look perfectly correlated. Nicolas Cage films released vs swimming pool drownings — genuinely correlated, causally meaningless.
"This drug doubles your risk of a rare side effect!" Sounds terrifying. If the baseline risk is 1 in 100,000, doubling it gives 2 in 100,000. Absolute risk increase: 0.001%. Both statements are true.
3D pie charts distort the apparent size of slices based on their position. The slice at the front appears larger than an identical slice at the back. There is no legitimate use for a 3D pie chart.
A graph showing "40% of people agreed" with no comparison to how many usually agree, or what the alternatives scored, is meaningless.
"9 out of 10 dentists recommend..." How many dentists? How were they selected? Were they paid? Sample methodology is almost never shown in advertisements.
🔍 Activity — Spot the Manipulation
Click each graph description to reveal what technique is being used and why it misleads. All based on real published graphs.
🔀 Simpson's Paradox — When the Whole Lies About the Parts
Simpson's Paradox occurs when a trend appears in different groups of data but disappears — or reverses — when the groups are combined. It's not a mathematical trick. It's a genuine phenomenon that has led to wrong conclusions in medicine, sport and criminal justice.
Data showed that 44% of male applicants were admitted vs 35% of female applicants — apparently discriminatory. But when broken down by department, women had a higher or equal acceptance rate in almost every individual department. The paradox: women disproportionately applied to more competitive departments. The aggregate data produced a false conclusion. The university was not discriminating — the data was.
A study found Treatment A had a 78% success rate overall, Treatment B had 83% overall. So B is better? No. For small kidney stones: A had 93%, B had 87%. For large stones: A had 73%, B had 69%. A is better for every size of stone. B's higher overall rate exists because it was used more on smaller (easier) stones. This is Simpson's Paradox — and it has led to wrong treatment recommendations in real medicine.
❓ Quick Check
Portfolio — Find and Expose a Misleading Statistic
Find a real statistic from a newspaper, advert or politician. Expose how it misleads.
Glossary — Lesson 1
🎯 By the end of this lesson you will…
✈️ Survivorship Bias — We Only See the Survivors
During World War II, the military analysed bullet holes on planes returning from missions to decide where to add armour. The bullet holes were concentrated on the wings and fuselage. Their initial plan: reinforce those areas. A statistician named Abraham Wald pointed out the fatal flaw.
The planes they were analysing were the ones that came back. The bullet holes showed where a plane could be hit and still survive. The areas with no bullet holes — the engines and cockpit — were precisely where they should add armour. Planes hit there didn't return to be counted. The sample was fundamentally biased toward survivors.
"Most successful entrepreneurs dropped out of university" — true for those who succeeded; we don't count the thousands who dropped out and failed. "Music was better in the 1970s" — we remember the best songs from 1970; we don't remember the thousands of terrible ones. "Old buildings are better built" — we see the ones that survived; the poorly built ones collapsed. Every time you only see the winners, you're seeing survivorship bias.
🏥 The Base Rate Fallacy — Why Positive Tests Aren't What They Seem
This is arguably the most dangerous statistical error in medicine — and understanding it could save your life.
A disease affects 1 in 1,000 people. A test for this disease is 99% accurate — it correctly identifies 99% of positive cases and correctly rejects 99% of negative cases. You test positive. What is the probability you actually have the disease?
About 9%. Not 99%. Here's why: Test 100,000 people. About 100 have the disease — the test correctly finds 99 of them. The remaining 99,900 don't have the disease — but the test incorrectly flags 1% of them: 999 false positives. So of the ~1,098 positive tests, only 99 are real. That's about 9%. A 99% accurate test gives you a 91% chance of a false alarm when the disease is rare. Base rate matters enormously.
This calculation is called Bayes' Theorem and it is why mass screening programmes (testing everyone for rare diseases) can cause more harm than good through unnecessary treatment of false positives. It is also why breathalyser and drug test results should never be treated as definitive without understanding the base rate. Medical professionals routinely get this wrong — studies show doctors, not just patients, often don't account for base rates correctly.
🧮 Activity — Bayes' Theorem Calculator
Enter the three numbers for any test scenario. The calculator shows the actual probability of a true positive — which is almost always lower than people expect.
🔬 Test Scenario Calculator
📊 Why Polls Fail — And What Margin of Error Really Means
Polls are everywhere — but almost nobody reads them correctly. Here's what you actually need to know:
A poll says Party A is on 42% ± 3%. This means the true figure could be anywhere between 39% and 45%. If Party B is on 40% ± 3%, the overlap is enormous — you cannot conclude either is ahead. But headlines routinely say "Party A leads Party B" based on exactly this situation. The margin of error is almost never shown on news graphics.
The Literary Digest poll in 1936 surveyed 2.4 million people and predicted Alf Landon would beat Roosevelt — one of the biggest polling failures in history. Why? Their sample came from car owners and telephone subscribers — disproportionately wealthy, disproportionately Republican. Sample size doesn't fix a biased sample. The 2016 Brexit referendum and US Presidential election were both called wrong by polls for similar reasons: non-representative samples and undercounting of certain demographics.
🔀 Activity — Statistical Paradox Explorer
Click each statistical paradox to understand it — then write your own explanation in plain English.
Portfolio — Build Your Own Simpson's Paradox
Create a worked example using your own invented numbers. This is the hardest and most impressive portfolio piece so far.
Glossary — Lesson 2
🎯 By the end of this lesson you will…
🎰 The Gambler's Fallacy
A coin lands heads 10 times in a row. Most people feel strongly that tails is "due." This feeling is the gambler's fallacy — and it is completely wrong. The coin has no memory. Each flip is an independent event. The probability of tails on flip 11 is exactly 50%, regardless of what came before.
Our brains are pattern-recognition machines. We evolved to notice streaks and sequences because they often do signal something meaningful — clouds before rain, animal tracks before prey. Randomness genuinely feels unnatural to us. But in truly random systems, the past has no causal connection to the future.
A 2016 study found that asylum decisions made by immigration judges were affected by the gambler's fallacy. Judges were significantly less likely to grant asylum to an applicant if the previous two or three applicants had been granted it — as if they felt approval was "due" to go the other way. People's lives were affected by a mathematical error in a judge's intuition.
🏀 The Hot Hand — When It's NOT a Fallacy
For years, psychologists said the "hot hand" in basketball — the feeling that a player who has just scored several times is more likely to score again — was an illusion, another form of the gambler's fallacy. Then in 2015, new analysis overturned this conclusion.
Gilovich, Vallone and Tversky (1985) studied NBA data and concluded the hot hand was a cognitive illusion. But Miller and Sanjurjo (2015) found a statistical flaw in the original study. When corrected, the data actually showed the hot hand is real in basketball — players genuinely do shoot better during streaks. The difference: basketball involves skill (where recent performance can predict future performance), while a coin involves pure randomness. Context matters.
🚪 The Monty Hall Problem
You are on a game show. Three doors: behind one is a car, behind two are goats. You pick Door 1. The host opens Door 3 — a goat. He offers you the chance to switch to Door 2. Should you?
Switching gives you a 2/3 chance of winning. Staying gives you 1/3. When you first chose, you had a 1/3 chance of being right. The other two doors together had a 2/3 chance of containing the car. The host's action — who always opens a goat door — doesn't change your original odds. It concentrates the 2/3 probability onto the single remaining door. This was published in 1990 and provoked thousands of angry letters from mathematicians insisting it was wrong. It isn't.
🎲 Activity — Test Your Probability Intuition
Answer each question with your gut instinct first — then check. The point is to feel how wrong our intuition can be, not to get the answer right.
Choose ONE probability misconception from this lesson. Explain it clearly enough that a friend who hasn't studied it would understand — and be convinced. Use a concrete example with real numbers.
Glossary — Lesson 3
🎯 By the end of this lesson you will…
🎯 What is Expected Value?
Expected value (EV) is the average outcome you would get if you repeated a decision many times. It is calculated by multiplying each possible outcome by its probability, then summing them:
EV = (Outcome 1 × Probability 1) + (Outcome 2 × Probability 2) + …
Simple example: You flip a coin. Heads: win £3. Tails: lose £1. EV = (£3 × 0.5) + (−£1 × 0.5) = £1.50 − £0.50 = +£1.00. This is a positive EV game — you should play it as many times as possible.
The UK National Lottery jackpot is typically around £5 million. The probability of winning is 1 in 45,057,474. A ticket costs £2. EV = (£5,000,000 × 1/45,057,474) + (£0 × 45,057,473/45,057,474) − £2 ≈ −£1.89 per ticket. For every £2 you spend, you get approximately 11p back on average. The lottery is a deeply negative EV activity. This is not a secret — it's how the lottery funds public projects.
Car insurance has negative expected value — on average, you pay more in premiums than you receive in claims. So why buy it? Because the utility of avoiding a catastrophic loss (writing off your car, a £50,000 medical bill) is worth more than its expected value. Expected value assumes money has equal utility at all amounts — but losing £1,000 when you only have £1,000 is catastrophically worse than losing £1,000 when you have £1,000,000. Insurance is rational because it protects against ruin, not because it has positive EV.
🧮 Activity — Expected Value Calculator
Enter outcomes and probabilities for any scenario. The calculator gives you the expected value and tells you whether to take the bet.
🎰 Why Casinos Always Win — The House Edge
European roulette has 37 numbers (0–36). You bet £1 on a single number. If it wins, you get £36 back (35:1 odds). The probability of winning is 1/37.
EV = (£35 × 1/37) + (−£1 × 36/37) = £0.946 − £0.973 = −£0.027. For every £1 you bet, you lose 2.7p on average. This seems small — but applied across millions of bets, it guarantees the casino profits. The house edge is simply negative expected value built into the game design. All casino games have it. The longer you play, the more certain your loss becomes.
If you have found a positive EV bet, how much should you stake? The Kelly Criterion gives the mathematically optimal answer: stake the fraction of your bankroll equal to your edge divided by the odds. It maximises long-term growth while preventing ruin. Professional gamblers and hedge funds use it. The formula is: f = (bp − q) / b, where b = odds, p = probability of winning, q = probability of losing.
Portfolio — Real Decision Analysis
Apply expected value to a real decision. Choose one of the suggestions or your own.
Glossary — Lesson 4
🎯 By the end of this lesson you will…
🔒 The Prisoner's Dilemma
Two suspects are arrested. They cannot communicate. Each is offered the same deal. The payoffs depend on what both choose:
| Prisoner B: Stay Silent | Prisoner B: Betray | |
|---|---|---|
| Prisoner A: Stay Silent |
1 year
A gets
1 year
B gets
|
10 years
A gets
0 years
B gets
|
| Prisoner A: Betray |
0 years
A gets
10 years
B gets
|
5 years
A gets
5 years
B gets
|
The Nash equilibrium is Betray/Betray — both get 5 years. Why? From A's perspective: if B stays silent, betraying gets A 0 years (better than 1). If B betrays, betraying gets A 5 years (better than 10). Betraying is always better regardless of what B does. B's logic is identical. So both betray — and both get 5 years — even though both staying silent would give 1 year each. Individually rational decisions produce a collectively terrible outcome.
♟️ Activity — Play the Prisoner's Dilemma
You are Prisoner A. The computer plays Prisoner B using a random strategy. Make your choice — then see what happened. Play multiple times and notice: does the Nash equilibrium always occur?
🌍 Real-World Game Theory
The Prisoner's Dilemma structure appears everywhere in human affairs:
If both countries disarm, both are safer and save enormous resources. But if one disarms and the other doesn't, the disarmed country is vulnerable. So both arm — the Nash equilibrium. Both spend billions on weapons neither wants to use, in a situation both would prefer to escape.
Two supermarkets could both charge higher prices and earn good profits. But if one cuts prices, it gains customers — so the other must cut too. Both end up with lower margins than they'd prefer, locked in a race to the bottom. ASDA and Tesco have been in exactly this equilibrium for decades.
Each country benefits from others reducing emissions while free-riding on their effort. Reducing emissions is costly; the benefits are shared globally. The Nash equilibrium — without binding agreements — is for everyone to emit more than the global optimum. This is precisely why international climate agreements are so difficult to maintain: the game theory pushes every nation toward defection.
🔍 Activity — Spot the Game
Click each scenario to discover the underlying game theory structure. Then identify the Nash equilibrium and whether it's collectively good or bad.
Portfolio — Game Theory Analysis
Choose a real-world situation and analyse it using game theory. Be specific.
Glossary — Lesson 5
🎯 By the end of this lesson you will…
⚖️ How Algorithms Can Be Biased
An algorithm is a set of mathematical rules for making a decision. It sounds neutral — no human prejudice, just numbers. But algorithms are built by humans, trained on human-generated data, and optimise for goals chosen by humans. Bias enters at every stage.
If you train a hiring algorithm on 10 years of past hiring decisions — and those decisions were biased against women — the algorithm learns to replicate that bias. Amazon built exactly this system and had to scrap it in 2018 when they discovered it was systematically downgrading CVs that contained the word "women's" (as in "women's chess club"). The algorithm was not programmed to discriminate — it learned discrimination from the data.
Even if you remove race from a predictive algorithm, it can still discriminate through proxy variables — other variables that correlate with race. Postcode is a classic example. Postcode correlates with race in many areas. An algorithm that uses postcode as a variable — even for a legitimate reason like predicting commute reliability — may systematically disadvantage people from certain racial backgrounds. Removing the explicitly protected variable doesn't remove the discrimination.
Algorithms optimise for whatever they are told to optimise for. YouTube's algorithm was optimised for watch time — so it learned that outrage and extremism kept people watching longer and began recommending increasingly extreme content. Facebook's engagement algorithm was optimised for engagement — it learned that anger drove more engagement than contentment. The algorithm did exactly what it was designed to do. The goal was wrong.
🔬 Activity — Algorithmic Bias Cases
Read each real algorithmic bias case. The accuracy bars show the documented performance gap by demographic group.
🤔 The Impossible Problem: Three Definitions of Fairness
Here is the most difficult insight in this lesson: there is no single definition of "fairness" — and three reasonable definitions are mathematically impossible to satisfy simultaneously.
Among people the algorithm scores as "high risk," the same proportion should actually reoffend regardless of race. If 60% of high-risk Black defendants reoffend, 60% of high-risk white defendants should too.
The algorithm should be equally likely to wrongly label an innocent person as "high risk" regardless of race.
The algorithm should be equally likely to miss someone who will actually reoffend, regardless of race.
Mathematician Alexandra Chouldechova proved that when two groups have different base rates of reoffending (which they do, due to historical inequality in policing and prosecution), it is mathematically impossible to satisfy all three definitions simultaneously. Any fair algorithm must choose which type of unfairness to accept. This is not a technical problem — it is a values problem wearing a mathematical costume. The question of which definition of fairness to use is a political and ethical question, not a mathematical one.
💬 Discussion — Who is Responsible?
Portfolio — Algorithmic Bias Investigation
Research one real algorithmic bias case (not covered in this lesson) and report on it.
Glossary — Lesson 6 + Unit Review
Peer Review — Unit Portfolio
Swap portfolios. Check all 6 pieces against this rubric.
Unit Complete
You can now read statistics critically, calculate expected value, understand why rational actors produce irrational outcomes, and investigate algorithmic injustice. These are tools most adults don't have.