Module 12 of 15

Probability Distributions

Averages hide spread, skew and extreme risks.

AMathematical Foundation

A probability distribution describes how likely each possible value of a variable is. Two distributions can share a mean but differ in spread (standard deviation), skew (asymmetry) and tails (probability of extreme values). Income-like quantities are often right-skewed: many small values and a long tail of very large ones. A log-normal distribution, where log(X) is normal with parameters μ and σ, is a common model; its mean exp(μ + σ²/2) exceeds its median exp(μ), and the gap grows with σ.

X\sim\text{LogNormal}(\mu,\sigma):\quad \text{median}=e^{\mu},\quad \text{mean}=e^{\mu+\sigma^2/2},\quad P(X<\text{mean})=\Phi(\sigma/2)

In words: For a log-normal variable, the median is e to the mu, the mean is e to the mu plus sigma squared over two, and the probability of being below the mean is Phi of sigma over two.

$\mu,\sigma$: mean and standard deviation of log(X)
$\Phi$: standard normal cumulative distribution function

Assumptions

The log-normal is a model; real distributions of earnings or audience size may have even heavier tails (power laws).
Summary statistics should match the decision: medians describe typical cases, means describe totals, tail probabilities describe risks.

BIntuitive Explanation

If ten people earn $30,000 each and one earns $3 million, the average is about $300,000, yet ten of the eleven earn a tenth of that. In winner-take-most careers, the average tells you mostly about the winners. The question an aspiring entrant should ask is: what happens to someone near the middle, and how likely is the top?

CConnection to Debate

Use distributions to challenge “average” claims. Average creator income, average graduate earnings under outcome-contingent tuition, or average grade under a cap all hide who is above and below. Distribution thinking also clarifies grading debates: a cap on the share of top grades imposes a relative distribution regardless of the absolute performance of a cohort.

DReal-World Application

Grading on a fixed distribution
Competing interpretation
Harvard's 2026 policy limits A grades to 20% of enrolment in each course (with flexibility for up to four more); Princeton's earlier policy capped A grades at 35% until 2014. A cap converts an absolute standard into a relative one: the cut-off for an A moves with the cohort. Students in unusually strong cohorts or small classes are most affected, because the share of strong students in a small group varies more by chance (see Sample Size). Princeton students often read their cap as a quota that made them compete for a limited number of A's.
Sources
- FAS Passes a Grade CapAbigail S. Gerstein, Amann S. Mahajan (The Harvard Crimson), 2026-05-20
- Princeton University to Eliminate ‘Grade Deflation’ PolicyPlanet Princeton, 2014-10-06
Skewed earnings in esports and creator careers
Competing interpretation
Recorded esports prize money is concentrated at the top: each of the ten highest earners has more than $5 million in tournament winnings (Esports Earnings). Prize lists capture only tournament winnings, not salaries or sponsorship, and only players who have won something. For a young aspirant the relevant quantities are the median outcome and the probability of reaching a sustainable income, which these figures do not show; a mean pulled up by a long right tail can be far above what a typical aspirant earns.
Sources
- Top Players by Overall EarningsEsports Earnings (esportsearnings.com)

These examples describe real policies and research. The numbers in the interactive below and in the practice questions are hypothetical teaching data unless a question says otherwise.

EInteractive Visualization

Compare a symmetric and a right-skewed distribution with the same mean: medians, tail probabilities and top shares.

Illustrative example — hypothetical data.Two hypothetical earnings distributions with the same average. Units: thousands of dollars per year.

Average (mean) outcome60

Inequality of the skewed distribution (log-scale σ)1.00

Higher means a longer right tail: a few huge outcomes, many small ones.

“Star” threshold200

Symmetric

Median

P(above 200)

0.00%

Right-skewed

Median

P(above 200)

4.42%

Share earning below the mean

69%

Share of all earnings held by top 1%

Both distributions average 60. In the skewed one, the typical (median) person gets 36, and 69% of people earn less than the average. The chance of passing the star threshold is 4.4%, versus 0.00% under the symmetric shape.

When a debate cites an “average” for careers like esports, creator work or startups, ask for the median and the tail probabilities. The mean is pulled up by a few extreme winners.

FDebate Practice

GLimitations

A fitted distribution is a model and can understate extreme tails. Distributions describe outcomes, not causes, and say nothing about whether an unequal outcome is fair; that is a normative judgment.

AMathematical Foundation

X\sim\text{LogNormal}(\mu,\sigma):\quad \text{median}=e^{\mu},\quad \text{mean}=e^{\mu+\sigma^2/2},\quad P(X<\text{mean})=\Phi(\sigma/2)

In words: For a log-normal variable, the median is e to the mu, the mean is e to the mu plus sigma squared over two, and the probability of being below the mean is Phi of sigma over two.

$\mu,\sigma$: mean and standard deviation of log(X)
$\Phi$: standard normal cumulative distribution function

Assumptions

The log-normal is a model; real distributions of earnings or audience size may have even heavier tails (power laws).
Summary statistics should match the decision: medians describe typical cases, means describe totals, tail probabilities describe risks.

BIntuitive Explanation

CConnection to Debate

DReal-World Application

Grading on a fixed distribution
Competing interpretation
Harvard's 2026 policy limits A grades to 20% of enrolment in each course (with flexibility for up to four more); Princeton's earlier policy capped A grades at 35% until 2014. A cap converts an absolute standard into a relative one: the cut-off for an A moves with the cohort. Students in unusually strong cohorts or small classes are most affected, because the share of strong students in a small group varies more by chance (see Sample Size). Princeton students often read their cap as a quota that made them compete for a limited number of A's.
Sources
- FAS Passes a Grade CapAbigail S. Gerstein, Amann S. Mahajan (The Harvard Crimson), 2026-05-20
- Princeton University to Eliminate ‘Grade Deflation’ PolicyPlanet Princeton, 2014-10-06
Skewed earnings in esports and creator careers
Competing interpretation
Recorded esports prize money is concentrated at the top: each of the ten highest earners has more than $5 million in tournament winnings (Esports Earnings). Prize lists capture only tournament winnings, not salaries or sponsorship, and only players who have won something. For a young aspirant the relevant quantities are the median outcome and the probability of reaching a sustainable income, which these figures do not show; a mean pulled up by a long right tail can be far above what a typical aspirant earns.
Sources
- Top Players by Overall EarningsEsports Earnings (esportsearnings.com)

These examples describe real policies and research. The numbers in the interactive below and in the practice questions are hypothetical teaching data unless a question says otherwise.

EInteractive Visualization

Compare a symmetric and a right-skewed distribution with the same mean: medians, tail probabilities and top shares.

Illustrative example — hypothetical data.Two hypothetical earnings distributions with the same average. Units: thousands of dollars per year.

Average (mean) outcome60

Inequality of the skewed distribution (log-scale σ)1.00

Higher means a longer right tail: a few huge outcomes, many small ones.

“Star” threshold200

Symmetric

Median

P(above 200)

0.00%

Right-skewed

Median

P(above 200)

4.42%

Share earning below the mean

69%

Share of all earnings held by top 1%

When a debate cites an “average” for careers like esports, creator work or startups, ask for the median and the tail probabilities. The mean is pulled up by a few extreme winners.