1. Statistics and Probability Basics

Time series comes later. First, read the distribution of a single number. The mean is the center, and volatility is the spread. Move the curve and watch: a higher mean shifts the whole curve right, and a larger volatility flattens it.

• The Return Distribution: Mean and Volatility — Suppose you buy one share. Next year's return is unknown. But its mean and its spread can be stated. With a mean of $8\%$ and volatility of $15\%$, returns wobble around $8\%$ by roughly $15\%$. The formula below is the shape of that distribution. Move the cutoff and watch how the probability of falling below it changes. For example, suppose you invest 100,000 won of allowance in a stock. With a mean return of $8\%$, you expect about 108,000 won next year. But with volatility of $15\%$, it could be 93,000 won or 123,000 won. The expectation and the wobble must be read separately. Source: Fama, E. F. (1965). The Behavior of Stock-Market Prices. Journal of Business, 38(1), 34-105. That paper showed real stock returns have fatter tails than a normal distribution, so the normal curve in this scene is only an approximation.
• As the Sample Grows, the Mean Finds Its Place — The previous scene showed the center of a distribution. Now estimate that center from a sample. With 10 draws, the sample mean can sit far from the true mean. With 100 or 1000 draws, it hugs the true mean. That is the law of large numbers. The curve below shows how the mean moves as each new draw joins the sample. For example, the average of 10 die rolls can sit far from 3.5. A run of sixes pulls it up sharply. The average of 1000 rolls lands very close to 3.5. The power of luck over the average weakens as the sample grows. Source: Bernoulli, J. (1713). Ars Conjectandi, the book where the law of large numbers was first proved.