FOUNDATION 06 / Business & data
Statistics, probability & uncertainty
Summarize data, build a simple scenario model and distinguish an estimate from a guarantee.
By the end of this lesson
- Distinguish mean, median and dispersion.
- Calculate expected return in a two-outcome model.
- Identify sampling and correlation traps.
IBBC Capital Finance Foundations · Original content drafted with AI assistance. IBBC Capital is responsible for editorial review and content. General education, not an official CFA Institute curriculum or credential.
Center is only part of the picture
The arithmetic mean sums observations and divides by their count. The median is the middle sorted value, or the average of the two middle values for an even count. A few extreme observations can move the mean substantially.
For 2, 3 and 100, the mean is 35 and the median is 3. Neither number alone describes the entire distribution. Inspect range, frequency, unusual observations and the data collection process.
Expected value is a weighted average
For mutually exclusive, exhaustive scenarios, E[R] = Σ pᵢRᵢ, with probabilities nonnegative and summing to one. Expected return need not be a return that actually occurs in any individual scenario.
Assigning probabilities requires evidence and judgment. A spreadsheet’s precise decimal output does not make the input probabilities reliable. Use scenario sensitivity rather than treating a single expected value as certainty.
Spread and standard deviation
For a complete probability model, variance = Σ pᵢ(Rᵢ−E[R])²; standard deviation is its square root. This is a population model, not the n−1 sample-variance estimator. Keep percentage and decimal units consistent.
Standard deviation measures dispersion around a mean, not every form of risk. Asymmetry, illiquidity, extreme losses and model error can matter even when a single volatility number looks modest.
Association is not causation
Correlation summarizes linear association and lies between −1 and +1 when defined. A zero correlation does not rule out nonlinear dependence. Correlation can change across regimes and is not proof that one variable causes another.
Check sample size, missing observations, survivorship bias and whether you selected only a favorable period. Separate in-sample description from out-of-sample prediction. More historical data do not automatically remove structural change.
Two equally likely outcomes
- Scenario A: +10%; scenario B: −10%; probability 50% each.
- Expected return = 0.5×10%+0.5×(−10%) = 0%.
- Variance in decimal units = 0.5×0.1²+0.5×(−0.1)² = 0.01. Standard deviation = 10%.
- Zero expected return does not mean no risk: neither scenario has a zero return.
Try before looking at the explanation
Change one input in the practice tool below. Predict the direction first, calculate second, then explain why it changed.
Work the numbers
Common mistakes to catch
- Using scenario weights that do not sum to one.
- Interpreting correlation as proof of cause.
- Mixing percent units and decimal units in a variance calculation.
Knowledge check · 6 original questions
Answer all six. At least five correct marks the lesson complete for self-study. You can retry; this is not a credential.
Keep these ideas
- Report both the center and the spread.
- A model is conditional on its data and assumptions.