GSOE9830 Chap.10 Sensitivity and Probability Distributions
Sensitivity and Probability Distributions
Sensitivity analysis changes an input and measures the effect on an output such as NPV. A tornado chart ranks tested drivers by impact, but its bars do not show likelihood and one-at-a-time changes can miss correlation. Probability distributions add likelihood, requiring non-negative probabilities that sum to one. The cumulative distribution then answers threshold questions such as the chance cost remains within budget.
Every sensitivity run should reset the model to its base case before another input changes, otherwise the bars become unlabelled combined scenarios. Bar widths are comparable only when their tested ranges are stated and defensible. Threshold probability begins by translating at most, below or at least into the correct set of ordered outcomes.
A cumulative total must never decrease and must reach one after the largest outcome, providing fast checks on the distribution and the inclusion boundary.
What this chapter covers
- 01
Base case
- 02
One-way sensitivity
- 03
Tornado chart
- 04
Probability mass
- 05
Cumulative probability
- 06
Correlation limit
Worked example · free
Rank NPV sensitivity and calculate a threshold probability
- +1Price NPV spread is $100,000.
- +1Capital-cost NPV spread is $36,000.
- +1Price is more influential over the tested ranges.
- +1P(price at most $100) is 0.20 plus 0.50, or 0.70.
Key terms
- Sensitivity analysis
- A what-if method that measures output response to an input change.
- Tornado chart
- A ranked display of output ranges caused by tested input ranges.
- Probability mass
- The probability assigned to an exact discrete outcome.
- Cumulative distribution
- The probability that a variable is at or below a threshold.
Sensitivity and Probability Distributions FAQ
What does the widest tornado bar mean?
It identifies the input that changes the output most over the tested range. It says nothing by itself about how likely either endpoint is in the decision setting.
How do I validate a discrete distribution?
Check that every probability is at least zero and that all probabilities sum to one before calculating expected value, variance or a cumulative probability from the model.
What does one-way sensitivity miss?
It can miss variables changing together, correlation, nonlinear interactions and the likelihood of each tested value. Scenario or simulation methods can address some of those limits.
Exam move
Start risk analysis from a validated base-case model. Record the base value of price, volume, capital cost, operating cost, timing and any other tested driver. For one-variable sensitivity, change only one input to its stated low value, recalculate the output, change it to the high value and then restore the base before moving to the next driver. This reset prevents later bars from becoming accidental combined scenarios.
Build a table with low-input output, base output, high-input output and total spread. Order the tornado bars by spread, not by the input’s numerical size. A wide bar identifies an influential assumption under the tested range. It does not prove the adverse value is likely, and bars are not comparable when their input ranges were chosen on inconsistent bases. State every range and explain any asymmetry around the base case.
Use scenarios for combinations and label them separately. A downside scenario can change price, volume and cost together, while a one-variable bar cannot capture correlation. Do not place a combined scenario beside isolated bars without explaining the different construction. The purpose is to distinguish leverage in the model from a coherent joint state of the world.
Validate a discrete probability distribution before calculating with it. Every probability must be non-negative and the total must equal one, apart from minor stated rounding. A missing mass of 0.10 is a model error, not a rounding adjustment. Identify the missing outcome or correct the probabilities before computing cumulative chance, expected value or variance. Translate threshold wording into an event set.
For at most or less than or equal, include all outcomes up to and including the boundary. For less than, exclude the boundary. For at least, include the boundary and larger outcomes. Write the included values beside the sum, then use the complement as an independent check when fewer outcomes lie on the opposite side. Build a cumulative table in ascending outcome order.
The cumulative probability must never decrease, must stay between zero and one, and must finish at one after the largest outcome. Interpret the result with the model and threshold attached, such as a 75% chance cost does not exceed $10m under the stated distribution. Do not call that the chance of exactly $10m. Keep sensitivity and probability conclusions separate until evidence connects them.
A driver can be highly sensitive but tightly distributed, or weakly sensitive but very uncertain. Risk priority depends on both impact and likelihood. Maintain an error log for previous sensitivity input not reset, incomparable ranges, bar width called probability, distribution not summing to one, boundary omitted and exact probability confused with cumulative probability.
Before submission, state whether the result describes leverage, a scenario or a probability, and show the range or event set that supports the statement.