Sensitivity Analysis — I can perform sensitivity reasoning (how changing an input affects an output) at my level. - Grade 11 Math

SECTION 1

Historical Context & Motivation

Whenever people build a mathematical model — whether it predicts the trajectory of a rocket, the profit of a business, or the spread of a disease — they rely on input values that are often estimates or approximations. Sensitivity analysis is the practice of asking: "What happens to the output if one of those inputs changes?" This question has been central to applied mathematics for centuries, and it remains one of the most important skills in modeling today.

The roots of sensitivity reasoning go back to early scientists who wanted to understand how measurement errors could ripple through their calculations. As engineers began designing bridges, aircraft, and financial systems, the need for systematic methods grew dramatically. Today, sensitivity analysis is a standard tool in fields ranging from economics to climate science.

1700s • Error Propagation in Astronomy

Astronomers like Gauss studied how small errors in telescope measurements could shift predicted planetary positions. This was one of the earliest forms of sensitivity reasoning.

1940s • Operations Research in World War II

Military planners used mathematical models to allocate resources. They performed 'what-if' analyses to see how changes in supply levels affected mission success, formalizing sensitivity analysis as a discipline.

1960s • Linear Programming & the Simplex Method

Economists and engineers began systematically testing how changes in constraints or coefficients affected optimal solutions in linear programs, giving sensitivity analysis its modern mathematical form.

2000s–Present • Climate & Data Science Models

Modern climate models contain thousands of input parameters. Scientists use sensitivity analysis to identify which parameters matter most, guiding where to invest research effort.

The central question sensitivity analysis addresses is deceptively simple: which inputs matter the most, and how much can they shift the output? As you learn to answer this question, you gain the ability to evaluate any model critically — deciding which assumptions to trust and which to question.

SECTION 2

Core Principles & Definitions

Before diving into calculations, it helps to understand the core ideas that make sensitivity analysis work. These principles apply whether you are working with a simple formula or a complex simulation. Mastering them will give you a framework for thinking about any model.

Input vs. Output

Every model has inputs (the values you feed in, like price or temperature) and outputs (the results, like profit or distance). Sensitivity analysis studies the link between them.

One-at-a-Time (OAT) Testing

The simplest approach: change one input while holding all others fixed, and observe how the output responds. This isolates the effect of each individual variable.

Sensitivity (Rate of Change)

The sensitivity of an output to an input is the ratio Δoutput ÷ Δinput. A large ratio means the output is highly sensitive to that input.

Percentage Change for Fair Comparison

Because inputs have different units (dollars vs. hours vs. degrees), comparing them requires percent change: a 10% increase in price versus a 10% increase in quantity.

Critical Inputs

An input is critical if a small change in it causes a disproportionately large change in the output. Identifying critical inputs is the main goal of sensitivity analysis.

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KEY TAKEAWAY

Think of a model like a recipe. Some ingredients (inputs) barely change the taste if you adjust them — a pinch more salt won't ruin a stew. But other ingredients are critical — double the chili flakes, and the dish becomes inedible. Sensitivity analysis tells you which ingredients matter most so you can focus your attention where it counts.

SECTION 3

Visual Explanation

The diagram below shows a simple model: a lemonade stand where profit depends on price per cup and number of cups sold. We vary one input at a time by ±20% and watch how profit responds. The steeper the line, the more sensitive the output is to that input.

The pink line (Price per Cup) is steeper than the cyan line (Cups Sold), meaning profit is more sensitive to price changes than to changes in volume. At ±20%, price shifts profit by $40, while cups sold shifts it by only $20.

Notice that both lines pass through the same base-case point ($40 profit) in the center. As we move left or right — decreasing or increasing each input — the outputs diverge. The steeper the line, the more sensitive the output is. This type of graph is sometimes called a tornado diagram (when drawn horizontally with bars) or a spider plot (when drawn as lines radiating from a center point). Both are standard tools for visualizing sensitivity.

SECTION 4

Mathematical Framework

To perform sensitivity analysis precisely, you need a few key formulas. These let you quantify exactly how much an output changes when an input shifts. You do not need calculus for these — just careful arithmetic and algebra.

ABSOLUTE CHANGE

ΔOutput = Output_new − Output_base

ΔOutput ("delta output") is the difference between the new output (after changing an input) and the original base-case output. A positive value means the output increased; a negative value means it decreased.

PERCENT CHANGE IN OUTPUT

% Change in Output = (ΔOutput ÷ Output_base) × 100

Expressing changes as percentages lets you compare the impact of different inputs on an equal footing, even when the inputs have different units.

SENSITIVITY RATIO

Sensitivity Ratio = (% Change in Output) ÷ (% Change in Input)

If the sensitivity ratio equals 1, the output changes at the same rate as the input. If the ratio is greater than 1, the output is amplified — small input changes cause large output changes. If the ratio is less than 1, the output is dampened.

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Why Percent Change?

Imagine you are comparing two inputs: price ($3.00 per cup) and cups sold (100 cups). A $0.30 change in price (10%) and a 10-cup change in sales (10%) are very different in raw numbers, but percent change puts them on the same scale for comparison.

LINEAR MODEL FORM

Output = a × x₁ + b × x₂ + c

Many models at this level are linear. In a linear model, the coefficients a and b directly tell you the sensitivity: each unit increase in x₁ changes the output by exactly a units.

SECTION 5

Types of Sensitivity Analysis

There are several ways to approach sensitivity analysis depending on the complexity of your model and the question you are trying to answer. At the high-school level, two methods dominate: one-at-a-time (OAT) analysis and scenario analysis. The diagram below illustrates how they differ.

Left: OAT analysis isolates each input's effect. Right: Scenario analysis bundles multiple input changes into best-case, base-case, and worst-case scenarios to show the full range of possible outcomes.

Feature	OAT Analysis	Scenario Analysis
Inputs changed	One at a time	Several at once
Main strength	Isolates individual effects clearly	Shows realistic combined effects
Main limitation	Ignores interactions between inputs	Harder to pinpoint which input drives the change
Best used when	You want to rank inputs by importance	You want to plan for best/worst outcomes

Comparison of OAT and Scenario Analysis

SECTION 6

Worked Example

A school club sells custom t-shirts to raise money. Their profit model is:

T-SHIRT PROFIT MODEL

Profit = (Selling Price − Cost per Shirt) × Number Sold − Fixed Costs

Base-case values: Selling Price = $20, Cost per Shirt = $8, Number Sold = 150, Fixed Costs = $200.

The club wants to know: Is their profit more sensitive to selling price or to the number of shirts sold? We will use a 10% increase in each input to find out.

Sensitivity of T-Shirt Profit

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Step 1 — Calculate Base-Case Profit

Profit_base = ($20 − $8) × 150 − $200 = $12 × 150 − $200 = $1,800 − $200

Profit_base = $1,600
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Step 2 — Increase Selling Price by 10%

New Selling Price = $20 × 1.10 = $22. New Profit = ($22 − $8) × 150 − $200 = $14 × 150 − $200 = $2,100 − $200 = $1,900. The absolute change is $1,900 − $1,600 = $300.

ΔProfit = +$300 (an 18.75% increase)
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Step 3 — Increase Number Sold by 10%

New Number Sold = 150 × 1.10 = 165. New Profit = ($20 − $8) × 165 − $200 = $12 × 165 − $200 = $1,980 − $200 = $1,780. The absolute change is $1,780 − $1,600 = $180.

ΔProfit = +$180 (an 11.25% increase)
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Step 4 — Compute Sensitivity Ratios

Sensitivity ratio for Selling Price = 18.75% ÷ 10% = 1.875. Sensitivity ratio for Number Sold = 11.25% ÷ 10% = 1.125. Both ratios are greater than 1, meaning profit is amplified compared to both inputs, but selling price has the larger effect.

Selling Price is the more critical input (ratio 1.875 vs. 1.125)
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Step 5 — Interpret the Results

A 10% increase in selling price boosted profit by 18.75%, while the same 10% increase in shirts sold boosted profit by only 11.25%. The club should focus on maximizing the price customers will pay — this input has a bigger leverage on profit. If the price estimate is uncertain, the club faces more risk than if the quantity estimate is uncertain.

Conclusion: Profit is most sensitive to selling price.

SECTION 7

Strengths & Limitations

Like any analytical tool, sensitivity analysis has clear strengths and important limitations. Understanding both will help you use it effectively and recognize when additional techniques might be needed.

Strengths	Limitations
Identifies which inputs have the greatest impact on the output, helping you prioritize where to gather better data.	OAT analysis assumes inputs change independently, but in reality inputs can be correlated (e.g., price and demand).
Requires only basic algebra — accessible without advanced math.	Provides no information about the probability of an input actually changing by a given amount.
Helps communicate risk and uncertainty to decision-makers in a clear, visual way.	For non-linear models, results can change depending on the size and direction of the input change.
Works with any model — formulas, simulations, or even mental models.	Does not guarantee that the model itself is correct — garbage in, garbage out.

Strengths vs. Limitations of Sensitivity Analysis

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KEY TAKEAWAY

Sensitivity analysis is like checking the weather forecast before a road trip. It tells you which conditions could change your plans the most — but it does not predict the actual weather. It is a tool for preparation and understanding, not a crystal ball.

SECTION 8

Connection to Advanced Theory

The sensitivity reasoning you are learning now is the foundation for more powerful techniques you will encounter in college-level courses. Calculus, statistics, and data science all build on this same core idea: understanding how changes in inputs propagate through a system to affect outputs.

What You Learn Now	What Comes Next
ΔOutput ÷ ΔInput (average rate of change)	Partial derivatives (∂f/∂x) — the instantaneous rate of change of a function with respect to one variable
Sensitivity ratio using percent changes	Elasticity in economics — measures the responsiveness of demand to price, for instance
Scenario analysis (best/worst/base)	Monte Carlo simulation — running thousands of random scenarios to map out a probability distribution of outcomes
Ranking inputs by impact	Global sensitivity analysis (Sobol indices) — advanced methods that account for interactions between inputs

From High School Sensitivity Analysis to College-Level Techniques

If you take AP Calculus, you will learn that the derivative of a function is essentially the limit of the sensitivity ratio as the input change approaches zero. In AP Statistics or college econometrics, you will encounter regression coefficients that quantify sensitivity in data-driven models. The intuition you build now — asking "what if this changes?" — is the same mindset that drives advanced research across every STEM field.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A farmer uses the formula Yield = Rainfall × Soil Quality × 0.5 to estimate crop yield. Without doing any calculations, which input do you think the yield is more sensitive to: a 10% increase in Rainfall or a 10% increase in Soil Quality? Explain your reasoning.

Show answer

In this model, Rainfall and Soil Quality enter the formula in exactly the same way — they are both multiplied together. Because they play symmetric roles, a 10% increase in either one will produce the same percent change in the output. Specifically, increasing Rainfall by 10% multiplies the entire yield by 1.10, and increasing Soil Quality by 10% also multiplies the yield by 1.10. The yield is equally sensitive to both inputs in this multiplicative model.

Question 2: BASIC CALCULATION

A catering company uses the model Total Cost = 15 × Guests + 200. The base case is 80 guests. Calculate the sensitivity ratio of Total Cost to the number of guests when guests increase by 10%.

Show answer

Base-case cost = 15 × 80 + 200 = $1,400. A 10% increase raises guests to 88. New cost = 15 × 88 + 200 = $1,520. ΔCost = $1,520 − $1,400 = $120. Percent change in cost = ($120 ÷ $1,400) × 100 ≈ 8.57%. The sensitivity ratio = 8.57% ÷ 10% ≈ 0.857. Since this is less than 1, the output is dampened relative to the input. This makes sense because the fixed cost ($200) absorbs some of the proportional change.

Question 3: INTERMEDIATE

A phone repair shop models monthly revenue as Revenue = Price × Repairs − Rent, where Price = $75, Repairs = 200, and Rent = $3,000. Perform an OAT analysis by increasing Price by 10% and then Repairs by 10%. Which input is revenue more sensitive to?

Show answer

Base revenue = $75 × 200 − $3,000 = $15,000 − $3,000 = $12,000. Increasing Price by 10% to $82.50: New revenue = $82.50 × 200 − $3,000 = $16,500 − $3,000 = $13,500. ΔRevenue = $1,500, which is a 12.5% increase. Sensitivity ratio = 12.5% ÷ 10% = 1.25. Increasing Repairs by 10% to 220: New revenue = $75 × 220 − $3,000 = $16,500 − $3,000 = $13,500. ΔRevenue = $1,500, which is also a 12.5% increase. Sensitivity ratio = 1.25. Revenue is equally sensitive to both inputs. This happens because Price and Repairs enter the formula symmetrically (as a product), and the fixed cost (Rent) does not change with either input.

Question 4: APPLIED

A student models the distance a car can travel on a full tank as Distance = (Tank Size ÷ Fuel Consumption Rate) × 1. Base values: Tank = 50 liters, Fuel Rate = 0.08 liters per km. The student is uncertain about both inputs: Tank Size could be off by ±5% and Fuel Rate could be off by ±15%. Perform a worst-case scenario analysis and compare it with the base case. What is the percent change in distance?

Show answer

Base distance = 50 ÷ 0.08 = 625 km. For the worst case, the tank is 5% smaller and fuel consumption is 15% higher. Worst-case tank = 50 × 0.95 = 47.5 liters. Worst-case fuel rate = 0.08 × 1.15 = 0.092 liters per km. Worst-case distance = 47.5 ÷ 0.092 ≈ 516.3 km. ΔDistance = 625 − 516.3 = 108.7 km. Percent change = (108.7 ÷ 625) × 100 ≈ 17.4% decrease. The combined effect of both unfavorable changes reduces distance by about 17.4%, which is substantial — the car might not reach a planned destination without refueling.

Question 5: CRITICAL THINKING

Consider the model Profit = (P − C)² × Q, where P = selling price, C = cost, and Q = quantity sold. Base values: P = $10, C = $4, Q = 100. Why might a standard OAT sensitivity analysis be misleading for this model compared to a linear model? Compute the sensitivity ratios for a +10% increase in P and a +10% increase in Q and discuss what you notice.

Show answer

Base profit = (10 − 4)² × 100 = 36 × 100 = $3,600. Increase P by 10% to $11: New profit = (11 − 4)² × 100 = 49 × 100 = $4,900. ΔProfit = $1,300, a 36.1% increase. Sensitivity ratio = 36.1% ÷ 10% = 3.61. Increase Q by 10% to 110: New profit = (10 − 4)² × 110 = 36 × 110 = $3,960. ΔProfit = $360, a 10% increase. Sensitivity ratio = 10% ÷ 10% = 1.0. The key insight is that Price enters through a squared term, so its effect is non-linear and amplified. A 10% increase gave a 36.1% profit jump, but a 10% decrease would give a different magnitude of change (not −36.1%). In non-linear models, sensitivity depends on the direction and size of the change, making OAT results valid only near the specific base case tested. This is why advanced methods use derivatives (instantaneous sensitivity) rather than finite percent changes.

SUMMARY

Lesson Summary

Summary

Sensitivity analysis is the systematic study of how changes in a model's inputs affect its outputs. The core tools include calculating the absolute change (ΔOutput), the percent change for fair comparison across different units, and the sensitivity ratio (percent change in output ÷ percent change in input) to determine whether an output is amplified or dampened relative to a given input.

Two key approaches are one-at-a-time (OAT) analysis, which isolates each input's effect, and scenario analysis, which examines best-case, base-case, and worst-case combinations. Inputs with the highest sensitivity ratios are critical inputs — they deserve the most attention when you are refining estimates or managing risk. While sensitivity analysis is powerful, remember that it does not tell you the probability of a change occurring, and results for non-linear models depend on the size and direction of the change tested.