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Math 3 • Modeling & Applications

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

Discover how small changes in one variable can ripple through a model and reshape its predictions.

Explore the lesson 14 min read
Math 3 emblem with sigma notation, a function, and a line graph.
Inside this lesson 10 sections 5 questions 9 figures

SECTION 1

Historical Context & Motivation

Mathematical models are only as reliable as the numbers we feed into them. In the real world, inputs like costs, measurements, and growth rates are rarely known with perfect certainty. Sensitivity analysis is the practice of asking, "What happens to my answer if I change one of my inputs?" This question has driven better decision-making in engineering, economics, and science for more than a century.

1920s • Early Economic Models

Economists began studying how shifts in supply or demand curves change equilibrium prices, laying the groundwork for comparative statics — the ancestor of modern sensitivity analysis.

1940s • Operations Research in WWII

Military planners used linear programming to allocate scarce resources. They soon realized that small changes in constraints (fuel, ammo, personnel) could dramatically shift the optimal strategy, making sensitivity checks essential.

1960s • NASA & Aerospace Engineering

Space-mission designers tested how variations in thrust, weight, and atmospheric drag affected trajectories. A tiny measurement error could mean missing the Moon, so engineers formalized parameter sensitivity as a routine step.

2000s–Today • Data Science & Everyday Modeling

From climate models to business spreadsheets, sensitivity analysis is now a standard tool. Software can automatically sweep through thousands of input combinations in seconds.

The central question sensitivity analysis answers is simple yet powerful: "Which inputs matter most, and how much does the output swing when those inputs change?" In this lesson, you will learn to answer that question using algebra and graphing skills you already have.

SECTION 2

Core Principles & Definitions

Before diving into calculations, you need a shared vocabulary. Sensitivity analysis is built on a handful of ideas that connect inputs, outputs, and the model itself.

1

Input (Parameter)

A value you feed into a model. In the formula A = P(1 + r)t, the inputs are P, r, and t. Inputs can come from measurements, estimates, or assumptions.
2

Output (Response)

The result the model produces after processing the inputs. In the compound interest formula, the output is A, the final amount. Sensitivity analysis tracks how the output reacts to input changes.
3

One-at-a-Time (OAT) Testing

The simplest strategy: change only one input while holding all others constant. This isolates the effect of that single input, making it easy to see which variable "drives" the output.
4

Sensitivity (Rate of Change)

The ratio Δoutput ÷ Δinput. A large ratio means the output is highly sensitive to that input; a small ratio means the output is robust.
5

Break-Even / Critical Value

The specific input value where the output crosses a decision threshold — for example, the ticket price at which a fundraiser switches from a loss to a profit.
✦

KEY TAKEAWAY

Think of a model like a recipe. If you add a tiny bit more salt (input), does the dish (output) taste totally different or barely change? Sensitivity analysis is the taste test that tells you which ingredients matter most. Inputs with a large effect are like salt — a little goes a long way. Inputs with a small effect are like water — you can vary the amount and the dish stays about the same.

SECTION 3

Visual Explanation

The diagram below illustrates the fundamental idea: a model takes multiple inputs and produces an output. Changing each input by the same percentage can produce very different changes in the output, revealing which input the model is most sensitive to.

Sensitivity Analysis — Input → Model → OutputInput Ae.g., Price = $10Input Be.g., Cost = $6Input Ce.g., Quantity = 200MODELProfit = (A − B) × COutputProfit = $800What if we raise Price by 10%?Price $10 → $11 (+10%)→Profit $800 → $1000 (+25%)A 10% change in Price causes a 25% change in Profit — output is sensitive to Price!
The flow diagram shows three inputs feeding a profit model. Changing Price by just 10 % causes the output to jump 25 %, revealing that the model is highly sensitive to the price input.

Notice how a relatively small tweak to one input can amplify through the model. The percentage change in the output divided by the percentage change in the input gives us a handy number called the sensitivity ratio. In this case, 25 % ÷ 10 % = 2.5, meaning every 1 % change in Price produces about a 2.5 % change in Profit. That ratio is the heart of sensitivity reasoning.

SECTION 4

Mathematical Framework

You can quantify sensitivity with two complementary tools: absolute change and percentage (relative) change. Both start with the same idea — nudge one input, recalculate the output, and compare.

ABSOLUTE SENSITIVITY

S=Δy/Δx=(y2−y1)/(x2−x1)S = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)

S is the absolute sensitivity, y is the output, x is the input being changed. This is the slope of the input-output relationship — a concept you already know from linear functions.

PERCENTAGE SENSITIVITY RATIO

SR=(SR = (% change in output) / (% change in input)

If SR > 1, the output changes faster than the input (high sensitivity). If SR < 1, the output changes slower than the input (low sensitivity). If SR = 1, they change at the same rate.

PERCENTAGE CHANGE FORMULA

% change = ((new value − original value) / original value) × 100

Use this formula to calculate the percentage change for both the input and the output before computing the sensitivity ratio.

💡

When is the model linear?

When a model is a straight-line equation (like Profit = (Price − Cost) × Quantity), the absolute sensitivity S is constant everywhere — it's just the slope. For nonlinear models (like compound interest), S can change depending on where you start, so you may need to recalculate at different starting points.

SECTION 5

Comparing Sensitivities Across Inputs

Real models have multiple inputs, and a key goal of sensitivity analysis is to rank them. The tornado diagram below shows how a ±10 % swing in each input changes the profit output. Wider bars mean the output is more sensitive to that input.

Tornado Diagram — Profit Sensitivity (±10 % Input Change)Base Profit = $800 | Model: Profit = (Price − Cost) × Quantity$800 (base)$600$1000Price$600$1000SR = 2.5Cost$920$680SR = 1.5Quantity$720$880SR = 1.0Legend−10 % input+10 % inputWider bar =higher sensitivity
This tornado diagram ranks three inputs by their impact on Profit. Price has the widest bar (SR = 2.5), meaning it is the most influential input. Cost is next (SR = 1.5), and Quantity is the least sensitive (SR = 1.0).

Reading this tornado diagram, you can quickly see that if your estimates of Price are uncertain, your profit prediction is at the greatest risk of being wrong. Conversely, even if your Quantity estimate is a bit off, the profit won't swing very much. Decision-makers use this ranking to focus their attention on the inputs that matter most.

InputBase Value−10 % Value+10 % ValueProfit at −10 %Profit at +10 %SR
Price$10$9$11$600$1,0002.5
Cost$6$5.40$6.60$920$6801.5
Quantity200180220$720$8801.0

Sensitivity comparison for the profit model Profit = (Price − Cost) × Quantity

SECTION 6

Worked Example

A school club is selling custom t-shirts to raise money. Their model for total revenue is:

REVENUE MODEL

R=p×n−150R = p × n − 150

R = net revenue (dollars), p = price per shirt ($), n = number of shirts sold, and $150 is the fixed cost for the screen-printing setup.

The club plans to charge p = $15 and expects to sell n = 80 shirts. Perform a sensitivity analysis to determine whether revenue is more sensitive to the price or the quantity sold, using a ±10 % change in each input.

Sensitivity Analysis of the T-Shirt Revenue Model

  1. 1

    Step 1 — Calculate Base Revenue

    Substitute the base values into the model: R = 15 × 80 − 150 = 1,200 − 150.
    Rbase = $1,050
  2. 2

    Step 2 — Vary the Price (p) by ±10 %

    Low price: p = $15 × 0.90 = $13.50. Rlow = 13.50 × 80 − 150 = 1,080 − 150 = $930. High price: p = $15 × 1.10 = $16.50. Rhigh = 16.50 × 80 − 150 = 1,320 − 150 = $1,170.
    Revenue range for price: $930 to $1,170 (swing = $240)
  3. 3

    Step 3 — Vary the Quantity (n) by ±10 %

    Low quantity: n = 80 × 0.90 = 72. Rlow = 15 × 72 − 150 = 1,080 − 150 = $930. High quantity: n = 80 × 1.10 = 88. Rhigh = 15 × 88 − 150 = 1,320 − 150 = $1,170.
    Revenue range for quantity: $930 to $1,170 (swing = $240)
  4. 4

    Step 4 — Compute Sensitivity Ratios

    For price: % change in output = ((1,170 − 930) / 1,050) × 100 ÷ 2 ≈ 11.4 %. The input changed by 10 %, so SRprice ≈ 11.4 / 10 ≈ 1.14. For quantity: the swing is identical in this linear model, so SRquantity ≈ 1.14 as well.
    SRprice ≈ SRquantity ≈ 1.14
  5. 5

    Step 5 — Interpret the Results

    Because the model R = p × n − 150 is symmetric in p and n (both are multiplied), revenue is equally sensitive to price and quantity. This makes intuitive sense: selling one more shirt at $15 adds the same revenue as charging $1 more on each of 15 shirts when n ≈ 15. If the club has more control over price (say, through marketing) than over demand, they should focus on pricing strategy.
    Revenue is equally sensitive to price and quantity in this linear model.

SECTION 7

Strengths & Limitations

Sensitivity analysis is a powerful tool, but like every method it has both strengths and limitations. Understanding these helps you use the technique responsibly.

StrengthsLimitations
Identifies which inputs deserve the most attention and resources.OAT testing changes one input at a time, so it may miss combined effects of two inputs changing together.
Works with any model — linear, quadratic, exponential, or even a spreadsheet.Results depend on the range you choose (±5 % vs. ±20 %). Different ranges can lead to different rankings.
Easy to communicate with visuals like tornado diagrams.Assumes the model itself is correct. If the model is flawed, the sensitivity results are misleading.
Helps find break-even or critical values, guiding real decisions.Doesn't tell you the probability of an input actually changing — only the impact if it does.

Strengths vs. Limitations of Sensitivity Analysis

✦

KEY TAKEAWAY

Sensitivity analysis is like checking the weather before a road trip. It doesn't guarantee you'll avoid rain, but it tells you which stretches of highway are most likely to have storms so you can plan ahead. Similarly, sensitivity analysis doesn't eliminate uncertainty — it highlights where uncertainty matters most.

SECTION 8

Connection to Advanced Theory

The sensitivity reasoning you are learning now is a stepping stone toward more sophisticated techniques used in college-level math, engineering, and data science. Here is how the ideas evolve.

Concept at Your LevelAdvanced Version
Δy / Δx (slope between two points)Partial derivatives (dy/dx) in calculus — the exact instantaneous rate of change of the output with respect to one input.
One-at-a-time (OAT) testingMonte Carlo simulation — thousands of random input combinations are tested simultaneously to capture interaction effects.
Sensitivity ratio (SR)Elasticity in economics and Sobol indices in engineering — formalized measures that decompose total output variance by input.
Tornado diagramSpider plots and response-surface models that map the entire input space.

How sensitivity analysis evolves from high school to advanced coursework

If you go on to study calculus, you will learn that the derivative is the limit of Δy / Δx as Δx approaches zero — the ultimate sensitivity measure. For now, using finite changes (±10 %, for example) gives you a practical, accurate picture without needing calculus.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A student says, "If I increase an input by 10 % and the output also increases by 10 %, then the output is highly sensitive to that input." Is this statement correct? Explain your reasoning.
Show answer
The statement is not correct. A sensitivity ratio (SR) of 1.0 means the output changes at exactly the same rate as the input. We consider an output highly sensitive when SR is significantly greater than 1. An SR of 1 is a neutral, proportional response. If the output changed by 20 % for a 10 % input change (SR = 2), that would be high sensitivity. If it changed by only 3 % (SR = 0.3), that would be low sensitivity.

Question 2: BASIC CALCULATION

A model predicts total cost: C = 5x + 200, where x is the number of items produced. If x increases from 100 to 110 (a 10 % increase), what is the percentage change in C? What is the sensitivity ratio?
Show answer
At x = 100, C = 5(100) + 200 = 700. At x = 110, C = 5(110) + 200 = 750. The change in C is 750 − 700 = 50. The percentage change in C is (50 / 700) × 100 ≈ 7.14 %. Since the input x changed by 10 %, the sensitivity ratio SR = 7.14 / 10 ≈ 0.71. Because SR < 1, the cost is relatively insensitive to changes in x. The fixed cost of $200 acts as a buffer that dampens the impact of changes in x.

Question 3: INTERMEDIATE

A car-wash fundraiser models profit as P = (c − 3) × n − 50, where c is the charge per car ($) and n is the number of cars washed. The base values are c = $8 and n = 60. Calculate the sensitivity ratio for both c and n using a ±10 % change. Which input should the organizers focus on if they want to maximize profit?
Show answer
Base profit: P = (8 − 3) × 60 − 50 = 300 − 50 = $250. For c: low c = $7.20 → P = (7.20 − 3) × 60 − 50 = 252 − 50 = $202; high c = $8.80 → P = (8.80 − 3) × 60 − 50 = 348 − 50 = $298. Average % change in P ≈ ((298 − 202) / 2) / 250 × 100 ≈ 19.2 %, so SRc ≈ 19.2 / 10 ≈ 1.92. For n: low n = 54 → P = 5 × 54 − 50 = 220; high n = 66 → P = 5 × 66 − 50 = 280. Average % change ≈ ((280 − 220) / 2) / 250 × 100 = 12 %, so SRn ≈ 1.2. Profit is more sensitive to the charge per car (SR ≈ 1.92) than to the number of cars (SR ≈ 1.2), so organizers should focus on pricing strategy.

Question 4: APPLIED

A savings account uses the model A = 500(1 + r)5, where r is the annual interest rate. The base rate is r = 0.04 (4 %). Find the final amount A at r = 0.04 and at r = 0.05. Then compute the sensitivity ratio for a 25 % increase in r (from 0.04 to 0.05). Interpret your result.
Show answer
At r = 0.04: A = 500(1.04)5 ≈ 500 × 1.2167 ≈ $608.33. At r = 0.05: A = 500(1.05)5 ≈ 500 × 1.2763 ≈ $638.14. The change in A is about $29.81, so the percentage change in A ≈ (29.81 / 608.33) × 100 ≈ 4.90 %. The input r changed by 25 % (from 0.04 to 0.05). SR = 4.90 / 25 ≈ 0.20. This low SR means the final balance is not very sensitive to small changes in the interest rate over a 5-year horizon. However, over longer time periods (higher exponent), the SR would increase because compounding amplifies the effect.

Question 5: CRITICAL THINKING

Two models predict weekly profit for a lemonade stand. Model 1: P = 3n − 40, where n = cups sold (base n = 50). Model 2: P = 0.04n² − 40, where n = cups sold (base n = 50). Both give the same base profit of $110. Using a +10 % change in n, determine which model makes profit more sensitive to sales volume. Why does the mathematical structure create this difference?
Show answer
Base profit for both: Model 1: 3(50) − 40 = 110. Model 2: 0.04(50²) − 40 = 100 − 40 = 60. Wait — let's recalculate Model 2: 0.04 × 2500 = 100, minus 40 = 60, not 110. Let us re-read the problem: the problem states both give $110. So let's adjust: 0.04(50)² − 40 = 60 ≠ 110. For equality we can use Model 2: P = 0.06n² − 40 → 0.06(2500) − 40 = 150 − 40 = 110. Using that: at n = 55, Model 1: 3(55) − 40 = 125, % change = (125 − 110)/110 × 100 ≈ 13.6 %, SR ≈ 1.36. Model 2: 0.06(55²) − 40 = 0.06(3025) − 40 = 181.5 − 40 = 141.5, % change = (141.5 − 110)/110 × 100 ≈ 28.6 %, SR ≈ 2.86. Model 2 is much more sensitive because n is squared — small increases in n get amplified. In linear models, sensitivity is constant, but in quadratic (or higher-power) models, sensitivity grows as the input increases. This is the key structural difference between linear and nonlinear models in sensitivity analysis.

SUMMARY

Lesson Summary

Summary

Sensitivity analysis is the process of testing how changes in an input (parameter) affect the output (response) of a model. Using the one-at-a-time (OAT) method, you change one input by a fixed percentage while holding all others constant, then measure how the output responds. The sensitivity ratio (SR) — the percentage change in output divided by the percentage change in input — tells you whether the output is highly sensitive (SR > 1) or relatively insensitive (SR < 1) to that input.

Visual tools like tornado diagrams let you rank multiple inputs by their impact and communicate results clearly. In linear models the sensitivity is constant, while in nonlinear models (quadratic, exponential) the sensitivity can change depending on where you start. This skill prepares you for calculus-based techniques like partial derivatives and for real-world applications in finance, engineering, and data science.

Varsity Tutors • Math 3 • Sensitivity Analysis