Models and assumptions — Foundations for Math, Statistics & Analytics Careers

What you'll learn

State a model's assumptions and limits
Recognise common assumptions such as independence, stationarity and linearity, and when they break
Use sensitivity analysis to show which assumption matters most

State a model's assumptions and the limits of where it can be trusted.

Why this matters

Every number an analyst produces comes from a model: a simplified mathematical description of something real. Simplification is the point, because a model that includes everything is as complicated as the world. But every simplification is an assumption, and when an assumption stops being true the model can fail badly. The statistician George Box put it memorably: "All models are wrong, but some are useful." The professional skill is knowing which assumptions you made and where the model stops being useful.

What a model is

A model has three parts:

Inputs (data and parameters), such as last year's claim counts or the average time to serve a customer.
Structure: the mathematical relationship, such as "demand falls in a straight line as price rises" or "customers arrive at random at a steady average rate".
Outputs: the prediction, price, schedule or estimate you use to make a decision.

Assumptions live in all three. Inputs assume the data is accurate and relevant. Structure assumes the relationship has a particular shape. Outputs assume the future resembles the period the model was built on.

Common assumptions to look for

Assumption	What it means	When it breaks
Independence	One event does not affect another	Floods damage many houses at once; infections spread
Stationarity	The process does not change over time	New laws, new products, a pandemic
Distribution shape	Data follows, say, a normal (bell-shaped) distribution	Rare large losses ("fat tails") are more common than a normal curve predicts
Linearity	Doubling an input doubles the effect	Effects saturate or accelerate
Range of validity	The model was fitted on certain values	Predicting far outside them (extrapolation)
Representative data	The sample reflects the population	Survey only reaches people with landlines

Worked example 1: a simple claims model

An insurer models the number of home insurance claims per month as random, independent events with an average of 50. Under this model, months with more than about 65 claims are rare.

Stated assumptions: claims are independent; the average rate is stable; the policy count does not change much.

Where it breaks: a major storm causes 2,000 claims in one week. The claims were not independent: one event caused all of them. Insurers handle this by modelling catastrophes separately, buying reinsurance and holding capital. The everyday model is still useful for ordinary months, as long as everyone knows it does not cover catastrophes.

Worked example 2: a straight-line forecast

An analyst fits a straight line to five years of a café's monthly sales, which rose steadily from 2,000 to 3,000 cups. The line predicts 6,000 cups in ten years.

Assumptions: growth continues at the same rate; nothing limits demand; the past five years are typical.

Limits: the café has 40 seats and fixed opening hours, so physical capacity caps sales. Extrapolating a trend far beyond the data ignores that cap. A better statement is: "If recent growth continues, sales could reach about 3,200 cups next year. Beyond that, seating capacity becomes the limit."

Worked example 3: an optimisation model

An OR analyst's staffing model assumes each nurse can handle the same number of patients per hour. It recommends cutting one evening shift. In reality, evening patients are sicker and take longer. The model's output was correct for its assumptions, and wrong for the hospital. Checking assumptions with the people who do the work, here the nurses, would have caught it.

How to state assumptions and limits

Good analysts write them down next to the result, in plain language:

What we assumed: the main assumptions, ideally three to five.
Why it is reasonable: evidence such as past data or expert input.
What would change the answer: which assumption matters most.
Where not to use it: conditions outside the model's valid range.

Sensitivity analysis makes this concrete: change one input or assumption at a time and see how much the output moves. If a small change in an uncertain input flips the decision, say so.

Common mistakes

Hiding assumptions. Decision-makers cannot judge a result whose assumptions they never saw.
Extrapolating. A model fitted on one range of data is weakest outside that range.
Confusing fit with truth. A model can match past data well and still predict poorly.
Assuming independence by default. Many real risks move together, which matters most in bad times.
Never revisiting. Conditions change; models need regular checks against new data.

Quick recap

A model is a deliberate simplification; every simplification is an assumption.
Check independence, stationarity, distribution shape, linearity, range and representativeness.
State assumptions, their justification, what would change the answer and where not to use the model; test sensitivity.

Try it

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Key terms (5)

model: A simplified mathematical description of something real, made of inputs, structure and outputs.
assumption: A simplification a model relies on, such as independent events, which can make the model fail when it stops being true.
Stationarity: The assumption that a process does not change over time.
extrapolation: Using a model to predict far outside the range of data it was fitted on, where it is weakest.
Sensitivity analysis: Changing one input or assumption at a time to see how much the model's output moves.