What is a scientific model? Why all models are wrong but useful

A scientific model is a simplified representation of a real system, built to explain how it works or to predict what it will do. It might be a physical object, an equation, a computer program or a diagram. Every model leaves things out on purpose, the way a subway map drops streets and distances so that the route stands out. A good model is judged not by how complete it is but by how well it does the job it was built for.

In short

• A model is a deliberate simplification: it keeps the parts of a system that matter for a question and drops the rest.
• Models come as physical objects, equations, computer simulations and diagrams, and scientists test them by comparing their predictions with observations.
• No model is the full truth, which is George Box's point that "all models are wrong, but some are useful": the question is where each one stops working.

What counts as a scientific model?

A model stands in for something harder to study directly. An atom is too small to see, the climate too big to experiment on, a disease too dangerous to give to people on purpose. So scientists build something they can handle, a set of equations, a scale model, a strain of laboratory mouse, and study that instead, on the assumption that what they learn carries over.

That assumption is what makes a model scientific. A model makes claims about the real system, and those claims can be checked against evidence. A model of a bridge predicts how far it will bend under a load; engineers can measure the bend. A model is in this sense a structured hypothesis: a precise statement of how a system is thought to work, worked out far enough to produce numbers.

What a model leaves out is a choice, not a flaw. A map that showed every tree would be as big as the land and just as hard to read. The Stanford Encyclopedia of Philosophy describes how scientists idealize on purpose, treating a planet as a point mass or a gas as tiny bouncing balls, because the simpler picture makes the important behavior visible.

What kinds of models do scientists use?

The main kinds differ in what they are made of, not in what they are for.

Kind	What it is	Example	Good for	Where it misleads
Physical or scale	A real object standing in for another	A wind-tunnel model of a wing; the double-helix model of DNA built from metal plates	Seeing shape and testing forces directly	Small things don't always behave like large ones
Mathematical	Equations relating quantities	Newton's law of gravity; a population growth equation	Exact predictions and general rules	Only as good as its assumptions
Computer simulation	A program that steps equations forward in time, often in many small pieces	Weather forecasts; climate models; crash simulations	Systems too complex to solve by hand	Errors in setup can hide inside millions of steps
Conceptual	A picture or description of how parts relate	The water cycle diagram; the atom as a tiny solar system	Teaching and organizing ideas	Easy to mistake for literal truth
Model organism	A living species studied to learn about others	Fruit flies, laboratory mice, yeast	Genetics, disease and drug research	Mice are not people, and many results don't transfer

How does a model make a prediction? A population in numbers

Take a population of 100 rabbits in a meadow. The simplest model says each year the population grows by 50%: next year's number is this year's times 1.5. That is exponential growth, and it predicts 150 rabbits after one year, 225 after two, and 5,767 after ten.

A slightly better model adds one fact the first one ignored: the meadow can feed only so many rabbits, say 1,000, called the carrying capacity and written K. In this logistic model, growth slows as the population nears the limit. Each year the population N grows by 0.5 × N × (1 − N/K). The term in brackets is close to 1 when the meadow is nearly empty and close to 0 when it is nearly full.

Year	Exponential model	Logistic model (K = 1,000)
0	100	100
1	150	145
3	338	289
5	759	511
7	1,709	752
10	5,767	951

Step through year 1 in the logistic model: 100 × 0.5 × (1 − 100/1,000) = 100 × 0.5 × 0.9 = 45, so the population goes from 100 to 145. By year 10 the bracket is small, and the population creeps toward 1,000 instead of exploding.

Both models are wrong; the second is wrong in a more useful way. The two agree for the first couple of years and then part sharply, which is exactly where field counts can tell them apart. Neither includes foxes, harsh winters or disease, and a real rabbit population would rise and fall with all of them. A biologist would add those only if the question needed them.

How did the model of the atom change?

The atom shows how models are replaced as evidence comes in, each one kept for what it still does well. In the early 1800s, John Dalton pictured atoms as tiny solid balls, one kind for each element. After J. J. Thomson discovered the electron in 1897, he proposed in 1904 that atoms were a spread of positive charge with electrons dotted through it, the "plum pudding". In 1911, Ernest Rutherford showed from the way particles bounced off gold foil that nearly all the mass sits in a tiny central nucleus. In 1913, Niels Bohr put the electrons in fixed orbits with fixed energies, which explained the exact colors of light that hydrogen gives off (Wikipedia). From 1926, quantum mechanics replaced the orbits with orbitals, clouds of probability for where an electron may be found.

Bohr's model is wrong: electrons don't circle the nucleus like planets, and the model fails for atoms with more than one electron. Yet it is still taught, because it gets hydrogen's energy levels right and gives the right idea of electron shells. What is an atom? covers what the current picture says.

What did George Box mean by "all models are wrong"?

The British statistician George Box made the point several times, and its best-known wording comes from a 1987 book he wrote with Norman Draper: "Essentially, all models are wrong, but some are useful" (Wikipedia). He meant that a model is never an exact copy of reality, so the right question is not whether it is true but whether it is close enough for the purpose, and where it breaks down.

Newton's laws are the classic case. They are wrong at speeds near light's and in very strong gravity, where relativity is needed. They are also accurate enough to have guided spacecraft to the Moon. Knowing a model's limits is part of knowing the model.

The common misconception runs in both directions. Some people treat a model's output as fact; others dismiss any model as "just a model". Both miss what a model is for. Its value lies in its tested track record within a stated range.

How are climate models tested?

Climate models are computer simulations that divide the atmosphere and oceans into a grid and apply the laws of physics to each cell. In the 1960s Syukuro Manabe led the development of physical models of Earth's climate, showing how more carbon dioxide raises surface temperatures, work recognized by the 2021 Nobel Prize in Physics.

One test is to check old projections against what happened. A study led by Zeke Hausfather, published in 2020, compared 17 model projections made between 1970 and 2007 with global temperatures observed through 2017. Ten closely matched, and 14 did once the researchers allowed for differences between the emissions the models assumed and those that actually occurred, with no sign that the models as a group ran too hot or too cold, as NASA summarized. The same summary notes the limits: models differ widely in how strongly they respond to added carbon dioxide, and many uncertainties remain in the details. What is climate change? sets out what the evidence as a whole shows.

Check yourself

• Without looking back, say what the logistic model adds to the exponential one and why the two predictions part after year 2.
• Why is Bohr's model of the atom still taught, even though it is wrong?
• What would you ask about any model before trusting its prediction?

Sources

• Models in Science — Stanford Encyclopedia of Philosophy
• All models are wrong — Wikipedia
• Bohr model — Wikipedia
• Study confirms climate models are getting future warming projections right — NASA
• The Nobel Prize in Physics 2021, press release — Royal Swedish Academy of Sciences, 2021
• Scientific modelling — Wikipedia