The way we understand and make sense of variation in the world affects decisions we make.

Part of understanding variation is understanding the difference between deterministic and probabilistic (stochastic) models. The NZ curriculum specifies the following learning outcome: “Selects and uses appropriate methods to investigate probability situations including experiments, simulations, and theoretical probability,** distinguishing between deterministic and probabilistic models.**” This is at level 8 of the curriculum, the highest level of secondary schooling. Deterministic and probabilistic models are not familiar to all teachers of mathematics and statistics, so I’m writing about it today.

## Model

The term, model, is itself challenging. There are many ways to use the word, two of which are particularly relevant for this discussion. The first meaning is “mathematical model, as a decision-making tool”. This is the one I am familiar with from years of teaching Operations Research. The second way is “way of thinking or representing an idea”. Or something like that. It seems to come from psychology.

When teaching mathematical models in entry level operations research/management science we would spend some time clarifying what **we** mean by a model. I have written about this in the post, “All models are wrong.”

In a simple, concrete incarnation, a model is a representation of another object. A simple example is that of a model car or a Lego model of a house. There are aspects of the model that are the same as the original, such as the shape and ability to move or not. But many aspects of the real-life object are missing in the model. The car does not have an internal combustion engine, and the house has no soft-furnishings. (And very bumpy floors). There is little purpose for either of these models, except entertainment and the joy of creation or ownership. (You might be interested in the following video of the Lego Parisian restaurant, which I am coveting. Funny way to say Parisian!)

Many models perform useful functions. My husband works as a land-surveyor, and his work involves making models on paper or in the computer, of phenomenon on the land, and making sure that specified marks on the model correspond to the marks placed in the ground. The purpose of the model relates to ownership and making sure the sewers run in the right direction. (As a result of several years of earthquakes in Christchurch, his models are less deterministic than they used to be, and unfortunately many of our sewers ended up running the wrong way.)

Our world is full of models:

- a map is a model of a location, which can help us get from place to place.
- sheet music is a written model of the sound which can make a song
- a bus timetable is a model of where buses should appear
- a company’s financial reports are a model of one aspect of the company

## Deterministic models

A deterministic model assumes certainty in all aspects. Examples of deterministic models are timetables, pricing structures, a linear programming model, the economic order quantity model, maps, accounting.

## Probabilistic or stochastic models

Most models really should be stochastic or probabilistic rather than deterministic, but this is often too complicated to implement. Representing uncertainty is fraught. Some more common stochastic models are queueing models, markov chains, and most simulations.

For example when planning a school formal, there are some elements of the model that are deterministic and some that are probabilistic. The cost to hire the venue is deterministic, but the number of students who will come is probabilistic. A GPS unit uses a deterministic model to decide on the most suitable route and gives a predicted arrival time. However we know that the actual arrival time is contingent upon all sorts of aspects including road, driver, traffic and weather conditions.

## Model as a way of thinking about something

The term “model” is also used to describe the way that people make sense out of their world. Some people have a more deterministic world model than others, contributed to by age, culture, religion, life experience and education. People ascribe meaning to anything from star patterns, tea leaves and moon phases to ease in finding a parking spot and not being in a certain place when a coconut falls. This is a way of turning a probabilistic world into a more deterministic and more meaningful world. Some people are happy with a probabilistic world, where things really do have a high degree of randomness. But often we are less happy when the randomness goes against us. (I find it interesting that farmers hit with bad fortune such as a snowfall or drought are happy to ask for government help, yet when there is a bumper crop, I don’t see them offering to give back some of their windfall voluntarily.)

Let us say the All Blacks win a rugby game against Australia. There are several ways we can draw meaning from this. If we are of a deterministic frame of mind, we might say that the All Blacks won because they are the best rugby team in the world. We have assigned cause and effect to the outcome. Or we could take a more probabilistic view of it, deciding that the probability that they would win was about 70%, and that on the day they were fortunate. Or, if we were Australian, we might say that the Australian team was far better and it was just a 1 in 100 chance that the All Blacks would win.

I developed the following scenarios for discussion in a classroom. The students can put them in order or categories according to their own criteria. After discussing their results, we could then talk about a deterministic and a probabilistic meaning for each of the scenarios.

- The All Blacks won the Rugby World Cup.
- Eri did better on a test after getting tuition.
- Holly was diagnosed with cancer, had a religious experience and the cancer was gone.
- A pet was given a homeopathic remedy and got better.
- Bill won $20 million in Lotto.
- You got five out of five right in a true/false quiz.

The regular mathematics teacher is now a long way from his or her comfort zone. The numbers have gone, along with the red tick, and there are no correct answers. This is an important aspect of understanding probability – that many things are the result of randomness. But with this idea we are pulling mathematics teachers into unfamiliar territory. Social studies, science and English teachers have had to deal with the murky area of feelings, values and ethics forever. In terms of preparing students for a random world, I think it is territory worth spending some time in. And it might just help them find mathematics/statistics relevant!

“The regular mathematics teacher is now a long way from his or her comfort zone. The numbers have gone, along with the red tick, and there are no correct answers. This is an important aspect of understanding probability – that many things are the result of randomness.”

How true! I work with a lot of qualitative researchers, and although most of my fellow statisticians tend to scoff at their practices, I think they have got a massive head start on us with this. They train their students to think about the philosophy of what they are trying to do, what they believe could be a reasonable model for the world, and what could affect the results they get, before they start collecting data. If only we could get there too. “Statistics is no substitute for thinking” is my latest catchphrase. But you are right that it is much easier to teach and assess simple rules…

Hi Robert – thanks for that. I hadn’t thought about qualitative research for a while. My Phd used mixed methods, starting with qualitative research. As part of that I wrote a statement of bias, where I examined my own views on the topic. It was very helpful. Many quantitative researchers see themselves and their numbers as objective, which I believe to be impossible. That would be a good topic for a post! It would be interesting to explore how maths teachers feel about qualitative and quantitative research.

Another fascinating twist is that much of the research into statistics education is qualitative – to understand how people think about things like probability and inference.

Would f-Test, T-Test and Z-Test be considerede stochastic models?

Good question. The tests themselves are not exactly models, but are based on stochastic models. For example the t-test creates a model of where we would expect a value for a test statistic to fall. This uses the t distribution. We then compare our observed value with the model value to see if what we have observed fits in the model. Pretty much anything to do with statistical analysis is based on stochastic (or probabilistic) models.