How to Refine a Mathematical Model in a PSMT

To refine a mathematical model in a Maths Methods PSMT, you need to identify a specific weakness in your initial model using mathematical evidence, propose a justified change and demonstrate that the refined model genuinely improves the accuracy, reliability or applicability of your solution. Refinement isn't about adding equations for the sake of it, it's about responding to what your evaluation reveals. If your initial model works well within the defined domain and answers the original question, you don't need to refine it.

This guide walks through the refinement process step by step with a worked example, a comparison table and a checklist to help you decide whether refinement is needed and how to communicate it clearly.

What is a Model Refinement?

In your Methods PSMT, mathematical modelling involves developing an initial model for your given data and real-life context, arriving at a solution to the take, identifying where the model doesn't quite work or is limited and then considering whether a change would produce a better result. That process of testing and improving is what refinement means.

In a PSMT, refinement might involve:

• Adjusting the parameters or coefficients of your model based on new evidence

• Changing the domain to better reflect the real-world constraints

• Testing a different function family when the initial one doesn't fit the data well

• Developing a piecewise model when a single function can't capture the full behaviour

• Removing or adjusting data points with clear justification

• Revisiting an assumption that turned out to be unrealistic

Do You Actually Need to Refine Your Model in Your Maths PSMT?

This is the first question to ask honestly. Refinement is only warranted when there's evidence that your initial model has a meaningful weakness that a realistic change can address.

Refinement is likely needed when:

• The residual plot shows a clear pattern, suggesting the relationship isn't what you assumed

• Your model's predictions deviate significantly from observed values at certain points

• The model produces unreasonable outputs within the valid domain

• An assumption you made in the formulate section clearly didn't hold

• A different function type would better capture the shape of the data

  
  

Refinement probably isn't needed when:

• Your model fits the data well and the residuals are roughly random

• The initial model answers the original question within an acceptable margin of error

• The only reason to change is to improve R² by a small amount

• A more complex model would reduce interpretability without meaningfully improving accuracy

A Worked Example: NBA Player Performance Model

☑️ Task context: 

You're given data on NBA basketball players' points per game across a season and asked to develop a model that could guide a team's draft selection for next season.

📈 Initial model: 

You fit a linear regression model to the data showing points per game over the course of the season, producing the equation y = 0.4x + 18.2 with an R² of 0.71.

🎯 Testing the initial model:

When you plot the residuals, you notice they follow a curved pattern rather than being randomly distributed around zero. This suggests the linear model isn't capturing the full shape of the relationship. You also notice that performance tends to peak mid-season and decline toward the end, which a linear model can't represent.

🔎 Evidence for refinement:

The residual plot shows a systematic pattern. The linear model overestimates performance in the early and late season and underestimates it mid-season. This is a meaningful weakness that a different function type could address.

✅ The refinement:

You test a quadratic regression model, producing y = -0.03x² + 1.2x + 14.6 with an R² of 0.89. The residuals are now roughly randomly distributed.

💬 Communicating the refinement:

✍️ Conclusion: 

The quadratic model is preferred because the residual analysis and improved R² value together demonstrate a better fit to the data. The model now captures the mid-season performance peak that the linear model missed, making it more useful for draft decision-making.

Types of Refinement and When to Use Them

What You Can and Can't Change

You can change:

• The type of function used to model the relationship

• The parameters or coefficients based on additional evidence

• The domain of the model

• The structure of the model, for example from a single function to a piecewise model

• Which data points are included, with clear justification

You shouldn't change:

• Data points simply because they make R² lower

• The model type without evidence that the original was inadequate

• Assumptions retroactively to avoid a limitation that already exists

• The model in ways that make it less interpretable without demonstrating a meaningful accuracy gain

How to Communicate Your Refinement Clearly 💬

A well-communicated refinement tells the marker:

1. What the problem with the initial model was and how you identified it

2. What you changed and why that change addresses the problem

3. How the refined model performs compared to the original

4. What the refined model still can't do, that is, its remaining limitations

✅ Strong sentence starters for each stage:

Identifying the problem:

• "Analysis of the residuals from the initial model reveals a [pattern], suggesting that [issue with the model]."

• "The initial model underestimates / overestimates [outcome] at [point or range], with a percentage error of [value]%, which indicates [specific weakness]."

Proposing the change:

• "To address this, the model was refined by [change], as this better accounts for [specific feature of the data or context]."

• "A [new function type] was tested because [reason it's more appropriate for this context]."

Comparing the models:

• "Comparing the two models using [R², residuals, percentage error], the refined model shows [specific improvement]."

• "The refined model reduces the average prediction error from [original] to [refined], representing an improvement of [value]%."

Acknowledging remaining limitations:

• "While the refined model improves [aspect], it remains limited by [remaining issue], which would require [further development] to address fully."

Before and After: PSMT Model Refinement Checklist

Before refining:

☐ I've identified a specific weakness in the initial model using mathematical evidence

☐ The weakness is meaningful, not just a minor inaccuracy

☐ I've explained what the evidence shows, such as residual pattern, percentage error or poor fit at certain values

☐ I've identified a realistic change that directly addresses this weakness

During refinement:

☐ I've explained what I changed and why the change is mathematically justified

☐ I haven't removed data points without a clear, documented reason

☐ The refined model still makes sense in the real-world context

☐ I've used technology to test and verify the refined model

After refinement:

☐ I've compared the original and refined models using the same criteria

☐ The comparison shows a genuine improvement, not just a different result

☐ I've acknowledged what the refined model still can't do

☐ I've verified the refined results using a second method or substitution

☐ I've updated my assumptions and limitations to reflect the refined model

Getting Extra Tutoring Support For Your Maths Methods PSMT

If you're unsure whether your model needs refining, what to change or how to justify it in a way that meets the ISMG criteria, working through it with an in-person or online tutor before you submit can prevent you from either forcing unnecessary refinement or missing a genuine opportunity to improve your mark. Our Maths Methods tutors work with Year 11 and Year 12 students through every section of the PSMT evaluation including refinement, verification and limitations. Book a free trial lesson to get started.

Frequently Asked Questions

How do I know when my mathematical model needs to be refined?

Your model needs refinement when there's mathematical evidence that it's not performing well enough within the context of the task. Common signals include a residual plot that shows a systematic pattern rather than random scatter, significant percentage errors between predicted and observed values, a model that produces unreasonable outputs within the valid domain, or an assumption that clearly didn't hold and has affected the accuracy of the result. If your initial model fits the data well and answers the original question, refinement isn't necessary.

What parts of my PSMT model can I change during refinement?

You can change the type of function used, the parameters or coefficients, the domain of the model, the structure from a single function to a piecewise model, or which data points are included with clear justification. What you shouldn't do is change data points to improve a statistic without justification, change the model type without evidence the original was inadequate, or add complexity without showing it produces a meaningfully better result.

How can I compare my original and refined mathematical models?

Use the same criteria for both models. This typically includes R² value, a residual plot, the average percentage error between predicted and observed values and whether the model produces reasonable outputs across the valid domain. Present the comparison in a table so the improvement is clear to the marker. The refined model should perform better on at least one of these measures, and any remaining weaknesses should be acknowledged.

Do I need to refine my mathematical model in every PSMT?

No. Refinement should only be included when there's evidence that your initial model has a meaningful weakness that a realistic change can address. If your initial model fits the data well, produces reasonable predictions and answers the original question, forced refinement adds complexity without adding value and may actually reduce the clarity of your report. The ISMG rewards genuine, evidence-based refinement, not refinement for its own sake.

How should I justify and communicate the refinements I made?

Explain what was wrong with the initial model using specific mathematical evidence, describe what you changed and why that change addresses the problem, compare the original and refined models using the same criteria and acknowledge what the refined model still can't do. Use the sentence starters in this article to structure each stage. For further guidance on communicating your evaluation clearly, see How to Evaluate a Maths Methods PSMT Model.

Other related articles:

• What Is a PSMT in QCE Maths Methods?

• Maths Methods PSMT Structure Explained

• How to Write Assumptions in a Maths Methods PSMT

• How to Write Limitations in a Maths Methods PSMT

• Maths Methods PSMT Common Mistakes

• How to Evaluate a Maths Methods PSMT Model

• Observations vs Assumptions in a Maths Methods PSMT