Mathematical modelling

Definition. A statistical model replaces a complex real-world situation with a small set of random variables and a probability distribution. The model captures what we care about (often the variation between observations) and ignores everything else.

Why model? Reality is messy: every light bulb has its own lifetime, every leaf its own length, every commute its own duration. We cannot describe each individual case. Instead we ask: 'what is the typical pattern, and how does it vary?' A statistical model is the answer.

Examples.

• Lifetimes of light bulbs $X$ modelled as $X \sim N(\mu, \sigma^{2})$.
• Number of phone calls per hour at a help-desk modelled as Poisson with mean $\lambda$.
• Outcome of a coin toss modelled as Bernoulli$(p)$.

Key vocabulary.

Term	Meaning
Population	the entire set of items/events we care about
Sample	a finite subset of the population we observe
Parameter	a number that fixes the model (e.g. $\mu$, $\sigma$, $\lambda$, $p$)
Statistic	a number computed from a sample (e.g. $\bar{x}$, $s$) — used to estimate parameters

The four stages.

1. Real problem. Identify the variables of interest and the question being asked. Example: 'What is the probability that a light bulb lasts more than $1500$ hours?'
2. Construct (or choose) a model. Pick a probability distribution and state the assumptions. Example: 'Assume lifetimes $X \sim N(\mu, \sigma^{2})$ with $\mu$ and $\sigma$ to be estimated from data.'
3. Analyse. Collect a sample and apply the model — estimate parameters, compute probabilities, make predictions. Example: from $200$ measurements get $\bar{x} = 1200$, $s = 480$, hence estimate $P(X > 1500)$.
4. Interpret and refine. Compare the model's predictions with reality. If they agree, the model is adequate. If not, refine — change the distribution, transform the data, or relax an assumption — and return to step 2.

Visual.

   Real problem
        │
        ▼
   Statistical model ◄────────┐
        │                     │  (refine)
        ▼                     │
   Analyse / collect data     │
        │                     │
        ▼                     │
   Interpret & compare ───────┘

Why the feedback loop matters. A model is rarely right on the first try. Edexcel routinely awards a mark for explicitly mentioning that the cycle repeats — i.e. the model is refined in light of evidence.

Advantages.

• Simplification: huge amounts of data are reduced to a small number of parameters.
• Quantifiable prediction: probabilities and confidence statements can be computed.
• Generalisation: predictions can be made about cases not yet observed.
• Cost and speed: a model is cheap to apply once built — no need for a full census.
• What-if analysis: changing a parameter (e.g. doubling the production rate) is easy in the model.

Limitations.

• Approximation only: the model omits details that may matter (rare events, structural change).
• Distributional misfit: if the chosen distribution does not match the real-world shape (e.g. positive skew vs the normal), predictions can be badly wrong.
• Assumption fragility: independence, identical distribution, constant rate — every assumption can fail.
• Non-stationarity: the underlying process may change over time (new competitor, regulation change) so past data no longer predicts the future.
• Extrapolation risk: predicting outside the range of the data is unsafe.

Exam discipline. When asked for advantages or limitations, state them in context. 'A normal model allows negative lifetimes which is physically impossible' scores far higher than 'the model is not perfect'.

Why list assumptions? They are the model's load-bearing walls. If they fail, the predictions fail.

Common S1 assumptions and their failure modes.

Assumption	Used in	Failure mode	Symptom
Independent observations	most models	Repeated measurements on same item	Patterned residuals
Identically distributed	most models	Mixing two populations (e.g. day & night demand)	Bimodal histogram
Normal distribution	regression, normal	Skew or heavy tails	Skewness $\neq 0$
Constant probability $p$	Bernoulli, binomial	Trial-to-trial drift	$\hat p$ shifts across the sample
Fixed mean / variance	summary statistics	Trend in time	Mean changes by sub-group

Checking assumptions. Most S1-level checks are visual:

• Histogram against a fitted curve.
• Box plot for symmetry and outliers.
• Comparison of mean and median ($\bar{x} \approx \text{median}$ $\Rightarrow$ symmetric).
• Pearson's skewness coefficient $\dfrac{3(\bar{x} - \text{median})}{\sigma}$ — near zero indicates symmetry.

Relaxing assumptions. If an assumption fails, the model can be refined:

• Transform the variable (e.g. work with $\ln X$).
• Adopt a different family (e.g. exponential instead of normal).
• Add a parameter (e.g. allow two means for two sub-groups).
• Collect more data to confirm the failure is real, not chance.

Setting. A factory records lifetimes (in hours) of $200$ bulbs: $\bar{x} = 1200$, median $= 1050$, $s = 480$, histogram shows a long right tail.

Step 1 — propose a model. 'Lifetimes are normally distributed.'

Step 2 — check assumptions.

• Mean $= 1200$, median $= 1050$ $\Rightarrow$ mean $>$ median $\Rightarrow$ positive skew.
• Pearson's skewness $= \dfrac{3(1200 - 1050)}{480} = 0.94$. Substantially positive.
• The normal range $\bar{x} \pm 3s = 1200 \pm 1440 = (-240, 2640)$ — the lower bound is negative, but lifetimes cannot be negative.

Step 3 — interpret. The data is positively skewed and bounded below by zero. The normal assumption fails on two counts: shape and support.

Step 4 — refine.

• Option A. Work with $Y = \ln X$ and check whether $Y$ is approximately normal. This is the 'log-normal' model.
• Option B. Use a non-negative skewed distribution such as the exponential.
• Option C. Collect more data and look for sub-groups (e.g. premium vs budget bulbs).

Step 5 — re-test. Compute the same checks (mean vs median, skewness) on the refined model and iterate until they look reasonable.

Exam answer template.

The data is positively skewed, so the assumption of normality fails. Furthermore, the normal model assigns positive probability to lifetimes below $0$, which is impossible. We refine by transforming the data ($Y = \ln X$) or by using a skewed distribution such as the exponential, then repeat the modelling cycle.