← Apply probability distributions

7 Comparing a model against real data

Every year's paper eventually asks the same question in different clothing: does your probability MODEL actually match what the DATA shows? A calculation with no comparison to the data, or a comparison with no calculation, both fall short of the top grade.

The method, every time

• Find the observed proportion directly from the sample data (a raw count from a table, or reading bars off a histogram) — this needs no distribution at all, just arithmetic on what was actually measured.
• Find the model's predicted probability for the same event, using the distribution you chose (or were given).
• State a verdict: which is bigger, by how much, and whether that difference is large enough to matter for the context — "the model overestimates/underestimates" is the required form of words.

Worked example Observed proportion vs. a normal model Of 100 recorded transactions, 33 involved a spend of more than 50. A normal model with μ = 27.75, σ = $46.22 gives P(X > 150) = 0.3151. Compare the two and comment on the fit. Observed proportion: 33/100 = 0.33. Model prediction: P(X > 150) = 0.3151 (normal model). Compare: 0.33 (observed) is slightly higher than 0.3151 (normal model) — the normal model slightly UNDERESTIMATES the proportion of high spenders. If a second, alternative model (e.g. triangular) gives a prediction closer to 0.33, say so explicitly: whichever prediction is numerically closer to the observed 0.33 is the better-fitting model for this data. ✓ The normal model (0.3151) slightly underestimates the true proportion (0.33); whichever alternative model lands closer to 0.33 fits this sample better.

Why a model can fit poorly even with a "sensible" choice

• A normal model is symmetric (mean = median = mode) and has no fixed endpoints — it always assigns some probability, however small, outside any given range. If the real data is skewed or bounded (spend can't go below $0), the normal model will systematically mis-predict the tails.
• A triangular model has fixed endpoints and can be skewed (mean ≠ mode), so it often fits bounded, skewed real-world data (like spend, or age) better than a normal model does — even though the normal model may look like the more "standard" choice.
• Naming this structural reason (bounded vs unbounded, symmetric vs skewed) is what separates an Excellence answer from one that only compares two numbers.

✕

A verdict needs a reason, not just a bigger/smaller comparison

Stating "0.33 is bigger than 0.3151 so the model underestimates" is only PART of a full-credit answer. The top band additionally wants a REASON grounded in the features of the distributions — for example, that the normal model assumes symmetry while the real spending data is skewed, or that a triangular model's fixed endpoints match a real-world minimum/maximum that the normal model does not respect. A bare numeric comparison with no structural reason caps out below Excellence.

Practise just this