Choosing a model and naming what it ignores

A mathematical model describes a real situation using maths. Start with the question you need to answer. Define each variable and its units, then choose a rule. In T = 6 + 5d, T might mean time in minutes and d distance in kilometres. The 5 then means minutes per kilometre. Record what the rule assumes and what it leaves out.

For snow depth D in cm after h hours, D = 4 + 3h assumes the depth increases by 3 cm every hour. Steady snowfall gives this increase only if melting, settling and other depth changes are absent or negligible. If snowfall stops after one hour and nothing else changes the depth, it stays at 7 cm. The formula would instead predict 10 cm after two hours.

Depth after two hours, in cm, under the two stated snowfall conditions.

A sensitivity test compares the effects of plausible uncertainty changes. Change one input at a time, hold all other inputs fixed and compare the same output at the same time. For C = 16 + 7n pounds at four days, a £3 starting-fee rise adds £3. A separate £2 daily-price rise adds £8. The daily-price change matters more in this test. Changing both together would not show their separate effects.

Increase in the four-day cost, in pounds, for each separate change.

Percentage growth multiplies a starting-value error too. With 5% annual growth, multiply by 1.05 each year. If two forecasts differ only in their starting counts by 40 people, their gap becomes 40 × 1.05 = 42 after one year, then 42 × 1.05 = 44.1 after two. A rate change also affects repeated multiplication. Neither input is always more sensitive: specify the changes and forecast date before comparing.

Forecast gaps in people. Both forecasts use the same 5% annual growth.

A model may apply only over a limited interval. A tank starts with 28 litres and drains at 4 litres per hour. R = 28 - 4h gives the litres remaining until it empties. It reaches zero at seven hours. After that, the formula gives negative amounts, which cannot describe remaining water. Use this draining rule from zero to seven hours. With no added water, the amount then stays at zero.

The negative result shows that this draining rule has been used beyond its useful interval.

Check a prediction against an observation by subtracting the smaller value from the larger. A prediction of 150 visitors and a count of 142 differ by 8 people. If a planning decision accepts a gap of at most 10, this check meets that tolerance. A larger gap needs investigation. One check supports a conclusion about the checked conditions, not every future use.

Visitor counts. Their gap is compared with the decision's allowed error.

Using weather to predict attendance does not mean weather is its only cause. A major event may affect attendance too. If a stated event effect explains a missed prediction, that identifies a relevant factor the model leaves out. It does not establish whether the event's effect is larger than the weather's effect. Check similar event days before relying on the model there.

A good fit to past data is useful evidence, but it does not guarantee future performance. Check whether future conditions match the assumptions and compare predictions with relevant new observations. Also distinguish exact arithmetic from an exact prediction: calculating perfectly with an estimated input does not remove uncertainty in that estimate.

Report a prediction with its units and conditions. For a fixed-fee cost model, multiply the monthly price by the number of paid months, then add the joining fee. State that the result assumes those prices stay fixed and any excluded extra charges do not apply. A calculation supports that conditional prediction, not a promise about every possible bill.