Maths IA · Statistics and probability · topic 21 of 40
Waiting times: does a Poisson model fit? · IB Maths IA idea: guide, sample graph and step-by-step plan
your own data = strong engagementYou've waited for a bus that took forever. Is that bad luck, or exactly what a random process predicts? Count, time and test.
- ✓ you can stand in a queue with a stopwatch
- ✓ you like testing a model against reality
- ✓ you want your own data
The big idea
← the surprise is when it FAILSIf arrivals are random and independent, counts in equal time intervals follow a Poisson distribution and the gaps between arrivals are exponential. Your job is to collect your own arrival times, estimate the rate, and test whether the model fits. A good fit is not guaranteed: timetables, bunching and lunch rushes break the assumptions, and finding out how they break it is your best material. You compare counts with a Poisson model using a goodness-of-fit test, and waiting times with an exponential curve. Poisson is AI HL content (an extension on AA) and the exponential distribution is in neither course. At HL, justify the link between the two rather than quoting it.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Collect at a canteen and at a bus stop, or at two times of day.
→ You compare estimated rates and discuss why one place fits the Poisson model better, with a test for each.Fit an exponential curve to gaps between arrivals and use its survival function.
→ You answer the passenger's question: what is the chance of waiting more than 5 minutes? That links the model to the context.Derive why exponential gaps give Poisson counts, or the reverse, from first principles.
→ This is where HL-level sophistication and rigour come from; quoting the result earns less.Re-count with 1-minute, 2-minute and 5-minute intervals.
→ You see how the choice affects the fit and the test, and have to decide which interval is justified.Compare the chance of waiting 3 more minutes given you have already waited 2 with the chance of waiting 3 from the start.
→ You test a surprising property directly against your data, a strong personal-engagement moment.Collect data at a timetabled stop and compare with a free-flow queue.
→ The poor fit becomes the finding; your reflection is on why regular arrivals look different (try an Erlang model).Your Maths IA research question: run the check
Write your own question first. Then make sure it ticks every box:
- Names the place, the days and the time window of the observation.
- Says what you count or time: arrivals per interval, gaps between arrivals, or both.
- Names the model you are testing and how you will decide whether it fits.
- Is realistic: enough arrivals (at least 100 gaps or 60 intervals) to test properly.
- Does not ask for a guaranteed 'it fits' answer.
The picture to draw
Record each arrival time. From the same data you get two views: counts per interval and gaps between arrivals. Each has its own model.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why this queue or stop; what question a waiting person would ask; your prediction of the rate before collecting data.
How you recorded times, where you stood, when, how many observations, ethics and anything you excluded.
Define the random variable and the rate, state the assumptions (independent, constant rate, one at a time) and say how each could fail here.
How gaps connect to counts. At HL, a derivation or justification, not just a quoted result.
A table of observed and expected counts, with the rate estimate, and a graph of both.
Hypotheses, expected counts, combining small categories, degrees of freedom (note the parameter you estimated), test statistic and p-value.
A histogram of gaps with the fitted curve, and probabilities you can compare with the data.
Does the model describe your queue? Answer in numbers and in plain words.
Which assumption failed first, effect of rush periods, sampling choices, and a next step with a reason. Include short reflection after each stage.
Sources for the distributions and for any timetable data.
What your Maths IA graph could look like
density, not counts!Made-up histogram densities from one afternoon, to show the form of a strong graph. example only
- Plot density, not raw counts, so changing the bin width does not change the picture.
- Horizontal bars show the bin width, vertical bars the counting uncertainty on each bin.
- Two models drawn together so the reader can see which one your data prefers.
- State how the rate 0.5 was estimated from the data, then the curve is yours, not a guess.
- Another graph to consider: observed against expected counts per interval, or a plot of ln(survival) against time, which is a straight line if gaps are exponential.
Your game plan
stand there for 20 min firstTick steps off as you go. We'll remember on this device.
0 of 10 done
- Pick a placeChoose a queue or stop you can watch for 30 to 60 minutes at a time without disturbing anyone. Check you are allowed to be there.
- Learn the two distributionsPoisson counts, exponential gaps, and what each assumes. Work out by hand what each predicts for a rate of 2 per minute.
- Trial runTime 20 minutes. Can you record each arrival accurately? What counts as an arrival? Fix the rules.
- Collect properlyRecord times of arrival to the second. Aim for 100 or more gaps, ideally over more than one session.
- Estimate the rateFrom the mean count per interval and from the mean gap. Do the two agree?
- Build expected countsUse the Poisson model for the chosen interval. Check categories are big enough for a chi-squared test, and merge if not.
- Test the fitCarry out the goodness-of-fit test, noting degrees of freedom. Interpret the p-value for the aim, not just 'reject' or 'accept'.
- Fit the waiting timesDraw the histogram of gaps with the fitted curve. Compute a probability from the model and compare it with the data.
- Reflect and extendWhich assumption failed? Is there a time pattern? Choose one extension and say why.
- Write, then cutKeep one worked example per method, then tables. Define each symbol once, then check graphs have titled axes before the teacher's draft.
Maths you'll need
- Probability distributions (AI) →
- Probability distributions (AA) →
- Hypothesis testing and chi-squared →
- Using your GDC for statistics →
Traps to avoid in this Maths IA
- Collecting too little data, so the chi-squared test has almost empty categories.
- Forgetting to reduce the degrees of freedom for the estimated rate.
- Assuming a constant rate across a lunch rush.
- Choosing the interval length after seeing which one fits best.
- Counting groups arriving together as separate arrivals.
- At HL: quoting 'gaps are exponential' with no justification; HL criterion E wants the reasoning.
- Pasting a calculator output without saying what the p-value means for the queue.
More Maths IA ideas like this
- Polls: how big a sample shrinks the margin of error?●●○
- Benford's law: does your data follow it?●●○
- Tie-breaks: do league rules change who wins?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.