A Statistical Investigation into Why ‘Between 8 a.m. and 1 p.m.’ Does Not Mean What People Think It Means
Author: Una Likely
Field: Probability and Statistics
Research Stream: F.A.R.T. S. (Science Division)
Abstract
Appointment windows are routinely expressed as ranges.
A customer may be informed that an engineer, delivery driver, inspector or other authorised person will arrive ‘between 8 a.m. and 1 p.m.’
The conventional interpretation is that the visit may occur at any point within those five hours.
This interpretation is mathematically convenient and observationally weak.
A study of 214 appointment windows found that arrival probability was not evenly distributed across the stated period. More importantly, it changed according to what the person waiting had decided to do.
The effect was particularly pronounced when the activity involved leaving the house for less than ten minutes.
The findings suggest that an appointment window should not be treated solely as a period of time. It is better modelled as a relationship between time, preparedness and the increasing need to do something elsewhere.
1. Background
The investigation began after twelve staff members independently reported that expected visitors arrived during brief absences from otherwise uninterrupted periods of waiting.
In one case, an individual stayed home for four hours and forty-three minutes before walking about 90 metres to post a letter.
The engineer arrived during minute three of the resulting seven-minute absence.
The event was initially dismissed as bad luck.
Eleven further examples made this statistically improbable.
F.A.R.T. S. therefore examined whether appointment arrivals were distributed randomly throughout their stated windows.
2. Method
Participants were given genuine domestic appointment windows ranging from three to six hours.
Their behaviour was recorded in broad categories:
State A: Passive availability
The participant remained at home and undertook activities that could be interrupted immediately.
State B: Conditional activity
The participant began a task such as cooking, showering, gardening or making a telephone call.
State C: Brief absence
The participant left the property for an activity expected to take no more than fifteen minutes.
State D: Resignation
The participant concluded that the visitor was probably not coming and began behaving accordingly.
We compared arrival times with the participant’s behavioural state.
The visitor was not informed of the experiment.
This was necessary because previous attempts to study appointment behaviour after informing the visitor produced unusually punctual results.
3. Results
During State A, arrival probability remained low.
Participants could sit beside the front door with their telephone fully charged, paperwork prepared and shoes on without producing a significant increase in visitor activity.
State B produced a measurable change.
The beginning of a shower was associated with a substantial increase in the probability of a knock at the door.
Cooking showed a weaker effect unless the participant had reached a stage at which leaving the food unattended would be inconvenient.
Telephone calls were particularly effective when made to organisations that used automated menus.
The strongest association occurred during State C.
Across the study, 38 participants left home briefly during an active appointment window.
Fourteen received an attempted visit during that absence.
This is considerably higher than would be expected if arrivals were evenly distributed across the available time.
The duration of the absence mattered less than the inconvenience.
A nine-minute visit to a nearby shop produced more arrivals than forty minutes spent sitting in the garden, even though the garden is technically outside the house.
The visitor therefore appears to respond not to physical absence but to practical unavailability.
I am aware of the implications of that sentence.
I have checked the calculations.
4. The Appointment Pressure Model
The evidence supports a revised model in which arrival probability is influenced by three variables:
- elapsed time within the appointment window;
- degree of customer inconvenience;
- confidence that there is still enough time to do something quickly.
The third variable appears particularly important.
Participants rarely left home during the first hour because they considered departure too risky.
They also avoided leaving near the end because arrival seemed imminent.
The critical period occurred when the participant thought:
‘I should have time.’
This phrase preceded 61 per cent of unsuccessful brief absences.
It should therefore be regarded as a warning rather than an assessment.
5. The Resignation Effect
State D produced a separate phenomenon.
When an appointment window neared its end without a visit, participants commonly began dismantling their waiting arrangements.
They changed clothes, started meals, went upstairs or placed their telephone somewhere they could no longer hear it.
Visitor arrival increased sharply during this transition.
The important point is that resignation did not require the appointment window to have ended.
It required only that the participant had emotionally stopped believing in it.
This presents a measurement problem because emotional resignation is difficult to record precisely.
For future studies, we recommend using the moment at which a participant says ‘Typical’ as a practical indicator.
6. Practical Implications
The findings do not support the common assumption that a five-hour appointment window provides five hours of equivalent arrival opportunity.
For the person waiting, the window includes long periods when almost nothing happens and several short periods when leaving the room becomes operationally dangerous.
Customers who want to minimise risk should avoid starting any activity that would become inconvenient if interrupted.
Unfortunately, maintaining this condition for five hours is itself inconvenient.
The study therefore offers no useful behavioural solution.
It does, however, permit more accurate disappointment.
7. Recommendation
Appointment providers should consider replacing conventional time windows with probability-based guidance.
For example:
Expected attendance: 8 a.m.–1 p.m.
Most likely arrival: shortly after you decide you cannot reasonably wait any longer.
This would not improve punctuality.
It would improve the description.