# Statistical Reliability of a Particular Train Arriving at a Particular Station on Time

I need to make an appointment in a distant city and I want to take the train and have a good probability of arriving on time. How can I find the probability of the Amtrak train arriving on time, 15 minutes late, 30 minutes late or similar statistics?

Specifically, and as an example, I am planning to take public transit to an appointment I will make in Oxnard California traveling from Santa Maria. So, I'd like to know how to find out the reliability of the Amtrak 777 Surfliner in arriving on time south-bound to Oxnard.

Amtrak has a train status button on their reservations page. It takes a little work but it will give past arrival times.

• Welcome, can you please provide a few clues or a better link than a blank page that needs the reader to do a little digging? Commented Jul 7, 2021 at 17:13
• Your question seems to be a straightforward application of statistics, not a problem that requires out-of-the-box thinking to resolve. Commented Jul 7, 2021 at 17:57
• Having worked in the industry, most railway networks I know of, worldwide, are legally obliged to publish punctuality statistics of this nature. Just look them up, or issue a Freedom Of Information request to get them. Commented Jul 7, 2021 at 20:31

My hack is to find something interesting in the distant city that you would like to see.

Suppose you think you will need one hour to find and see that point of interest. Take a train that will arrive at least one hour sooner than you would need to make the appointment.

• If the train is on time (or early) then visit the interesting place before your appointment.

• If the train is late, go directly to the appointment, and save the tourist activity for later (or skip it).

This is hedging your bets, and is a more reliable way to build in extra time and do something interesting too.

The Single Event Probability Calculator uses the following formula (among others): P(E) = n(E) / n(T) = (number of outcomes in the event) / (total number of possible outcomes)

The answer will be a number between 0 (not probable - The train did not arrive at the desired or acceptable time) and 1 (the train arrived each time it was scheduled within your specified time).

Reliability (statistically speaking) is another term for consistency. If the train arrives several times within your specified time period, the train arrival is consistent. Again, Reliability will be a percentage between 0 (unreliable) and 100 (reliable).

You should specify a tolerance - the amount you will accept for being early as opposed to late or absent. Then, work backwards to give you the amount of acceptable time to wait for your appointment by being early. What is the range of arrival times (variability) of the transportation method you've chosen? Incorporate that into your calculation since you cannot affect this but must tolerate this as part of the system. If the train has no record of being more than X minutes late, use that or a multiple of that into your tolerance.

If the train arrives once a day, you might want to arrive a day early to ensure you have ample time to arrive at your appointment.

If all this is relevant (It is), you might want to check on the validity of your assumptions and intentions.

Such calculations are relatively easy to compute. The difficult part is the interpretation of the results and adopting a course of action based on the calculations.

For example: What if the probability is 0.66 and is 99.9% reliable?

Good luck.

• The interpretation of statistics seems pretty simple here. If you want a 95% chance of arriving by noon to make the appointment, you just need to take the latest train that historically has arrived by noon at least 95% of the time. I don't really understand what you mean by a "probability of 0.66" in the last line - how could a train arrive before the specified time 99.9% of the time, but only have a 66% probability of getting you there on time? Commented Jul 7, 2021 at 16:51

You could make a histogram of all the historical arrival times, see how early/ late they are by how much time, and compute a 95% confidence interval of the deviations from being "on time." You could convert these deviation times to standard deviations by normalizing them (i.e., converting each to a z-score). Each z-score will have a corresponding percentile. You can then compute that, for example, xx percent of arrivals are late by more than 15 minutes, yy percent by more than 30 minutes, etc.

I assume that arriving early is acceptable but arriving (too) late is not. You could potentially take an earlier train to ensure early arrival. Depending on how important your appointment is, and how tolerable it is to be late, you may want to aim for an arrival time that is not likely to be greater than, say, the 95th percentile for example (or a z-score of 1.96); in other words you would want 95% of the past arrivals to be no later than this amount of time. On the east coast and midwest the arrival times are pretty much guaranteed to be late (i.e., zero % chance of being on-time or early) in my experience, but on the west coast it might be different nowadays. The historical arrival times you mention is valuable data. Even if the "average" train is on-time, it is the deviations from on-time that you are interested in to make your estimate.