# Probability Distribution for "Time Spent"?

What is a good probability distribution for the "Time Spent" doing something?

Say I had data as follows:

Time
23.2ms
232.1ms
21312.ms


Would a Gaussian work well? Or would something else be better?

• This isn't really answerable in this form without, say, a histogram of your actual data. From these three points it looks like a Gaussian is not a good fit, though a log-normal might be (or something else that operates on log scale). Mar 27, 2016 at 15:21
• I see. This is just a hypothetical question. As in, what would be the best distribution. I suppose we do need to look at the characteristics of the distribution itself.
– user46925
Mar 27, 2016 at 15:32

The question of "Time spent", or "Time until event", is the focus of the field of survival analysis. Many people immediately think about censoring when they think of survival analysis (suppose you had a 4th attempt that was ongoing but not yet completed, so you just knew that "time spent" was greater than the current time), but this is not necessary.

The simplest distribution used to fit this type of data is the exponential distribution. This requires the strong assumption of constant hazards or memoryless function. In a nut shell, this means that the probability the event will occur in the next time interval is independent of how much time has already been spent. Two standard generalizations of the exponential distribution that are commonly used as well are the Weibull distribution and gamma distribution. These do not require the constant hazard assumption. As noted earlier, for extremely skewed distribution, log-normal can be used as well.

• Thank you! I've found out about the Weibull distribution as well (typically used for survival analysis), which seems to work for this case.
– user46925
Mar 27, 2016 at 15:55

# Rule of thumb

## If your histogram looks centralized => model it with a Gaussian

Example: Period of pregnancy, most women give birth between the 36th and 40th week, and the number of births decreases as you get further from the center.

## If your histogram declines as time passes => model it with an exponential or lognormal distribution

Example: Time of a phone call to a support center, most issues can be resolved quickly, and The longer a call is, the rarer it is.

• Not a bad guideline. +1
– user46925
Mar 27, 2016 at 18:20
• Please consider using HUGE BOLD LETTERS less often, this doesn't look good at all and makes an immediate bad impression of your answer. Mar 27, 2016 at 18:55
• There's a lot in between the distributions exponential and log-normal. Mar 27, 2016 at 19:16
• That's why I wrote "rule of thumb" and not "silver bullet" or "gold standard method" Mar 27, 2016 at 19:21