# Return Period Units

#### CivilSigma

Problem Statement: I obtain monthly maximum temperature values for the last 30 years, and I fit the data to a statistical model. Now, I want to find the 50-year return period maximum temperature.
Relevant Equations: P(X>x) = 1/R

I was having a discussion with my "Prof" for one of my classes about return period and I would like a second opinion on what he told me.

The problem:

I obtain monthly maximum temperature values for the last 30 years, and I fit the data to a statistical model. Now, I want to find the 50-year return period maximum temperature.

For me, that simply means finding P(X>x)= 1/50 = 0.02 , with no significance given to the units of time.

But, my prof insists that if we are looking for the 50-year return period, we need to actually find the 50*12= 600 month return period, i.e P(X>x) = 1/600.

I completely disagree with this as it makes no sense. The entire notion of return period is originally independent of units, and is just a way to communicate a percentage of exceedance in a more humane way.

Thank you,

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#### DaveE

The entire notion of return period is originally independent of units,...
Well, I'm confused. The return period for a 100 year flood is 100 years, or 1200 months, or 1 century. Of course there are units. The units are time, it's even in the name "return PERIOD". For a different data set the units could be something else. Like "per number of samples" for discrete events like rolling dice.
Why he thinks you have to do it in months instead of years is a mystery based on your description.
You will get a better explanation from a web search...

#### CivilSigma

Well, I'm confused. The return period for a 100 year flood is 100 years, or 1200 months, or 1 century. Of course there are units. The units are time, it's even in the name "return PERIOD". For a different data set the units could be something else. Like "per number of samples" for discrete events like rolling dice.
Why he thinks you have to do it in months instead of years is a mystery based on your description.
You will get a better explanation from a web search...
From my understanding, the notion of return period is just to describe the probability of exceedance.

If I say the 50-year maximum wind speed is 100 km/hr, this speed can happen today, in 10 years, 20, or even in 100 years.

From a statistical point of view, the units should not matter. If you calculate that for a given temperature, x=50 C, P(X>50)= 0.01, then the return period is 1/0.01 = 100 "years". Moreover, even if your analysis is using monthly or hourly data, it should not matter because the calculation is independent of time.

#### DaveE

then the return period is 1/0.01 = 100 "years"
Yes, it is true that you don't have to really do much with the units while you are doing the computation (unless someone wants the answers in months instead of years ). But the units are there and are fundamental. You have even included them in your answer above. Suppose you want to know how often to change the oil in your car and the answer is 6. Wouldn't you wonder if they mean 6 months, 6K miles, 6,000 km?
The broader point is that in the real world numbers either come with units to describe physical quantities, or they are dimensionless, which has as much meaning as unit of measure. Dimensionless answers are a special sort that tell you you are dealing with a ratio of like quantities. Always include the units, even if it is trivial to do or you think it's obvious.

Edit: I've often thought that we should have something like a "null" unit to clearly communicate that the number is dimensionless and not that we forgot to include the units. It is actually quite common amongst people who specialize in a field to assume the units and not talk about them, like physicists who assume c=1. That may be convenient in their practice, but it can be a barrier to understanding for others if they aren't clear when they communicate their results.

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#### DaveE

I would also suggest that you go back to your prof and ask for more clarification. The conversion from months to years is trivial. You know that, and he knows that too. So I think there is likely to be some misunderstanding between the two of you. The fact that you are confused about your discussion isn't likely to be about the fact that there are 12 months in a year, there must be something else he was trying to communicate.

#### pasmith

Homework Helper
I would also suggest that you go back to your prof and ask for more clarification. The conversion from months to years is trivial. You know that, and he knows that too. So I think there is likely to be some misunderstanding between the two of you. The fact that you are confused about your discussion isn't likely to be about the fact that there are 12 months in a year, there must be something else he was trying to communicate.
The sampling frequency must be relevant. If you have monthly figures then you have to allow for the fact that 50 sample values span $\frac{50}{12} = 4\frac{1}{6}$ years rather than 50 years.

Probabilities themselves are dimensionless, but that doesn't mean that dimensions can be ignored completely.

"Return Period Units"

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