# Normal distribution negative values

DrunkenPhD
Hi
I read at this link http://www.eecs.berkeley.edu/~aude/papers/TRB2012_stat_traffic.pdf [Broken]
something like bus travel times can be normally distributed.
Sounds strange to me because normal distribution presumes even negative values
Anything I am missing here?
Regards

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## Answers and Replies

Staff Emeritus
Homework Helper
Negative values for what?

The standard normal distribution has a mean value μ = 0. Different mean values will shift the distribution to the left or right of the vertical axis, just like the standard deviation σ makes the curve skinny or broad.

DrunkenPhD
Thank you SteamKing.
But in case of not standard normal distribution smth like N(μ,σ) where both μ,σ are not equal to 0, I cannot really see smthg like -5 minutes which can be a value of simulation N(μ,σ)
Any ideas about that?
Regards

Staff Emeritus
Homework Helper
Even in statistics, sometimes you gotta employ common sense.

-5 minutes can only be a realistic value if you allow for time travel to the past, which busses might make you feel is happening.

The pdf's used in the study appear to be constructed so that for values of travel time <= 0, the probability is identically 0, which is what would happen in the real world. The mean travel times are also >= 0, which implies that the pdf is shifted to the positive side of the vertical axis.

Soveraign
I run into this issue with my research when modeling particle detector responses with a normal distribution since it will on occasion try to produce a negative energy particle (clearly not realistic). There are options but they produce biases. One is to just cap the low end at zero (or nearly zero), which produces a positive bias. Another might be to cap the range from 0 to 2x the mean. This removes the bias but now the RMS is smaller than expected.

An option I've considered but haven't yet implemented would be to use something like a log-normal distribution. It has the feature of not going negative, but approaches a normal distribution when the mean is sufficiently larger than the sqrt(variance).

Homework Helper
"But in case of not standard normal distribution smth like N(μ,σ) where both μ,σ are not equal to 0, I cannot really see smthg like -5 minutes which can be a value of simulation N(μ,σ)
Any ideas about that?"

Remember that a crude but simple descriptive property of normal distributions is that almost everything is contained between $\mu - 3\sigma$ and $\mu + 3 \sigma$.

If your model (and that's what it is, a model, a mathematically convenient description of some phenomenon) has the distribution portion normal and mean and standard deviation restricted so that, say, $\mu - 6 \sigma$ and $\mu + 6 \sigma$ take in only "reasonable" values - values that are realistic - you won't have the type of non-sense simulation problems you describe.

On the other hand, if you see a situation in which the minimum possible measurement is 0, and are told the mean is 8 with standard deviation 5, you should know a normal distribution can't apply, as 0 is "too close" to the mean. Using a normal distribution there is a sure way to encounter problems.

Homework Helper
Gold Member
Hi
I read at this link http://www.eecs.berkeley.edu/~aude/papers/TRB2012_stat_traffic.pdf [Broken]
something like bus travel times can be normally distributed.
Sounds strange to me because normal distribution presumes even negative values
Anything I am missing here?
Regards
In case the other responses didn't make this clear, what you said above is correct. Any normal distribution, regardless of its mean and standard deviation, has infinite tails and therefore nonzero probability of a negative outcome. So some physical phenomenon which cannot be negative (such as bus waiting times) cannot truly be normally distributed. However, it can be approximately true if the mean and standard deviation are chosen so that the probability of a negative outcome is very small. This is what statdad is referring to above.

Be aware, though, that even if you choose ##\mu## and ##\sigma## such that ##\mu - 3\sigma > 0##, you will still have a 0.15% probability that the outcome will be less than ##\mu - 3\sigma##. So if you run enough trials, you WILL get a nonsensical result. If you increase the margin to ##6\sigma##, then the chance of an outcome below ##\mu - 6\sigma## is only about one in a billion. This may or may not be an issue depending on how many times the experiment will be performed.

In practice, if you're writing simulation software where the outcome MUST be non-negative (e.g. if you're going to perform a square root), but the distribution MIGHT generate a negative number, you need to add an operation such as "if x < 0 then set x = 0" to prevent disaster.

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