# Normal distribution negative values

Hi
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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SteamKing
Staff Emeritus
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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.

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)
Regards

SteamKing
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.

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)

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.

jbunniii
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Gold Member
Hi