Normal distribution negative values

In summary: 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).Any ideas about that?Any normal distribution, regardless of its mean and standard deviation, will have infinite tails and therefore nonzero probability of a negative outcome. So some physical phenomenon which cannot be negative (such as
  • #1
DrunkenPhD
10
0
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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  • #2
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.
 
  • #3
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
 
  • #4
Even in statistics, sometimes you got to 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.
 
  • #5
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).
 
  • #6
"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 [itex] \mu - 3\sigma[/itex] and [itex] \mu + 3 \sigma [/itex].

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, [itex] \mu - 6 \sigma [/itex] and [itex] \mu + 6 \sigma [/itex] 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.
 
  • #7
DrunkenPhD said:
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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1. What is a normal distribution with negative values?

A normal distribution with negative values is a statistical distribution that is symmetric and bell-shaped, with the majority of values being clustered around the mean and decreasing as they move away from the mean in both positive and negative directions. This means that there are some negative values in the distribution, which is why it is referred to as "normal distribution negative values."

2. Can a normal distribution have negative values?

Yes, a normal distribution can have negative values. In fact, the standard normal distribution, also known as the z-distribution, has negative values on the left side of the mean. This is because the distribution is centered around the mean, which can be a positive or negative number, and the negative values represent the values that are below the mean.

3. What does it mean if a data set follows a normal distribution with negative values?

If a data set follows a normal distribution with negative values, it means that the data is approximately symmetric and follows a bell-shaped curve. This is a common distribution for many natural phenomena and is often used in statistical analyses. The presence of negative values in the distribution indicates that there is variability in the data, with some values falling below the mean.

4. How do negative values affect the shape of a normal distribution?

Negative values do not necessarily affect the shape of a normal distribution. The shape of a normal distribution is determined by its mean and standard deviation, not by the presence of negative values. However, the presence of negative values can indicate that there is a difference between the mean and median of the data, which can affect the skewness of the distribution.

5. Are there any limitations to using a normal distribution with negative values?

There are no inherent limitations to using a normal distribution with negative values. However, it is important to ensure that the data actually follows a normal distribution before using it in statistical analyses. Additionally, if the data has a large number of negative values, it may be more appropriate to use a different type of distribution that better fits the data.

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