No mode or all mode

central-tendencydefinitionstatistics

Let's say I have a data set:- $${1,2,3,4,5}$$

I decide to find the mode though it would not be an appropriate measure of central tendency.
But still……..

Now this is an exceptional case and for exceptional cases , I have the following rule:-

When the each appears equal number of times or 1 time then all are the mode but sometimes we say that no observation is occurring frequently and hence there is no mode.

This statement looks extremely unclear as they do not describe the case when we do it sometimes.

So I have the following possibilities:-

  1. All are mode

  2. No mode

But the second one looks more accurate as there is no observation occuring frequently yet it is not consistent with the statement.

When the each appears equal number of times or 1 time then all are the mode.

So It's more probable The second one is correct.

But I am not sure what should be correct all mode or no mode as they seem to be in opposition of each other.

Best Answer

It would seem strange to me to say "The mode of a Poisson distribution with parameter $\lambda$ is $\lfloor \lambda \rfloor$, except when $\lambda$ is a positive integer in which case $\ldots$" and follow this with "$\ldots$ it has no mode."

Instead you might say "$\ldots$ both $\lambda$ and $\lambda-1$ are modes of the distribution" as Wikipedia currently does, or perhaps "$\ldots$ both $\lambda$ and $\lambda-1$ are the values with equal highest probability." In your example, you could say $1,2,3,4,5$ all occur equally often, and more often than any other value. They are in that sense modes of the dataset.

Note that the mode is not really a measure of central tendency. For the exponential distribution, the mode is $0$, the left hand end of its support, and there are other similar examples

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