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The normal distribution.

Q: What is the distribution when the mean median and mode are the same?

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No they are not the same in a unimodal symmetrical distribution and they will never be

They are all the same.

A normal distribution is symmetrical; the mean, median and mode are all the same, on the line of symmetry (middle) of the graph.

Yes, they are. A uniform distribution is one in which the probability of each outcome is the same and, as a result, the mean and median are the same. A uniform distribution should not be confused with a set of random variables, all with the same distributions - much less the same values!For example, the median of a Poisson distribution is not the same as its mean. So if you have a number of random variables (RVs), each with the same Poisson distribution, their mean and median will be different. This is true of any set of RVs whose distributions are asymmetric.And it is very easy to see that the mode need not be the same. The outcome of a single roll of a regular die is the uniform distribution over the numbers {1, 2, 3, 4, 5, 6}. The mean and median are 3.5 but the mode cannot be 3.5 since that is not a value that can ever be observed.

No, it is in general not true - for example for uniform distribution on [0,1] every number in the interval is a mode, but the mean is 1/2. The correct answer would be that a symmetric unimodal distribution has one mode equal to the mean (but may have modes elsewhere).

Related questions

Mean, median, and mode are all equal in a normal distribution.

Yes.

No they are not the same in a unimodal symmetrical distribution and they will never be

The mean is the same as the mode and median.

In a symmetric distribution, the mean and the median are the same. Otherwise there is no relation. In symmetric distributions with only one mode, the mode will coincide with the mean and median, but otherwise there is no relation.

In a normal distribution the mean, median and mode are all the same value.

Yes it is. The normal distribution is symmetrical around the mode. Therefore the median has to be the same :)

A normal distribution is symmetrical; the mean, median and mode are all the same, on the line of symmetry (middle) of the graph.

No, in a normal distribution they are the same.

They are all the same.

They are all equal . . . they are the same.(In an asymmetric distribution they are not equal.)

No. Normal distribution is uni-modal, specifically with the mean, mode, and median at the same value.