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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.
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What is the distribution when the mean median and mode are the same?
The normal distribution.
What is the shape of a distribution when the mean and the median are about the same value?
That would provide some evidence that the distribution is symmetric about the mean (or median).
Are the mean median and the mode the same in a unimodal symmetrical distribution?
No they are not the same in a unimodal symmetrical distribution and they will never be
In a perfectly symmetrical unimodal distribution is the mode and the median the same as the mean?
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).
How does the median affect the graph of a normal distribution?
A normal distribution is symmetrical; the mean, median and mode are all the same, on the line of symmetry (middle) of the graph.
