If I have the mean, median, and standard deviation of a distribution, can I determine the shape of the distribution? More specifically, I suppose, can I determine, say, where the top x% of the distribution is (it’s been too long since stats class to remember exactly how to ask this question, but I think you can get the idea). If it helps, these distributions would be bounded by zero on the low end.
Thanks…
(I just passed Precal, so cut me slack)
If the data is normally distributed around the median, then yes. There is a table (I’ll be darned if I’ll find it online, but I have a real copy) that shows compares t (the coefficient of the standard deviation) in relation to P (the percentage of data falling into that range). It’s tough for me to explain, but if you want to find the top 10% of the data (again, only if it’s normally distributed), you could find the t value corresponding to where P = .800. Take this t value, multiply it by the standard deviation, add this value to the median, and that is the lowest value for the top 10%. The range of the top 10% would then be [this newly found value] through [highest value].
NB: The reason I went with .800 and not .900 is because the P range slides both right AND left of the median. That leaves three sections (lowest 10%, middle 80%, and top 10%). Hope that clarifies things.
Well, if it’s bounded by zero it’s obviously not normal. Without knowing what class of distributions it is in (e.g. normal, poisson, etc.), there’s no way to know what the distribution without an infinite number of parmeters. Since you have only three, you’re out of luck. If you want to know where the top x% of the distribution is (the techniccal term is “x precentile”, but your way of putting it is clear enough), you can get an upper and lower limit, but that’s usually not going to be exact (for instance, you might find that it’s somewhere between 10 and 15).
If the data is normally distributed around the mean, the median equals the mean. If I’m not mistaken, the normal distribution has a standard deviation of one. (Is it not true that any distribution in which the mean equals the median with a standard deviation of one has to be normal?)
I have a nagging suspicion that with these three parameters, we can at least approximate the shape of the distribution. After all, the median and the mean give us two points on the curve (if it’s not a normal distribution–and in my specific case, they are different), and the median is the maximum point on the curve. Does the standard deviation provide information about the shape of the curve (convex or concave) between the median and the mean? If +/- one standard deviation contains two-thirds of the observations (or is that only true with a normal distribution), is that enough information?