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What do we mean by the distribution. What's being distributed?

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Data. It is the way data points are "distributed" or arranged.
You get a distribution when you count the number of things falling into a a categories. For example number of people in a class with brown eyes, black eyes, blue eyes, gray eyes etc. Then you plot these values on a graph with the eye colour on x axis and the number of people having the eye colour on the y axis. You now have a distribution. The x and y axis can represent anything, ranging from the eye colour and the corresponding number of people in a class, to the kinetic energy and number of atoms holding a particular energy value in a container.

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When they distribution they are talking about the probability being distributed.

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