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What is the purpose of a measure of center in Statistics?
Measures of center are a way to try capture the nonrandomness of a population or a random variable.This may sound a bit confusing,so I will walk through the 3 most common measures of center and explain how they try to do that.

Mean . The mean is the most common measure of center and it tries to capture what is , in average, the process value .It poses the simple question: If I kept repeating this process , sum all the outcomes and divide by the number of trial, what value should I expect? The mean shines in practical use when you can repeat the experiment. For example, suppose you have a lottery that cost 50 dollars to enter, has a 75% chance of losing all and 25% of earning 10.000 dollars. If you could only play this lottery just once, it doesn't matter that the mean is 2.500, a lot of people wouldn't join the lottery. However if could play this indefinitely , most people would play it.

Mode .The mode represent the most likely outcome of a process.If we take our last example, the mode of the lottery is 0.The mode is often used when every occurrence of the process matter or when the process doesn't repeat very often and you need "the best guess possible of the outcome"

Median The median is the value that separates the process in half, with the half most likely outcomes before it and halt after. It gives a point where you can say confidently: " Half the values will be less than this". The median is really interesting when you are dealing with social data, for example: If the average income of an American is 50k a year, but the median is 10k something is problematic.