When you have thousands of different pieces of data, to grasp it mentally, you need to replace them with some simplification. For example, instead of a thousand different weights you could imagine a thousand identical weights, such that the new set is somehow the same as the original set; and then you would focus on the individual weight from the new set.
What precisely does “somehow the same as the original set” mean? Well, it depends on what did the numbers from the original set do; how exactly they join together.
For example, if we speak about weights, the natural way of “joining together” is to add their weight. Thus the new set of the identical weights is equivalent to the original set if the sum of the new set is the same as sum of the old set. The sum of the new set = number of pieces × weight of one piece. Therefore the weight of the piece in the new set is the sum of the pieces in the original set divided by their number; the “sum/n”.
Specifically, if addition is the natural thing to do, the set 3, 4, 8 is equivalent to 5, 5, 5, because 3 + 4 + 8 = 5 + 5 + 5. Saying that “5 is the mean of the original set” means “the original set behaves (with regards to the natural thing to do, i.e. addition) as if it was composed of the 5′s”.
There are situations where some other operation is the natural thing to do. Sometimes it is multiplication. For example, if you multiply some original value with 2, and they you multiply it by 8, the result of these two operations is the same as if you would multiply it twice by 4. In this case it’s called geometric mean, and it’s a root of product.
It can be even more complicated, so it doesn’t necessarily have a name, but the idea is always replacing the original set with a set of identical values such that in the original context they would behave the same way. For example, the example above could be described as a 100% growth (multiplication by 2) and 700% growth (multiplication by 8), and you need to get a result 300% (multiplication by 4); in which case it would be “root of (product of (Xi + 100%)) − 100%”.
If there is no meaningful operation in the original set, if the set can be ordered, we can pick the median. If the set can’t even be ordered, if there are discrete values, we can pick the most frequent value as the best approximation of the original set.
When you have thousands of different pieces of data, to grasp it mentally, you need to replace them with some simplification. For example, instead of a thousand different weights you could imagine a thousand identical weights, such that the new set is somehow the same as the original set; and then you would focus on the individual weight from the new set.
What precisely does “somehow the same as the original set” mean? Well, it depends on what did the numbers from the original set do; how exactly they join together.
For example, if we speak about weights, the natural way of “joining together” is to add their weight. Thus the new set of the identical weights is equivalent to the original set if the sum of the new set is the same as sum of the old set. The sum of the new set = number of pieces × weight of one piece. Therefore the weight of the piece in the new set is the sum of the pieces in the original set divided by their number; the “sum/n”.
Specifically, if addition is the natural thing to do, the set 3, 4, 8 is equivalent to 5, 5, 5, because 3 + 4 + 8 = 5 + 5 + 5. Saying that “5 is the mean of the original set” means “the original set behaves (with regards to the natural thing to do, i.e. addition) as if it was composed of the 5′s”.
There are situations where some other operation is the natural thing to do. Sometimes it is multiplication. For example, if you multiply some original value with 2, and they you multiply it by 8, the result of these two operations is the same as if you would multiply it twice by 4. In this case it’s called geometric mean, and it’s a root of product.
It can be even more complicated, so it doesn’t necessarily have a name, but the idea is always replacing the original set with a set of identical values such that in the original context they would behave the same way. For example, the example above could be described as a 100% growth (multiplication by 2) and 700% growth (multiplication by 8), and you need to get a result 300% (multiplication by 4); in which case it would be “root of (product of (Xi + 100%)) − 100%”.
If there is no meaningful operation in the original set, if the set can be ordered, we can pick the median. If the set can’t even be ordered, if there are discrete values, we can pick the most frequent value as the best approximation of the original set.