Hölder mean
So, let's say you have a bunch of numbers like 2, 5, and 8. You want to find a way to talk about all these numbers together and figure out something about them. That's where the Hölder mean comes in!
The Hölder mean is a way of finding an average, or a middle number, from a group of numbers. But it's a little different from other types of averages you might have heard of, like the arithmetic mean (which is just adding all the numbers together and dividing by how many there are) or the geometric mean (which involves multiplying all the numbers together and then taking the nth root, where n is the number of numbers you started with).
The Hölder mean takes into account how big each number is, or its "weight," when calculating the average. So if you have two numbers, one that's really big and one that's really small, the Hölder mean will give more weight to the big number when finding the average.
To find the Hölder mean, you need to know two things: the numbers you're starting with, and a value called the "order" of the mean. The order is just a number that tells you how much weight to give each number.
For example, let's say you have the numbers 2, 5, and 8 again, and you want to find the Hölder mean with an order of 2. First, you raise each number to the power of 2 (since that's the order we're using): 2^2 is 4, 5^2 is 25, and 8^2 is 64. Then you add up all those numbers: 4 + 25 + 64 = 93. Finally, you take the square root of that sum and divide by how many numbers you started with: √93/3 = about 4.079. So that's the Hölder mean with an order of 2 for those three numbers.
And that's basically it! The higher the order, the more weight the mean will give to bigger numbers. The Hölder mean is just one way to talk about groups of numbers and find something meaningful from them.
The Hölder mean is a way of finding an average, or a middle number, from a group of numbers. But it's a little different from other types of averages you might have heard of, like the arithmetic mean (which is just adding all the numbers together and dividing by how many there are) or the geometric mean (which involves multiplying all the numbers together and then taking the nth root, where n is the number of numbers you started with).
The Hölder mean takes into account how big each number is, or its "weight," when calculating the average. So if you have two numbers, one that's really big and one that's really small, the Hölder mean will give more weight to the big number when finding the average.
To find the Hölder mean, you need to know two things: the numbers you're starting with, and a value called the "order" of the mean. The order is just a number that tells you how much weight to give each number.
For example, let's say you have the numbers 2, 5, and 8 again, and you want to find the Hölder mean with an order of 2. First, you raise each number to the power of 2 (since that's the order we're using): 2^2 is 4, 5^2 is 25, and 8^2 is 64. Then you add up all those numbers: 4 + 25 + 64 = 93. Finally, you take the square root of that sum and divide by how many numbers you started with: √93/3 = about 4.079. So that's the Hölder mean with an order of 2 for those three numbers.
And that's basically it! The higher the order, the more weight the mean will give to bigger numbers. The Hölder mean is just one way to talk about groups of numbers and find something meaningful from them.
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