HM-GM-AM-QM inequalities
Okay kiddo, let me explain HM, GM, AM, and QM inequalities to you in a simple way.
We'll start with some numbers: Let's say you have four numbers - 2, 4, 6 and 8. Now, if someone asks you what is the average (also known as the Arithmetic Mean or AM) of these four numbers, you would add them all up (2+4+6+8) and divide by 4, giving you an AM of 5.
But, there are three other types of means we can calculate from these four numbers:
1. Harmonic Mean (HM): This is calculated by finding the reciprocal of each number, adding them up, dividing by the total number of values, and then finding the reciprocal of that value. So, the HM of 2, 4, 6, and 8 would be [(1/2)+(1/4)+(1/6)+(1/8)] / 4, which gives us an HM of 3. Indeed, the HM is always less than the AM.
2. Geometric Mean (GM): This is calculated by finding the product of all the numbers, and then taking the nth root, where n is the total number of values. So, the GM of 2, 4, 6, and 8 would be [(2 x 4 x 6 x 8 )]^1/4, which gives us a GM of 4. Indeed, the GM is always less than or equal to the AM.
3. Quadratic Mean (QM) or Root Mean Square (RMS): This is calculated by finding the square root of the average of the squared values of all the numbers. So, the QM or RMS of 2, 4, 6, and 8 would be √[(2²+4²+6²+8²)/4], which gives us a QM of √(120/4) which is √30 or about 5.48. Indeed, the QM is always greater than or equal to the AM.
Now, here's the cool part: We can use these means to compare the data set. Specifically, if we arrange the numbers in ascending order, then we can say the HM is always less than or equal to the GM, which is always less than or equal to the AM, which is always less than or equal to the QM.
So, in the example we've been using, we can see that the HM (3) < the GM (4) <= the AM (5) <= the QM (√30 or about 5.48).
Does that make sense, kiddo?
We'll start with some numbers: Let's say you have four numbers - 2, 4, 6 and 8. Now, if someone asks you what is the average (also known as the Arithmetic Mean or AM) of these four numbers, you would add them all up (2+4+6+8) and divide by 4, giving you an AM of 5.
But, there are three other types of means we can calculate from these four numbers:
1. Harmonic Mean (HM): This is calculated by finding the reciprocal of each number, adding them up, dividing by the total number of values, and then finding the reciprocal of that value. So, the HM of 2, 4, 6, and 8 would be [(1/2)+(1/4)+(1/6)+(1/8)] / 4, which gives us an HM of 3. Indeed, the HM is always less than the AM.
2. Geometric Mean (GM): This is calculated by finding the product of all the numbers, and then taking the nth root, where n is the total number of values. So, the GM of 2, 4, 6, and 8 would be [(2 x 4 x 6 x 8 )]^1/4, which gives us a GM of 4. Indeed, the GM is always less than or equal to the AM.
3. Quadratic Mean (QM) or Root Mean Square (RMS): This is calculated by finding the square root of the average of the squared values of all the numbers. So, the QM or RMS of 2, 4, 6, and 8 would be √[(2²+4²+6²+8²)/4], which gives us a QM of √(120/4) which is √30 or about 5.48. Indeed, the QM is always greater than or equal to the AM.
Now, here's the cool part: We can use these means to compare the data set. Specifically, if we arrange the numbers in ascending order, then we can say the HM is always less than or equal to the GM, which is always less than or equal to the AM, which is always less than or equal to the QM.
So, in the example we've been using, we can see that the HM (3) < the GM (4) <= the AM (5) <= the QM (√30 or about 5.48).
Does that make sense, kiddo?