THE FLAVOURS OF AVERAGES

What is exactly average or mean?
Consider the numbers, 14,15, 16, 12. Add them and divide by 4. We get 14
14+15+16+12 = 56
also 14+14+14+14 =56
Hence 14 is the representative of the group of four numbers or 14 represents each number. That is what we call mean. This is especially called Arithmatic Mean.(AM) We think that only 'this average exists, but ther are variety of means we have to choose proper mean for each situation.

GEOMETRIC MEAN
Suppose your salary doubles every 5 years. 20k,40k,80k, 160k. What is average increase for every five years. Here, these numbers have multiplicative relationship not additive relationship. Hence multiply all the numbers and take fourth root of the answer . Since there are 4 numbers.
Fourth root of (20*40*80*160) is 56.56k
20*40*80*160 =10240000. Also the geometric mean 56.56k truly represents these four numbers. The average increase in salary for every five years is 56.56k.
What is the true mean of 12 and 3. The arithmatic mean of them is 7.5. But the geometric mean is 6. That is square root of (12*3) is 6.

also, 12/6 =2
6/3=2.
The ratio of 12 to 6 and 6 to 3 is one and the same.
Hence the GM is 6 is the true mean here. The aspect ratio in mobile screen and TV screen is found using GM.

HARMONIC MEAN
Consider these numbers 2,4,5,100. Here AM and GM will not bring out the best average. Because one number 100 is situated for away from the other three. Hence follow this method.
1. Find the reciprocal of all the numbers.
1/2=0.5; 1/4=0.25; 1/5=0.2; 1/100=.01
2. Add them
0.5+0.25+0.2+0.1 = 0.96
3. Divide the number of numbers by the total.
That is 4/0.96 =4.166
4.166 is harmonic mean. That is the true representative here. if you take AM, it will be 55. We know, that is not a good representative.
Hence, if there are outlying numbers, it is better to go for harmonic mean.

ROOT MEAN SQUARE
-2,-4,2,4. The AM for this group is 0. This 'mean' does not give any meaning. Now follow this procedure.
1. square all
4,16,4,16
2. Take average
4+16+4+16/4 = 10
3. Take square root
10 =3.162
Root mean square(RMS) 3.162 is the true mean of these numbers. By squaring, we eliminated negative sign. Hence we got a meaningful 'mean'.
Errors are always positive and negative. In calculating 'average of errors' RMS is employed. The alternating current -AC has positive and negative cycle. Here also RMS is employed and there are many more.
So there is a variety of averages. Choose the suitable one.
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