Arithmetic mean. Why does it work?

Question

I've been using the formula for the arithmetic mean all my life, but I'm not sure why it works. My current intuition is this one: The arithmetic mean is a number that when multiplied by the number of elements, gives you the sum of all the elements. Because of this fact, it can't be more than the maximum nor less than the minimum, and it should be located somewhat around the center. But I was wondering if there are other intuitions out there? Why does this formula work? If in passing you could talk about the weighted average as well, that would be nice too. Thanks

Answer 1

Another way to think about the arithmetic mean is to think of it as describing even shares of a whole, or the fair distribution of something. Suppose that I and my friends Jean and Sara go out to pick berries. We all go to different parts of the field, and I get lucky with a big berry bush where I collect two cups of berries. Jean isn't so lucky and only collects one cup, while Sara finds a huge bush and collects three cups of berries. When we meet up again, we find that all together we have six cups of berries. If we share those out evenly between the three of us, that makes two cups of berries for each of us. That's the same as (2+1+3)/3 = 6/3 = 2. By finding the total of our results and then dividing them evenly we each end up with the same amount, and that's what the arithmetic mean tells you.

A weighted mean makes some of the values we're considering more important than others. Teachers use this a lot in calculating grades. Suppose your teacher says that your grade for this unit is made up of homework, a quiz, and a test, where the quiz counts twice as much as homework and the test counts three times as much as homework. If you score a 70 on your homework, an 80 on the quiz, and a 90 on your test, your grade for the unit is (70*1 + 80*2 + 90*3)/(1+2+3). We took the score for each item, multiplied it by its weight, and then divided by the sum of the weights. (If we only divided by three, you'd get an average of 166, which doesn't make much sense!) This gives you an overall grade for the unit of 83, because your high test grade counted more than your low homework grade, so your teacher wants your grade to be closer to the test grade than the homework grade.

There's also another way to think of averages and weighted averages as if you were trying to find the balance point of a ruler with weights at different places along the ruler. If you're especially interested in physics that's a fun way to imagine it.