Why does dividing the final(chosen) height in a line by 2 give you the average height(y) in that line?

Question

I was taught, of course, that the average of something is the sum of all the parts divided by the number of those parts.

This is easy when it is a set of values(2, 6, 8) because its simply (2+6+8)/3 = 16/3 ≈ 5.33.

But when you have a line, you have an **infinite amount of values** divided by **infinity**, right?

So how is it even possible to take the average? And why can you simply divide it by 2 when its a line?

And then of course there is the question about the average of curves and whatnot…

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I feel so dumb, plz help me understand.

**Also this question comes from the idea that average velocity is half the final velocity**

In: 0

Answer 1

It’s a line, so it won’t change direction, we know that.

So if we take the sum of two points in the line, and divide by 2, we get the average between two points. We can also take 3 points in the line, add them together, and divide by 3, and we get the average between those points.

The start and end points are just two points on the line.

If I drive 30 miles in 1 hour, my average speed was 30mph. You know this even though I only told you how far I got from the start and how long it took to get there.

Answer 2

Note that in mathematics the word “average” doesn’t actually have a technical definition. It’s a colloquial term often associated with the “arithmetic mean”. In a sense, average is intended to capture a “measure of central tendency” and there are many ways to do this, such as using median, mode, geometric mean, harmonic mean, etc.

Having said that, “average speed” is just a midpoint between a starting value and a stopping value. That aligns with the idea of a measurement of “central tendency”, right? Average is intended to capture the “most typical” value of something. Sometimes you may have been faster, sometimes you may have been slower, but we can “typify” our speed with this single value.

Answer 3

In general, just think of the graph of a linear function. No matter the slope, half of the graph is gonna be above the average value and half is gonna be below. That makes the “average” value the mean of the starting and ending points