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…
​
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
I can only answer the last part of your question. I’m a physics teacher. Final velocity divided by 2 is the average velocity only if the object starts at rest. 0+vf/2 would be the average velocity
For a linear value, which is what this question is about, it’s enough to take average between start and end points
Avg = (a + b) / 2
When we start at 0, i.e. a = 0, it reduces to
Avg = (0 + b) / 2, which is
Avg = b / 2
For non linear functions (curves, etc), this doesn’t work.
Here’s the 5-year-old explanation. Imagine I’m on the ground floor about to walk up 10 steps. If I walk up them in a row my height off the ground goes from 0, to 1, to 2, steadily until I hit 10. I start at height zero and end at height 10. The average height is halfway up the stairs – 5, so 10/2. Similarly, (0+1+2+3+…10)/11 – not /10, since zero is a step too – equals 5.
Now, imagine I don’t walk up in order. I go up 2, down 2, up 3, down 1, up 3, but eventually make it up the stairs. My height goes 0, 2, 0, 3, 2, 5… and finally hits 10. This height over time isn’t a nice smooth line, it’s a jagged one with positive and negative slope. However, at the end I’m at the top of the stairs, and my average height was still 5. And if you do the math of 0+2-2+3-1+3… divided by however many steps you took, you’ll still get 5 as the average.
Averages of curve you learn about in Calculus through 2 new math operations. You learned multiply, divide, add and subtract, but there are 2 more. The operators which are used when numbers are constantly changing and not in a linear away. Enter Sir Isaac Newton. He was observing the motion of the planets and noticed you could not use any existing math to explain it, so he invented Calculus and the 2 operation, derivative and integral which operate on functions, not numbers.
Consider the path of a projectile. It follows a parabola, a curve. If I were to ask you at 2.5 seconds, how fast is the projectile coming down, with normal math you would be hard pressed to figure it. You could but it would a complicated process. With Calculus and the derivative, you take your function, the path of the projectile and take the derivative of it and now you have a function for the rate of change for the path. You could take the derivative again and get the rate of change, of the rate of change. We are essentially going from Position of the projective, and what do you call a change in position over time? Velocity, and what do we call a change in velocity over time, acceleration. So acceleration is the 2nd derivative of the function for the position of the projectile and integral is the ‘opposite’ of the derivative. If you have a curve, The integral of that curve is the sum of the area underneath the curve. It just so happens when you add up all the acceleration changes, you get the formula for the velocity, and when you add up all the velocity changes, you get the formula of the position.
So, the average of a curve would be to take the integral of the formula of that curve and then divide it by the difference in the starting and ending points of the curve, which looks like the top formula.
[https://calcworkshop.com/wp-content/uploads/average-value-theorem.png](https://calcworkshop.com/wp-content/uploads/average-value-theorem.png)
The average of the values in a line segment will be half of the starting value plus the ending value. For velocity call this (v_i + v_f )/2, which is the same as taking the average between just the final and initial velocities. This makes sense because to take the average of the whole line you can take the average between the initial and final values, then the average of the moment just after the initial value and the value just before the final value, and continue this until you get to the middle. All of these averages are the same, so the simplified formula works.
The definition of average is a number where “half the values are on one side and half the values are on the other side”.
50% of Americans are below average intelligence, and will always be, because that’s what average is. It’s the “mid point” in a set of values where half is above and half is below
The way to think of the average height of a line, or a curve, on a graph is to think about the area under the curve. The average height is the height of a horizontal line starting and ending at the same x-coordinates as the line or curve in question which has the same area beneath it.
For a straight line segment, that will always be the y-coordinate of the point in the middle of the line segment.
This would be much easier to explain with a drawing, but I don’t know how to add one here.
Draw a (sloped) line segment on a graph. Draw vertical lines at the x-coordinates that are the left and right ends of the line segment. Then draw a horizontal line through the midpoint of the sloped line, beginning and ending at the same x-coordinates. Notice that there is a triangle included under the sloped line which is not under the horizonal line, and another triangle under the horizontal line that is not under the sloped line. Notice that those two triangles are identical in size. You can see in that way that the area under the horizontal line is the same as the area under the sloped line.
The branch of mathematics that addresses problems like these is calculus. We’d have to get into that to explain why the area under the line (or curve) is the way to think about the average, as well as how you can compute the area under something that isn’t a straight line. I don’t think I can ELI5 calculus.
Let’s say I have the velocity of a car going from 0-10 MPH increasing 1 MPH every second. What is its average speed over the 10 seconds? I could do (0+1+2+3+4+5+6+7+8+9+10)/11 and get 5. I could also take 0(the initial velocity) + 10(the final velocity)/2 and it would be 5. Both are correct ways to do it, one is much easier. this works because the car is speeding up at a consistent rate. it works because it all cancels each other out. (0+10)/2 = 5. (1+9)/2=5 (2+8)/2 = 5. (3+7)/2 = 5 (4+6)/2 = 5. Leaving 5/1 = 5. Any number of points equally spread out along the line will always give 5 as the average speed. For example, (1.5+8.5)/2 = 5. We could do this for the whole .5 spectrum and determine that (0+.5+1+1.5+2+2.5+3+3.5+4+4.5+5+5.5+6+6.5+7+7.5+8+8.5+9+9.5+10)/21 = 5.
A bit less ELI5: If the car wasn’t speeding up consistently or decreasing sometimes and we wanted to know what the average speed was travelling throughout the entire trip, the longer method is better. In fact the more data points evenly spread out through the trips we use the better. To find a more exact answer we can use calculus as with an equation we could for example turn velocity into distance, determine the total distance travelled and divide that by the time it took the car to travel that distance.
You can think of average as “with what number do I need to replace all the numbers for the result to stay the same”. 3 + 6 + 12 = 7 + 7 + 7. This obviously depends on the operation, for adding it’s the arithmetic mean, for multiplying it’s the geometric mean (3 × 6 × 12 = 6 × 6 × 6) etc.
This generalizes fairly reasonably to the continuous case. If a car’s speed varies over time but by the time *t* it covers the distance *d,* then the average speed is the constant speed that, if maintained at all times during the period of time *t,* would cause the car to get to *d,* and that’s easy to calculate, *d/t.*
You’re almost there. You said it: “you have an infinite amount of values divided by infinity”. This is the math concept of Limits (which leads to Calculus, but we don’t need to go there). Simple case for a straight line: 3 points: (0, 1 ,2) /3 =1. Now take more, smaller steps: (0, 0.5, 1.0, 1.5, 2.0)/ 5 = 1. Keep going: (0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0)/9 =1. You are *approaching* the case you said:”an infinite amount of values divided by infinity”. We can see by the pattern, that as the number of points increases (even “to infinity”) the result is still 1. The Limit for this case is 1.
Latest Answers