- Topics At a Glance
- Left-Hand Sum
- Right-Hand Sum
- Comparing Right- and Left-Hand Sums
- Error in Left- and Right-Hand Sums
- Midpoint Sum
- Midpoint Sums with Shortcuts
- Over or Under Estimates
- Trapezoid Sum
- Trapezoid Sum with Shortcuts
- Over or Under Estimates
- Comparison of Sums
- Definite Integrals of Non-Negative Functions
- Definite Integrals of Real-Valued Functions
- Conditions for Integration
- General Riemann Sums
- Properties of Definite Integrals
- Single-Function Properties
- Talking About Two Functions
- Thinking Backwards
**Average Value****Averages with Numbers**- Averages with Functions
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

To find the average of a bunch of numbers, you add all the numbers and divide by how many there are.

Five students took a test. Their scores were 60, 70, 80, 85, and 87. The average of their scores is

In that example, if we add all the students' scores, we get 382. Now suppose we added up the students' scores and got 382, but that all the students got the same score on the test. In order for that to happen, each student would have had to score

You can think of averaging as redistributing points so that everyone gets an equal share.

Exercise 1

Six students took a test. Their scores added up to 474. If every student got the same score, what was that score?

Exercise 2

Eleven kids had 22 brownies to share. If each kid got the same amount of brownie, how many brownies did each kid get?