What is the difference between the normal average and weighted average?

What is the difference?For what kind of purpose do we use the weighted average method?Why don't we instead always use the normal average method?and also,how to get the weighted average of some numbers?Also,do we have first to get the mean(normal average) of numbers in order to get their weighted average?

Thank you so much.

Let's say that you are flipping a coin, and with heads you win a dollar, tails you lose a dollar. Then you'd expect that the average amount of money you'd win is zero.

Now, let's say it turns out the coin was weighted, so that heads came up half as often as tails. Then the amount you'd expect to win is:

$$\frac{2\cdot(-1)+1\cdot 1}{3} = -\frac{1}{3}$$

This is a weighted average of $-1$ and $1$, based on their frequencies.

Weighted average is important when you are dealing with frequencies or distributions. If you are given a set of data for grades in a math class and you are told that 10 students made a 90, 15 students made an 80, and 5 students made a 70 and asked to determine the average grade for the class, then you cannot use the normal average of (90+80+70)/3. You have to account for the fact that there are multiple instances of each grade. In effect, you weight each grade (90, 80, 70) by multiplying it by the number of instances (10, 15, 5 respectively). Then you total the weights and divide by the number of instances to calculate a weighted average.

Of course you can see from this simplistic example that you need not calculate the normal average to determine the weighted average. You have probably also noticed that if you write out all the grades and do a normal average you should get the same result. For 30 students that isn't much trouble but if you are collecting thousands of data points then that wouldn't be practical.

As for its uses, there are many times when it would be necessary to use. Suppose you doing an historical study of grades in a Calc 1 class and you wanted to know the average grade over the last 10 years the class was taught. You collect the average grade of each class and how many students were in that particular class over the last 10 years. It doesn't make sense to take a normal average of the average grades because each class had a different number of students taking the class. You would want to weight each class average using the number of students who took that class.

Another form of weighted average familiar to probably all high school students is how their grade is calculated. A teacher wants to place more emphasis on midterm and final test scores than on homework and unit tests. The teacher sets weights for each type of grade, maybe Midterm/Final - 70%, Homework - 5%, and Unit Tests - 25%. Then the teacher calculates the average of each type of grade and multiplies it by the weight to determine the average.

These are just a few simple examples. Anytime you are working with data that is unequal in some sense a weighted average is practical. Often times, it is when you are averaging averages, but really the possibilities for its use is endless.