Weighted Average

An average that has eaten waaaaaaay too many Tootie Bars, Cheesy Poofs, and Snacky Cakes.

A weighted average is an average that doesn’t allow each number in the average to count equally towards the average. Instead, each data point is weighted according to its overall contribution. Normally, in a regular average, we just add up the numbers and divide by the total number of data points. This “spreads” the average value equally across all the data points.

A weighted average first multiplies the data point by its weight then divides the sum of all the weighted values by the total amount of weights. Your school GPA is a classic weighted average, because classes that are for more credits, like a five-credit course compared to a three-credit course, count more toward your GPA.

Let’s say you took a three-credit math class and got a 3.0, a four-credit psych class and got a 2.5, and a five-credit social science course and got a 4.0. Your regular average would be a Shmoop-pleasing (3.0 + 2.5 + 4.0)3 = 9.533.17. Your actual GPA as a weighted average would be an even more Shmoop-pleasing (3.03 + 2.54 + 4.05)(3 + 4 + 5) = (9 + 10 + 20)(12) = 3912 = 3.25. Since the five-credit course counted more heavily in the weighted average, that 4.0 helped bump the weighted average up above the regular average.

The wrong way to manage this process? Here's a clue: Don't just add up all your grades and divide by, uh… how many grades you have. You're creating a weighted average, which admittedly sounds like an average that needs Jenny Craig...but is just an average that gives each value a different weight rather than the same weight.

Want to know how casinos are quite literally designed to make money, or rather to take your money and make it their money? It's because they understand how the probabilities of winning or losing a game all merge together to tell them what the likely results of playing that game a jillion times are. This long-term, cumulative result is the expected value of the game.

Like most colleges, Whassamatta U determines your GPA as a weighted average. Classes that are for more credits count more towards your GPA. Classes for fewer credits count less towards your GPA. Each letter grade in a course is converted to a number, which is then multiplied by the number of credits. The results are called grade points. 'Cause they're gonna get averaged together to create a grade point average. Clever, right? A regular average would just add up the four grades and divide by four. This way, each course weighs equally in the average. A weighted average takes those grade points that are weighted by the number of credits and divides by the total number of credits attempted, which is 14. And each course weighs a different amount in the final average. It's pretty cruel, but at least they have wicked fast WiFi in all the dorms.

Teachers also typically use weighted averages to determine your grade in a course. To determine your grade, you just need to multiply your score in a category by its percentage weight and add up all the answers. Boom! Instant weighted average grade.

It's usually easiest to leave scores in percent form, and convert the weights to decimals or fractions before multiplying. Poor Smelly McNeedsashower. He only managed a 67.06%, even with decent test and quiz scores. 25% for Personal Hygiene does seem kinda harsh, though.

Abigail D. Moneybags is a huge fan of making money in the stock market. She tends to buy more and more stock in a company if its value continues to increase over the years. She has been in on the stock for the makers of the Squatty Potty since day one. Those stocks have made her, um...flush with cash. Abigail bought 20 shares for $5 a share in 2000. Then 30 shares for $11 a share in 2005. Then another 15 shares for $18 a share in 2008. And finally, another 25 shares for $27 a share in 2015. She'd like to know the average price she paid per share.

A regular average, where we add up the four prices and divide by four, doesn’t take into account that she bought different numbers of shares at the four different prices. Abby needs a weighted average.

It turns out that Abby paid about $15.28 per share over the years, when we rightly account for the prices of the groups and the sizes of the groups. Mrs. Moneybags expects her information about her stocks to be A number 1...not number 2.

At the casino, the roulette wheel has 38 equally sized spaces. 18 are red. 18 are black. 2 are green. Sadly, none are rainbow-colored. Many gamblers believe betting on a color like red or black is the best way to make money in the casino. We can figure out how clever of a strategy this is by finding the expected value. The expected value is the average outcome we expect after many, many trials of spinning that wheel and seeing how much we lose, or occasionally...win. To find an expected value, we need to know all the possible outcomes and their associated probabilities. We can win or…lose. We also need to know the probability of each of those outcomes if there are 18 reds out of 38 spaces total.

We'll drop a fiver as a bet on red, and if we win, we get our fiver back plus another five. If we lose, they keep the five. To find the expected value, we just calculate another weighted average. We multiply the outcome by the probability, and add up the results, just like getting a grade from the breakdown in a syllabus.

Yeah, that's negative 0.263 dollars...or negative 26 cents. We expect to lose 26 cents if we keep dropping $5 on red (or black) over and over, all night long. We clearly can’t lose 26 cents on any one game.

We should, however, expect to walk out with less money than we walked in with if we play the one-color bets all night. Another spoiler alert: every single casino game has a negative expected value. Every. Single. One. Meaning that, over millions of plays, people lose…a lot.

It ain't just the casino games that sport negative expected values. Yup, lottery tickets, too. We know. We know. We're really tossing a wet blanket on stuff. We're gonna take the probability distribution that has to be printed on the back of each ticket and pair that up with the possible outcomes, or winnings, if we drop $2 on the ticket. When we find an expected value, otherwise known as a weighted average, we find that we should stop buying lottery tickets. We spend $2 buying this ticket over and over a whole mess of times, and expect by the end to be about $1.23 poorer than when we started.

Why is math such a buzzkill sometimes?

Not all expected values are negative. Black Fortress, a new fast food joint, has done a careful study of the number of cars in the drive-thru, and created a probability distribution showing how likely it is for there to be anything from 0 to the max of 6 cars in line.

The expected value can be found like every other one we've done. We multiply the number of cars by the corresponding probability and add up the answers. This story tells the Dark Lords in the Black Fortress that, over many, many counts of the number of cars in line, they can expect 2.33 cars to be in line.

Or that they can expect 2 or 3 cars to be in line since, uh...0.33 of a car isn't a thing. Unless you drive a Smart Car.