I've heard that you have some way of doing questions about averages that makes them really fast and really easy. And since I know since a lot of people asked you if you'd concentrate on the kinds of things that they give on the SAT and PSAT, at least until the PSAT is over in October, they do ask a lot of questions about averages. But a lot of them are not normal average questions, like average this group of numbers. They make them hard by putting in a weird twist, at least usually. So anyway it would be great if you would do something on averages this week or next week.
Thanks a lot,
(Junior High School Student with Decent Grades)
Well, there are a lot of shortcuts with averages if you think of them in a real-world kind of way and think about what the word "average" really means, but I am not sure which magic trick you might have heard I do, so I'll explain what I think is the most helpful concept to know.
First, understand this: Most people think of averages as inexact amounts, sort of like estimations. And in one sense that is exactly what they are: When one uses any form of an average as a predictor of FUTURE EVENTS, then indeed averages are very risky predictors and, mathematically, more likely to be wrong than right. EXAMPLE: Say 12-year-old Jonny is a new bowler. He picked up a bowling ball for the first time one afternoon and found that he had a knack for knocking down pins. Even more importantly, he found that he enjoyed the sport. But, as mentioned, he is still in an early stage of developing his technique, and so far has only played eight games. On two of those games he scored 70, and, as it turns out, his overall average for all eight games is exactly 70. Now, even though you hate betting in any form whatsoever, someone manages to talk you into filling out all the appropriate documentation and getting it signed and notarized by your legal guardians, both of whom happen to be over the age of twenty-one; and you do this JUST so you could pool your money with three friends to accumulate a full dollar and use it to place a lawful bet. The police and other appropriate authorities shake your legal guardians' hands and wish you luck. The thing you were required to bet on was the score 12-year-old Jonny would most likely bowl in the big game to begin in one hour. You did your calculations and checked your work and happen to be excited because you can't wait to win the first bet of your life so you can increase the amount you give to your favorite charity every Sunday.
And what did you bet? Well, when you are told that you can write down a twenty point range that you hope will include his score, you decided to play it 'safe' and bet that he would most likely score between 60 and 80. Sounds reasonable. Right?
Again: Averages are not reliable ways to predict future events. Yes he scored 85 on two separate occasions, but the real reason he fell in love with the sport is that ALREADY on two games he was super-focused and very lucky and scored 150 points on one of the games and 150 points on another. Then why, you may ask yourself, is that not reflected in his average? Wouldn't it seem that his average should be a bit higher than the 70 I mentioned above? Well, sometimes things go that way. But sometimes they do not. The problem for little Jonny is that the games in which he super-focuses and shows remarkable talent are tough on his back. He hasn't yet developed good technique and appropriate posture, so, at least so far, his pattern has been to quickly follow up those excellent games with additional games the following day – yet on those follow-up games he has trouble lifting the ball. In fact, prior to each of those games, he has trouble lifting his toothbrush and later realizes that this was the beginning of his lifelong battle with gum disease.
And so, you later learn, for each of the 'follow-up' games his scores were lower than one might guess. Of the eight games he has played, his chronologically arranged scores go like this: 50, 70, 150, 10, 150, 5, 70, and 55.
QUESTION: Is your prediction of 60-80 pretty close to a sure thing?
No, it is not. It stinks.
NOTE: as stated, for predictions of future events, averages tend to be unreliable. HOWEVER, in another sense, averages are PERFECTLY EXACT numbers and may correctly be considered 100% reliable. How is that? And, MOST IMPORTANTLY, how can this long nonsensical story actually lead you to a 'trick' that is 100% effective for zooming in on the correct answer on a major test in much less time than anyone could do so algebraically?
STAY TUNED. .. IT'S COMING TOMORROW IN PART TWO OF 'JONNY THE BOWLER, AND HOW HE LOST HIS SMILE'. AND I WILL PERSONALLY GUARANTEE YOU WILL BE SATISFIED WITH THIS 'TRICK'. AND IF YOU ARE NOT 100% SATISFIED, YOU WILL BE ENTITLED TO GET BACK YOUR FULL REFUND. (YEAH, THAT'S IT.)
IS JUST A DAY AWAY....
SEE YOU BACK HERE REAL SOON, I HOPE.