Averaging it out…

The long and short of averages

 You immediately recognized what I’m going to write, didn’t you? In case you didn’t (because of the slightly silly title), let me tell you.

This post is about one of the common topics of Math: Average.

 You sure know what average is. You also know what other names it goes by (arithmetical mean or simply mean). So I’ll only go through the basic idea in a flash and then move on to the important stuff.

Average of a given number of values is the sum total of all the values divided by the number of values.

So if you scored 24, 28 and 14 in 3 tests of Paleontology (whatever it is), the average will be calculated as

(24 + 28 + 14) /3

= 66/3

= 22.

 Cool; you’re done with the basic idea. Now you’d want to look at the variety of make-up this concept wears. I use the word make-up because underneath the different shades, the original idea remains unchanged in all questions.

Average can do something interesting. It can tell you the total of all the values.  In the above Paleontology example, you could view the question backwards (and correctly too) by saying that if the average of 3 tests is 22, then the total of the 3 tests must have been 22 x 3 = 66.

 Hence,

Average = Total/Number of Values

and

Total = Average x Number of Values

This means we can use the idea of average to find out totals too.

But just before going further, if you can visualize what this concept means, it will save you a good deal of trouble in some cases.

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 Average, in a certain sense, means the middle value. So when you see the five fingers, you know the biggest finger (the middle finger) is the average of the five fingers.

 Now look at the following numbers:

64, 67, 70, 73, 76

The middle number is 70, so 70 is the average of the given numbers. But that is subject to a requirement:

If a set of numbers have the same difference throughout, the average of the set of numbers will always be the middle value.

But what if the total number of values is even? Then you can’t have a middle-value.

For example with the four numbers 10, 20, 30, and 40, you don’t have a middle values.

Well, in that case take the two middle values and average them out. So here you have 20 and 30 as the two middle values and averaging out 20 and 30 gives 25.

So 25 is the average of 10, 20, 30 and 40.

And here’s a slight impish variation: find the average of the numbers 15, 35, 20, 10, 25, 30, 40.

At the first glance, it looks like we won’t be able to apply the middle-value trick here. But hold your horses; why not rearrange the numbers in ascending (increasing) order?

And yippee! You have a disciplined line of numbers:

10, 15, 20, 25, 30, 35, 40

And you don’t need to be told that 25 is the average.

This brings us to our first Strategy:

Strategy 1:

If you want to find the average of numbers whose difference remains constant throughout, all you need to do is to arrange them in ascending order. Once that is done, the middle value is the average.

If the number of values is even, choose the two middle-values. The average of these two middle values is your average.

Consider another question:

The average number of matches played by each of the 16 basketball players at a university was 26. When Rashid joined the university, the average number of matches played by each player now becomes 26.5. Find the number of matches Rashid has played.

Solution: Forget Rashid.

If there are 16 players and each has played an average of 26 matches, the total number of matches played is

16 x 26 = 416.

Enter Rashid.

Now you have a total of 17 players and each has played an average of 26.5 matches.

That means the total number of matches is

27 x 17 = 459

You have two totals 416 and 459. You also know they are different because of only one person. Rashid.

Hence Rashid’ number of matches is 459 – 416 = 43.

This question brings you a good peek into the world of short-cuts.

You don’t really need to do the long calculations I have shown here. All you need is a slightly different perspective.

Short-cut:

Without Rashid, the average of 16 players is 26.

With Rashid, the average of 17 players is 27.

Instead of matches, think of each player having a sum of money.

Without Rashid, the average of 16 players is $26.

With Rashid, the average of 17 players is $27.

That means Rashid donates 1$ to each of the 16 players.

That way Rashid donates 16$.

That way Rashid donates 16$ to others and keeps 27$ for himself.

So Rashid needs (16 + 27)$ = 43$ to do all this donating-and-average business.

That brings us to Strategy 2:

Strategy 2:

Assume you know the average of a certain number of values. Now a new identity is added so the average changes.

Part 1: The value of the new identity is equal to the new average plus the old number of values times the difference in the averages.

Note that if the average has decreased, the above rule reads as

Part 2: The value of the new identity is equal to the new average minus the old number of values times the difference in the averages.

Exercise:

1. Sean a different number of emails all the six days this week. He received 15 emails on Monday, 27 emails on Tuesday, 19 on Wednesday, 31 emails on Thursday, 11 emails on Friday and 23 emails on Saturday. Find the average number of emails Sean received per day.

(Hint: Strategy 1)

2. The average weight of 40 students in a class is 48 kgs. When a new student joins the class, the average weight becomes 48.2 kgs. Find the weight of the new students.

(Hint: Strategy 2, first part)

3. There are 50 parks in a state and the average size of each park is 20,000 sq. ft. A new park has just come up due to which the average size of each park is 19,980 sq. ft. Find the size of the new park.

(Hint: Strategy 2, second part)