Averages- Different approaches to calculate averages.

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Hi,
Hope you guys liked our last post about relative speed. Let’s discuss some unique approaches to calculate averages, trust me you will fall in love with calculating averages.

Let us take a case:

If there are three rooms A, B and C, such that A has 20 students, B has 10 students and C has 15 students,

avg

 

Then average number of students in the three rooms is –

(20 + 10 + 15) /3 = 15.

So, Average refers to the sum of observation divided by number of observation.

Average = Sum of observation/ Number of observation

 

We got 15 as average number of students in three rooms, what does that mean? What is 15 trying to say?
Let us see- we had three numbers (20,10,15) whose average is 15.

Let say, below is a line representing “15”(Average value).

avg2

In this figure you can see that 20 is above the average, 10 is below the average and 15 is on the average line. Let us see how much deviation does 20 and 10 have from the average number.

20 – (+5) (Positive deviation)
10 – (-5) (Negative deviation)

Net deviation = Positive deviation + Negative deviation

i.e (+5) + (-5) = 0

Ohhh!! Did we get a “0” . Isn’t that interesting, yes it is ! Average is such a number which equals the net deviation of the whole observation to zero.

So If there is no deviation in any of the numbers of the observation, then what does it mean ? It means that every number has the same value. So when you get 15 as average you can say that each classrooms A, B & C have equal number of students i.e 15.

So the above arrangement of rooms can also be represented as –

avg3

Now if I tell you to calculate the sum of this new observation what will you do ?
Since you need to sum 15 three times therefore, You will directly write 15×3=45 or can you represent this equation as 15=45/3 . Can you now observe this equation ? This is the same thing that you write as formula:

Average = Sum of observation/ Number of observation

Now did you get an idea that if you know the concept there is no need of remembering formulas because ultimately the fact is, these formulas were derived from the same concepts.

More on Averages:

Q.The average of five numbers is 29. If one number is excluded the average becomes 27. What is the excluded number?

Ans.  As we know that the average of all 5 numbers is 29. Now as we have seen above in the room and students example, here all those 5 numbers can be assumed to be 29. So, Let’s suppose all those 5 numbers are 29.

avg4

Now in the second arrangement, the numbers become 27, originally they were 29, now how do we calculate the lost/excluded number ?

Here is our observation –looking at the figure I can clearly observe that initially I had 5 boxes where as now I am left with four boxes, so I have lost one box and since each box was numbered 29, thus I have lost “29” .

Wait !  I just had one more observation, apart from one box in which I lost 29. I can also see that each of my boxes are now numbered “27” , which means I lost two from each of my original boxes. Total number that I lost from each of my boxes is (4 × 2 = 8) – Since now I had total 4 boxes.

Now can you calculate what is the total number I lost/excluded?
It’s (29+8 = 37) , excluded number was 37.

Textual method to solve above problem:

Average of five numbers is 29, so let us consider (A+B+C+D+E)/5 = 29
If one number is excluded average becomes 27 or (B+C+D+E)/4 = 27

A+B+C+D+E = 145                     (1)
B+C+D+E = 108                        (2)

Putting value of (2) in (1) we get A = (145 – 108) = 37 .

Let me know is it required to perform these mathematical operations or forming equations if you can just visualize certain things to calculate the answer while solving Averages?  Leave back your traditional approaches and start visualizing things, this will increase your speed and save your time when you are sitting for limited hour examinations.

Let us take one question for which you think you will require to form equations to calculate answer:

  1. The average weight of the teacher and six students is 12 kg which is reduced by 5 kg if the weight of  the teacher is excluded. How much does the teacher weigh ?

Ans. Forget about methods that you have been using, Keep your pens down and just visualize with me. There are 7 people whose average weight is 12, so you can consider each one of them weigh 12 kg. If the weight of the teacher is excluded, average weight reduces by 5kg i.e now you are left with 6 people who weigh 7kgs each. Just like the previous question calculate the weight that you have lost [12kg (teacher’s weight) + 30kg (5kgs each of 6 students)] = 42 kg. So the excluded weight was 42kg and who’s weight was excluded? –  TEACHER’s.

Thus the teacher weighed- 42 kg.

Wasn’t that simple to calculate. Stay connected we’ll come up with unique approaches for more topics !!

So here’s joke, let’s see how many of you can figure this out-

Three statisticians went out hunting, and came across a large deer. The first statistician fired, but missed, by a meter to the left. The second statistician fired, but also missed, by a meter to the right. The third statistician didn’t fire, but shouted in triumph, “On the average we got it!   

(Written by Yeshu)

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