Maths Puzzle 9 - Simple Way To Calculate The Mean Of A Group Of Numbers

Let's say we have to calculate the mean (average) of a group of numbers.

54

36

45

57

49

39

28

There is of course the simple way of summing up all the 7 numbers above and divide the sum by 7.

Let's now say we want to do it fast. Speed is always important. Now how do we find the average faster. And let's say a small inaccuracy is allowed. (Example is when we know the possible solutions and we have to choose one or when all we need to know is an approximate answer within +/- 3%).

What does the average of the numbers seem to be? Maybe about 40? ok. Let's take 40 as the base and find out the differences of these numbers from 40.

Orig  Difference

No.    From 40

54       14

36       -4

45        5

57       17

49        9

39       -1

28      -12

---------------

Total   28

The sum of the differences from 40 is 28 as shown above. The average of this sum of differences is 28/7=4

Add it to the original base (40).

The actual average = 40+4 = 44.

Now, couple of points to be noted here:

1. The sum of the differences is easier to calculate than the sum of the original numbers because the differences are usually smaller. Where the differences are large this method will not be easier.
2. To make the differences small, we have to choose a base which is close in value to all the original numbers. This is done by a quick scan of the numbers for which we need to take the average. 
3. And then we should choose such a number from which the difference is easy to calculate. For example we chose 40 as the base. We would get the same answer for the average of the original numbers even if we had chosen 4 or 400 or 41. But out of these 3 choices for the base, which is the easiest to work with? 40, isn't it?

Using 40 as a base made it easier to calculate the average of the numbers given at the top. 

Maths is fun. We use it in everyday life. We can calculate faster if we know some "tricks".