Let’s take a look at the very popular problem of finding the average of a list of numbers. The conventional method for this is to add all the numbers and then divide by the length of the list. For example:

1, 3, 5, 7

1 + 3 + 5 + 7 = 16
16/4 = 4

So the average is 4.

Now let’s say you add another number to the list, 9. What would you do? If you follow the above method, you will re-add all the numbers and divide again. Here that’s not a problem, but imagine you have a huge list with a lot of numbers coming in. Would you still add all the numbers and then divide?

That’s wasteful. We need to find a better algorithm for finding the average. To do that, we will need to use the previous average to calculate the new average. We will name the previous average prevAvg, the previous list length prevLength, the current list length currLength and the new number x. We want to find the current average, currAvg.

currAvg = (prevAvg*prevLength + x) / currLength

In Python code:

#Initial list, average and length
sequence = [1,3,5,7];
avg = 4;
length = 4;

#Add x to list, length increases by 1
x = 9;

avg = (avg*length+x)/(length+1);

This can speed up a lot your work, especially when you are dealing with big data and want to calculate, say, the mean.