General - Is there a way to calculate an average based off of an existing average and given new parameters for the resulting average?

Question

Lets say, we're calculating averages of test scores:

Starting Test Scores: 75, 80, 92, 64, 83, 99, 79

Average = 572 / 7 = 81.714...

Now given 81.714, is there a way to add a new set of test scores to "extend" this average if you don't know the initial test scores?

New Test Scores: 66, 89, 71

Average = 226 / 3 = 75.333...

Normal Average would be: 798 / 10 = 79.8

I've tried:

Avg = (OldAvg + sumOfNewScores) / (numOfNewScores + 1)

(81.714 + 226) / (3 + 1) = 76.9285

Avg = (OldAvg + NewAvg) / 2

(81.714 + 79.8) / 2 = 80.77

And neither come up the exact average that it "should" be. Is it mathematically possible to do this considering you don't know the initial values?

Answer 1

You have to know the number of test scores in the original set and the old average:

newAve = ((oldAve*oldNumPoints) + x)/(oldNumPoints+1)

Answer 2

Say, you have 2 blocks of scores:

1st: n scores with average = a1 
2nd: m scores with average = a2

then average of total scores equals:

a1*(1.0*n/(m+n))+a2*(1.0*m/(m+n))

In case you want just add 1 score (a2) to existing set formula becomes

a1*(n/(n+1))+ a2/(n+1)