Alright I have a question for the math-minded...?

Question

300 people have voted between 1 and 5, and the average of the votes is 4.83

How can I figure out the power of one vote? I understand that the value will change, depending on the number of votes. To make it simpler, how can I easily figure out the value of the 301st vote? For example, if my 301st vote is a 1, 2, 3, 4 or 5 .. to easily figure out how much that will change my average.

I can figure it out, but think there must be a formula I can put together to make it quicker.

Right now I'm multiplying the # of votes I have by my current average (so 300 * 4.83) , then adding to it what my next vote would be (1, 2, 3, 4 or 5) , then dividing the result by 301. Is that the easiest way to do it?

It would be nice to put together a formula where i can input the # of votes (V), my current average (A) and the amount of my next vote (?) and get a result of my new average

Answer 1

Kingsley's answer is correct, but I'll make it a little more concrete:

Suppose the average after 300 votes is 4.83 and you cast a vote of 2.0. Your vote is 2.83 less than the average, so the new vote will reduce the average by 2.83/301, or a bit less than 0.01. In other words,
new average = 4.83 + (2.0-4.83)/301 = 4.8206
= old_avg + (new_value - old_avg)/new_total_count

You can use this to mentally update batting averages during a baseball game. Suppose a player comes into a game with a batting average of 0.300 after 200 at-bats. If he gets a hit (which by itself would be an average of 1.000), his batting average will go up by 0.700/201, which is about 0.0035. If he makes an out (which corresponds to an average of 0.000), his average will go down by 0.300/201, which is about 0.0015.

Answer 2

If there are already V votes, the power of each vote is 1/V. If you add one vote, there are now V+1 total votes, and the power of the added vote is 1/(V+1); the new average is ∑v(i)*i/(V+1) where the sum goes from 1 to V+1. Separate this into two parts, the first being the sum to V only:

V
∑v(i)*i/(V+1) + v(V+1)/(V+1)
i=1

{v(i) is the value of vote i, and v(V+1) is the value of the added vote.}
(The summation limits should be understood to be the same in the following.)

Note that v(i)/(V+1) = [V/(V+1)]*v(V+1)/V, so the sum can be written

[V/(V+1)]*∑v(i)/V + v(V+1)/(V+1)

The summation above is the average vote before the last one was added in, A = ∑v(i)/V

[V/(V+1)]*A +v(V+1)/(V+1);

this is the formula for the new average in terms of the old average, the prior no of votes, and the value of the new vote.

It can be written

A(new) = [1/(V+1]*[A*V +v(V+1)], i.e, multiply the old average times the old number of votes, add in the new vote's value, then divide by one plus the old no of votes.

You can derive this without formulas also. If you multiply the average value by the number of votes, you get the sum of all the vote values (as if all the votes were the same and equal to the average value). Now you add the new value to that and get a new sum. The new average is this new sum divided by the total number of votes, which is now one more than before.

Answer 3

Well, your current formula would be represented as (AV + ?) / (V + 1), which is pretty much as simple as it can get, although if you were to split the formula into partial fractions, you would get A + (? - A) / (V + 1), so the net effect of the (V + 1)th vote on the average would be (? - A) / (V + 1). Hope that helps.