algorithm used to calculate 5 star ratings

I need to calculate 5-star ratings like the one on Amazon website. I have done enough search to find what is the best algorithm, but I am not able to get a proper answer. For example, if these are the ratings

5 star - 252
4 star - 124
3 star - 40
2 star - 29
1 star - 33

totally 478 reviews

Amazon has calculated this to be "4.1 out of 5 stars". Can anyone tell me how this figure is arrived at? I am not able to get this just by doing average.

Solution 1:

That's a weighted average, where you weigh each rating with the number of votes it got:

(5*252 + 4*124 + 3*40 + 2*29 + 1*33) / (252+124+40+29+33) = 4.11 and change

Solution 2:

If you are start calculation of overall rating from beginning then this formula will help you.

Formula

((Overall Rating * Total Rating) + new Rating) / (Total Rating + 1)

Example

suppose you have no ratings till now then formula is like, overall rating is "0" till now. total rating "0" and given rating is "4"

((0*0)+4)/1 = 4

If overall rating is "4.11" Total rating is "478" And new rating giving by one user is "2"

then formula is like

((4.11 * 478)+ 2)/479 // 479 is increment of new rating from 478

Solution 3:

a better way to do this,

rating = (sum_of_rating * 5)/sum_of_max_rating_of_user_count  

example:

total users rated: 6  
sum_of_max_rating_of_user_count: 6 x 5 = 30  
sum_of_rating: 25

rating = (25 * 5) / 30

Done!

Solution 4:

Yes, you can average them out:

(5 * 252 + 4 * 124 + 3 * 40 + 2 * 29 + 1 * 33) / 478 = 4.11