So, a company have 500 reviews and their current rating is 3.2 stars out of 5.
I'm trying to calculate how many 5 star reviews they'd need to get to reach an overall rating of 4 stars.
I'm basically just looking for an equation that would output this answer (using to track stuff in a spreadsheet).
Any help would be appreciated. Thanks!
An equation would look something like...
$$n = \frac{t(m_2 - m_1)}{c - m_2}$$
where
$m_1$ is the old average among $t$ number of ratings
$m_2$ is the new average among $t$ number of ratings
$t$ is the number of ratings used in the first average
$c$ is the score ($0-5$ stars) of the additional ratings used in the new average
$n$ is the number of the additional ratings used in the new average
In your specific application you have...
$$n = \frac{500(4 - 3.2)}{5-4}$$
You can see the math as follows: let $s_i$ be the $i$th score, or rating, for $i = 1,2,3,...,500$. We know that...
$$\frac{\sum_{i=1}^{500} s_i}{500} = 3.2$$
$$\sum_{i=1}^{500} s_i = 1600$$
We want to solve for $n$ in the following
$$\frac{5n+ \sum_{i=1}^{500} s_i}{500 +n} = 4$$
$$\frac{5n+ 1600}{500 +n} = 4$$
$$5n+ 1600 = 4(500 +n)$$
$$5n+ 1600 = 2000 + 4n$$
$$5n - 4n = 2000 - 1600$$
$$n = 400$$