# Formula for increase in review statistics

#### scrappyvegas

##### New member
So this may be posted in the wrong place but I am trying to work out a formula regarding reviews. If a restaurant has x number of reviews and they are sitting at 3.6 stars out of a possible 5 how many 5 star reviews would they need to get to a target of 4.6 for example. I hope this makes sense.

Thanks

#### Subhotosh Khan

##### Super Moderator
Staff member
So this may be posted in the wrong place but I am trying to work out a formula regarding reviews. If a restaurant has x number of reviews and they are sitting at 3.6 stars out of a possible 5 how many 5 star reviews would they need to get to a target of 4.6 for example. I hope this makes sense.

Thanks
To start - first calculate the sum of points the restaurant received from those reviews.

If the average is 3.6 from 'x' reviews, then total (sum of) points out of those reviews = 3.6 * x

continue....

#### scrappyvegas

##### New member
Hi

Yep I get what your saying but have no idea what to do next. I'm thinking I need to get to a point when say 15 five star reviews will increase the score by 0.1 There or 50 points in total from 0 rating to 5 star rating. if x is 1000 then the sum of 3.6 is 3600. The difference would be 1400. If I divide that by 50 does that give me how many 5 star reviews will move me up 0.1 points. i.e 28

#### ksdhart2

##### Senior Member
Let's start from the definition of average:

$$\displaystyle (\text{Average}) = \frac{(\text{Total # of points})}{(\text{# of reviews})}$$

Plugging in the known information gives us:

$$\displaystyle 3.6 = \frac{(\text{Total # of points})}{x}$$

$$\displaystyle (\text{Total # of points}) = 3.6x$$

That re-establishes what Subhotosh Khan's hint told you, but now what would we do? Well, we can use the exact same principle to derive when the average will be 4.6 points. Since we don't know how many new reviews we need, let's give that variable a name so we can talk about it. Call it $$k$$. Setting up the definition of average again:

$$\displaystyle 4.6 = \frac{3.6x + ???}{x + k}$$

We know that each new review is guaranteed to be a 5-star review, so how many new points would $$k$$ 5-star reviews add to the total? How does having this complete fraction help you solve for $$k$$? Note that your answer will not be a numerical one, but rather you'll find that $$k$$ is a function of $$x$$.