**CW: The most spicy of arithmetic**

**Kinks: Simulation**

Welcome to the latest edition of “Ratings are a Lie,” a series where I talk about how a novel’s rating may not accurately reflect its true quality, and prove it with pretty graphs. I will also once again be using ratings and reviews interchangeably as I see fit, because even though someone pointed out that “raters” is the option and not “ratingers” (or “bestowers of ratings”), I decided to not back down. It’s because I’m full of ***whimsy*****.**

The first post, “[Vibes and Virality](https://www.reddit.com/r/ReverseHarem/comments/1nr47dz/vibes_and_virality_the_effect_of_reader_priority/),” discussed new readers with different standards of quality/goodness than the expected population finding out about a book and swinging the rating. The second post, “[Size Really Does Matter](https://www.reddit.com/r/ReverseHarem/comments/1nuh4wr/ratings_are_a_lie_series_size_really_does_matter/),” investigated how many ratings are needed to be confident that the average actually reflects the “true” rating of a novel.  The third post, “[ARC-Angels and ARC-Demons](https://www.reddit.com/r/ReverseHarem/comments/1o6i1rg/ratings_are_a_lie_series_arcangels_and_arcdemons/),” took a look at how many typical readers were required to offset initial high ratings from the (occasionally biased) ARC reviews; a little shade may have been thrown at an author who has publicly complained about this subreddit before as well.  

This time, we’re going to look at how sites requiring ratings to be rounded to a certain metric (either full-stars or half-stars) make a difference, depending on how the reviewer chooses to round.  

**BACKGROUND**

Book review websites all have different stratification cutoffs for their ratings. Goodreads requires an integer number of stars only (no decimals allowed), [Romance.io](http://Romance.io) has half-stars, and Why-Choose allows for quarter-stars.

It’s possible even Goodread’s setup is entirely sufficient for accuracy, and that people automatically think in 1-5 stars now. I’m skeptical, though, because in my personal experience, not everything that’s rated 5 stars is equal. Nor is everything rated 4 stars equal. I even have seen reviewers on Goodreads where people break it down into quarter-star rankings.  

I did choose to limit people’s true ratings to quarter-star increments, because while I have used pure uniform distributions in the past, I think it’s more realistic here that people thing “I’d give that 3.75 stars” than “I’d give that 3.702238 stars.”  

***EXPERIMENTAL SETUP***

**Population Distribution**

*Note: As usual for this series, unless otherwise specified these numbers were chosen arbitrarily to illustrate a point, though may have some basis in reality (for example, 4.0 for the typical reader was chosen because the average rating for a romance novel, according to Google, is between 3.9 and 4.3).*  

We’ll take a population of 70% typical readers,  20%  positive readers, and 10%  negative readers.  The typical readers will give the book an average rating of 4.0, the positive readers will always give it an average of a 4.5, and the negative readers will give it an average of 2.0. For an infinite number of reviews and with no rounding,  that would give the book an average of a 3.9.

However, that does not mean that every typical reader gives it a 4.0; just that if we had an infinite number of typical readers, their average would be a 4.0 (same is true for positive and negative). Likewise, every possible group doesn’t have exactly 70% typical readers.

To account for that, we’re going to conduct experiments where we pick random readers off the street that are from our general population distribution of typical, positive, or negative readers. For typical readers, they’ll have rated the book uniformly between a 3.0 and a 5.0, in quarter-star increments. Positive readers will rate it uniformly between 4.0 and 5.0, in quarter-star increments. Negative readers will have rated it uniformly between 1.0 and 3.0, in quarter-star increments.

*Limitations*

I intentionally chose rating ranges where there would be a relatively equal chance of it being rounded up or down conventionally. To combat that, I also ran an experiment with the average ratings being the same, but less of a wide distribution: negative became between 1.5 and 2.5, positive stayed the same, and typical became between 3.5 and 4.5. The effects of this were that the full-star system, more of the options would round to the average, instead of a wider distribution. I refer to this as the non-balanced system.

 

**Types of Rounding**

I’m going to look at 3 main ways to round: always up, always down, and conventional.

1. Round-up will go up to the next highest allowable rating (so 4.25 would go up to a 5.0 in a full-star system and a  4.75 would go up to a 5.0 in a half-star system)

2. Round-down will go down to the next lowest allowable rating (so a 4.75 would go 4.0 in a full-star system, and a 4.75 would go to a 4.5 in a half-star system).

3. Conventional is the generally accepted rounding method: if a number is halfway or more between two options, it rounds up. For the half-star system, it’s the same as the always round up.

***EXPERIMENTAL DESIGN***

For each run, I collected 500 random readers from our population and found their true rating, then converted it using the described rounding scheme (up, down, and conventional for full-star, and conventional and down for half-star); 500 was chosen because it tended to give generally accurate to predicted rating results in previous simulations. I then averaged those results to determine the overall rating using each rounding scheme for that run.

I repeated the process 100 times, and averaged those results as well.

***RESULTS***

**Original System**

Actual Rating: 3.90

Full-Star (Conventional Rounding): 4.01  
Full-Star (Round-Up): 4.23  
Full-Star (Round-Down): 3.58  
  
Half-Star (Conventional Rounding): 4.01  
Half-Star (Round-Down): 3.80

**Non-Blanced System**

Actual Rating: 3.90

Full-Star (Conventional Rounding): 4.08

Half-Star (Conventional Rounding): 4.00

 

**COMMENTARY**

For the original rating distribution, half versus full star doesn’t seem to make a significant difference for the conventional rounding results; they’re a little higher than actual rating, but they ended up with the same approximate results.

For the non-balanced system, we saw an increase in the full-star conventional rating relative to the actual rating; this makes sense, as it is no longer possible to rate below 2 (for negative reviewers) or 4 (for typical reviewers). That’s another effect that may be explored in more detail in a later post. The issue wasn’t seen in the half-star system because everything cut out was balanced on both sides; 1.5 still rounds to 1.5 instead of rounding up to 2 (and is balanced by 2.5) for the negative reviewer, for example.

We also didn’t see major differences in rounding up/conventional versus rounding down in the half-star system; they both were biased in a predictable way (rounding up is higher, rounding down is lower), but both were about a 0.1 away from the actual rating.

 The significant changes really began to be seen when we got to the round-up and round-down for the full-star system, though, where we got a 0.3 point swing from the actual (so the rating varied by over half a point between the two options). The impact is probably higher for the round-down ratings (I think more people would be hesitant to read a book with a 3.6 rating than a 3.9 rating).

In practice, I imagine it’s rare to get a situation where everyone rounds up or everyone rounds down; things are probably a mix of the two. It’s just another example of how it can be dangerous to trust the ratings that get seen for more limited rating systems.

I hope y’all had as much fun as I did!

Ratings are a Lie Series: Getting a-Round (like a Playboy MMC)