Is Amazon's "average rating" misleading? If I understand correctly, book ratings on a 1-5 scale are Likert scores. That is, a 3 for me may not necessarily be a 3 for someone else. It's an ordinal scale IMO. One shouldn't really average ordinal scales but can definitely take the mode, median and percentiles.
So is it 'okay' to bend the rules since the large part of the population understands means than the above statistics? Although the research community strongly rebukes taking averages of Likert scale based data, is it fine to do this with the masses (practically speaking)? Is taking the average in this case even misleading to start with?
Seems unlikely that a company like Amazon would fumble on basic statistics, but if not then what am I missing here? Can we claim that the ordinal scale is a convenient approximation to the ordinal to justify taking the mean? On what grounds?
 A: Benefits of using the mean to summarise central tendency of a 5 point rating
As @gung mentioned I think there are often very good reasons for taking the mean of a five-point item as an index of central tendency. I have already outlined these reasons here.
To paraphrase:

  
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*the mean is easy to calculate
  
*The mean is intuitive and well understood
  
*The mean is a single number
  
*Other indices often yield similar rank ordering of objects
  

Why the mean is good for Amazon
Think about the goals of Amazon in reporting the mean. They might be aiming to


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*provide an intuitive and understandable rating for an item

*ensure user acceptance of the rating system

*ensure that people understand what the rating means so they can use it appropriately to inform purchasing decisions


Amazon provides some sort of rounded mean, frequency counts for each rating option, and the sample size (i.e., number of ratings). This information presumably is enough for most people to appreciate both the general sentiment regarding the item and the confidence in such a rating (i.e., a 4.5 with 20 ratings is more likely to be accurate than a 4.5 with 2 ratings; an item with 10 5-star ratings, and one 1-star rating with no comments might still be a good item).
You could even see the mean as a democratic option. Many elections are decided based on which candidate gets the highest mean on a two-point scale. Similarly, if you take the argument that each person who submits a review gets a vote, then you can see the mean as a form that weights each person's vote equally.
Are differences in scale use really a problem?
There are a wide range of rating biases known in the psychological literature (for a review, see Saal et al 1980), such as central tendency bias, leniency bias, strictness bias. Also, some raters will be more arbitrary and some will be more reliable. Some may even systematically lie giving fake positive or fake negative reviews. This will create various forms of error when trying to calculate the true mean rating for an item.
However, if you were to take a random sample of the population, such biases would cancel out, and with a sufficient sample size of raters, you would still get the true mean.
Of course, you don't get a random sample on Amazon, and there is the risk that the particular set of raters you get for an item is systematically biased to be more lenient or strict and so on. That said, I think users of Amazon would appreciate that user submitted ratings come from an imperfect sample. I also think that it's quite likely that with a reasonable sample size that in many cases, the majority of response bias differences would start to disappear. 
Possible advances beyond the mean
In terms of improving the accuracy of the rating, I wouldn't challenge the general concept of the mean, but rather I think there are other ways of estimating the true population mean rating for an item (i.e., the mean rating that would be obtained were a large representative sample asked to rate the item).


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*Weight raters based on their trustworthiness

*Use a Bayesian rating system that estimates the mean rating as a weighted sum of the average rating for all items and the mean from the specific item, and increase the weighting for the specific item as the number of ratings increases

*Adjust the information of a rater based on any general rating tendency across items (e.g., a 5 from someone who typically gives 3s would be worth more than someone who typically gives 4s).


Thus, if accuracy in rating was the primary goal of Amazon, I think it should endeavour to increase the number of ratings per item and adopt some of the above strategies. Such approaches might be particularly relevant when creating "best-of" rankings. However, for the humble rating on the page, it may well be that  the sample mean better meets the goals of simplicity and transparency. 
References


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*Saal, F.E., Downey, R.G. & Lahey, M.A. (1980). Rating the ratings: Assessing the psychometric quality of rating data.. Psychological Bulletin, 88, 413.

A: In my experience, the mean of rating-scale data is often the most closely correlated with the level of real-world metrics we try to associate with the rating scale. We have found a lot of linear relationships, and the average is therefore one of the better ways to summarize the data. That being said, as Jeromy pointed out, most ways of analyzing the central tendency of a rating scale will give similar results (rank orders, etc) most of the time. 
Also, I suspect Amazon is probably not all that concerned with the scientific validity one way or the other. Amazon's goal, in the end, is to get people to shop more on Amazon.com, and the way reviews help achieve that will probably not vary with whatever one-number summary is used. Good products will be rewarded, really bad products punished, and nervous purchasers will have a chance to review pros and cons in more detail.
A: Amazon ratings are misleading due to companies gaming the system. When customers are offered rebates and free merchandise in return for 5-star reviews, the "statistics" of what the ratings number is or means become moot. 
A: Everyone has good opinions on this. I don't really think I can add very much more. However, I will post this:

A: To be somewhat technical here, those ratings aren't actually a Likert scale; they're just ordinal ratings.  Now, having said that, your point is essentially correct.  However, I often think too much is made of this issue.  One thing to note is that it is typically understood that the average of a number of ordinal items can be approximately interval, and thus, when there are many ratings the mean becomes a more reasonable representation.  I have found this answer by @JeromyAnglim to be excellent (really, the question and all attendant answers there are worth reading).  For a more theoretical treatment, see here.  On a different note, I like Amazon, but I see no reason to expect statistical sophistication from them, especially in terms of basic site design--the point is usability by consumers, not to impress stats professors.  
A: You make a good point.  Taking the mean of ordinal numbers is somewhat misleading. Any summary of several rankings would suffer from the fact that my subjective 3 may really equate to your 4.  So combining different individual scores is probably the biggest problem. Interpreting the average of a 3 and a 4 as 3.5 is not nearly as egregious.
