Although the raw mean and median are shown under the detailed vote breakdown graph on the ratings pages, the user rating vote displayed on a film / show's page is a weighted average. In order to avoid leaving the scheme open to abuse, we do not disclose the exact methods used.
We can provide some more detail here to reassure you that our methods are both sound and fair. First of all, the same formula is applied universally across the database to all movies and shows without exception so there is no bias in when and where the scheme operates. Occasionally we receive mail from people who seem to assume that some favorite movie or show has been victimized by the weighted ratings whereas this is not the case. The objective of the scheme is to present a more representative rating which is immune from abuse by subsets of individuals who have combined together with the aim of influencing (either up or down) the ratings of specific movies or shows. This includes people involved in the production of a movie / TV show and their friends or fans trying to unduly raise the rating far above that of where the typical IMDb users would rate it.
The scheme combines a number of well-known and proven statistical methods, including a trimmed mean to reduce extreme influences and, most importantly a complex voter weighting system to make sure that the final rating is representative of the general voting population and not subject to over influence from individuals who are not regular participants in the poll. The scheme has been developed internally over the 25 years which the poll has been in operation and tuned on a regular basis to make sure it remains fair.
Most of the feedback we receive about our votes is based on the incorrect assumption that all votes cast by our users have the same impact on the final rating. This is not the case. Different votes may have different weight when they're used to calculate the final weighted rating. Most people think of the arithmetic mean when they hear the word average. When calculating an arithmetic mean, all votes are treated equally: the average is the sum of all the votes divided by the number of votes.
A weighted average, however, is defined as "an average that takes into account the proportional relevance of each component, rather than treating each component equally". There's nothing arcane or mysterious about it: it's a very simple, universally accepted statistical method, commonly used in a wide variety of fields (from financial analysis to student reports).
For example, an automobile magazine reviewing a new car may give it high marks in several categories (appearance, comfort, fuel efficiency, price, number of cup holders). However a model with high ratings in all those categories may still get a low overall score if it gets a low vote in just one or two other areas (like speed or safety): even the cheapest, nicest-looking and most fuel-efficient car isn't worth buying if the fuel tank explodes when you hit the brakes or if its top speed is only 15 miles per hour. Clearly, a high (or low) vote in some categories has more weight than the same exact vote cast in another category, so the final rating takes these differences into account.
Our calculations follow a similar principle: some votes have more weight than others. We will not disclose exactly if/when/how certain votes are weighted differently because the idea is, among other things, to give a more objective rating and neutralize attempts to artificially inflate or deflate the average user rating on a film or show.
We're confident that our system provides a reasonably faithful representation of our users' opinions. This scheme is applied uniformly to all films listed in the database, without exception; and since it has proved to be very effective we do not plan to abandon it, though we periodically revisit and tweak it to make it even more accurate and tamper-proof.