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What is OLAP? 

The term OLAP stands for On-Line-Analytical-Processing. OLAP provides for an "Online" interactive connection to business information.  The interaction that OLAP provides allows users to ask questions "Query" of the database. 

FAST - In general, OLAP queries are very fast. A typical OLAP database should return data back to the user within 5 seconds. This allows users the ability to "interact" with the data, since they don't have to wait long between queries. 

ANALYSIS - OLAP is excellent at numeric calculations. (In fact, you can think of an OLAP cube as a huge calculation engine.) "Speed of thought analysis" is what OLAP provides to users. 

This means that users can quickly and easily visualize the data by navigating up and down business hierarchies.

OLAP is key in providing analysis because it allows the user to look at any combination
of data that is available within the database without having to write complex queries, meaning you are in "Total control of you Business Data".

MULTIDIMENSIONAL - Dimensionality is the key to OLAP.  Multidimensionality allows users to analyze data across multiple axis.  An example of a 5 dimensional cube may be:

Within each dimension are "levels" which are hierarchies which organizes the dimension members. 
An example of a time hierarchy may be:

OLAP allows users the ability to look at any intersection among the dimensions. 

INFORMATION - Information includes all of the stored data and calculations within a cube. 

OLAP databases are often called "Cubes" because of their multidimensional nature. 
Organizations will oftentimes have numerous "Cubes" for each functional part of the business. 
It is not uncommon to see a single organization have the following cubes:

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