An overview of the Analyzing Patterns toolset
Identifying geographic patterns is important for understanding how geographic phenomena behave.
Although you can get a sense of the overall pattern of features and their associated values by mapping them, calculating a statistic quantifies the pattern. This makes it easier to compare patterns for different distributions or different time periods. Often the tools in the Analyzing Patterns toolset are a starting point for more indepth analyses. Using the Incremental Spatial Autocorrelation tool to identify distances where the processes promoting spatial clustering are most pronounced, for example, might help you select an appropriate distance (scale of analysis) to use for investigating hot spots (Hot Spot Analysis).
The tools in the Analyzing Patterns toolset are inferential statistics; they start with the null hypothesis that your features, or the values associated with your features, exhibit a spatially random pattern. They then compute a pvalue representing the probability that the null hypothesis is correct (that the observed pattern is simply one of many possible versions of complete spatial randomness). Calculating a probability may be important if you need to have a high level of confidence in a particular decision. If there are public safety or legal implications associated with your decision, for example, you may need to justify your decision using statistical evidence.
The Analyzing Patterns tools provide statistics that quantify broad spatial patterns. These tools answer questions such as, "Are the features in the dataset, or the values associated with the features in the dataset, spatially clustered?" and "Is the clustering becoming more or less intense over time?". The following table lists the tools available and provides a brief description of each.
Tool 
Description 

Calculates a nearest neighbor index based on the average distance from each feature to its nearest neighboring feature. 

Measures the degree of clustering for either high values or low values using the GetisOrd General G statistic. 

Measures spatial autocorrelation for a series of distances and optionally creates a line graph of those distances and their corresponding zscores. Zscores reflect the intensity of spatial clustering, and statistically significant peak zscores indicate distances where spatial processes promoting clustering are most pronounced. These peak distances are often appropriate values to use for tools with a Distance Band or Distance Radius parameter.  
Measures spatial autocorrelation based on feature locations and attribute values using the Global Moran's I statistic. 

MultiDistance Spatial Cluster Analysis (Ripley's kfunction) 
Determines whether features, or the values associated with features, exhibit statistically significant clustering or dispersion over a range of distances. 