Summary statistics are calculated by the Aggregate Points, Summarize Within, Summarize Nearby, Join Features, and Dissolve Boundaries tools.
Equations
Mean and standard deviation are calculated using weighted mean and weighted standard deviation for line and polygon features. None of the statistics for point features are weighted. The weight is the length or area of the feature that falls within the boundary.
The following table shows the equations used to calculate standard deviation, weighted mean, and weighted standard deviation:
Statistic  Equation  Variables  Features 

Standard Deviation  where:
 Points  
Weighted Mean  where:
 Lines and polygons  
Weighted Standard Deviation  where:
 Lines and polygons 
Note:
Null values are excluded from all statistical calculations. For example, the mean of 10, 5, and a null value is:
(10+5)/2=7.5
Points
Point layers are summarized using only the point features within the boundary areas.
A reallife scenario in which points could be summarized is in determining the total number of students in each school district. Each point represents a school. The Type field gives the type of school (primary school, middle school, or secondary school) and a population field gives the number of students enrolled at each school.
The figure below shows a hypothetical point and boundary layer, and the table summarizes the attributes for the point layer.
ObjectID  District  Type  Population 

1  A  Primary school  280 
2  A  Primary school  408 
3  A  Primary school  356 
4  A  Middle school  361 
5  A  Middle school  450 
6  A  Secondary school  713 
7  B  Primary school  370 
8  B  Primary school  422 
9  B  Primary school  495 
10  B  Middle school  607 
11  B  Middle school  574 
12  B  Secondary school  932 
The calculations and results for District A are given in the table below. From the results, you can see that District A has 2,568 students. When running a tool, the results would also be given for District B.
Statistic  Result District A 

Sum 

Minimum  Minimum of:

Maximum  Maximum of:

Mean 

Standard Deviation 

Lines
Line layers are summarized using only the proportions of the line features that are within the boundary areas.
Tip:
When summarizing lines, use fields with counts or amounts so proportional calculations make logical sense in your analysis. For example, use population rather than population density.
A reallife scenario in which you can use this analysis is determining the total volume of water in rivers within a specified boundary. Each line represents a river that is partially located inside the boundary.
The figure below shows a hypothetical line and boundary layer, and the table summarizes the attributes for the line layer.
River  Length (miles)  Volume (gallons) 

Yellow  3  6,000 
Blue  8  10,000 
The calculations for volume are given in the table below. From the results, you can see that the total volume is 9,000 gallons.
Note:
The calculations use the proportions of the lines within the boundary area. For example, the yellow line has a total volume of 6,000 gallons with two of its three total miles within the boundary. Therefore, the calculations are preformed using 4,000 gallons as the volume for the yellow line:
6000*(2/3)=4000
Statistic  Result 

Sum 

Minimum  Minimum of:

Maximum  Maximum of:

Mean 

Standard Deviation 

Polygons
Polygon layers are summarized using only the proportions of the polygon features that are within the boundary areas.
Tip:
When summarizing polygons, use fields with counts or amounts so proportional calculations make logical sense in your analysis. For example, use population rather than population density.
A reallife scenario in which you can use this analysis is determining the population in a city neighborhood. The blue outline represents the boundary of the neighborhood and the smaller polygons represent census blocks.
The figure below shows a hypothetical polygon and boundary layer, and the table summarizes the attributes for the polygon layer.
Census block  Area (miles²)  Population 

Yellow  6  3,200 
Green  6  4,700 
Pink  2.5  1,000 
Blue  8  4,500 
Orange  4  3,600 
The calculations for population are given in the table below. From the results, you can see that there are 10,841 people in the neighborhood and an average (mean) of approximately 2,666 people per census block.
Note:
The calculations use the proportions of the polygons within the boundary area. For example, the yellow polygon has a total population of 3,200 with four of its six total square miles within the boundary. Therefore, the calculations are preformed using 2,133 as the population for the yellow polygon:
3200*(4/6)=2133
Statistic  Result 

Sum 

Minimum  Minimum of:

Maximum  Maximum of:

Mean 

Standard Deviation 

Related topics
Use the following topics to learn more about summary statistics within a specific tool: