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: N = Number of observations xi = Observations x̄ = Mean	Points
Weighted Mean		where: N = Number of observations xi = Observations wi = Weights	Lines and polygons
Weighted Standard Deviation		where: N = Number of observations xi = Observations wi = Weights x̄w = Weighted mean N' = Number of non-zero weights	Lines and polygons

(10+5)/2=7.5

Points

Point layers are summarized using only the point features within the boundary areas.

A real-life 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.

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.

280+408+356+361+450+713
=2568

[280, 408, 356, 361, 450, 713]
=280

[280, 408, 356, 361, 450, 713]
=713

2568/6
=428

√((280-428)²+(408-428)²+(356-428)²+(361-428)²+(450-428)²+(713-428)²)/(6-1)
=150.79

Lines

Line layers are summarized using only the proportions of the line features that are within the boundary areas.

A real-life 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.

The calculations for volume are given in the table below. From the results, you can see that the total volume is 9,000 gallons.

6000*(2/3)=4000

4000+5000=9000

[4000, 5000]=4000

[4000, 5000]=5000

((2*4000)+(3*5000))/(2+3)
=(8000+15000)/5
=4600

√(2(4000-4600)²+3(5000-4600)²)/((2-1)/2(2+3))
=692.8

Polygons

Polygon layers are summarized using only the proportions of the polygon features that are within the boundary areas.

A real-life 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.

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.

3200*(4/6)=2133

2133+3133+400+3375+1800=10841

[2133, 3133, 400, 3375, 1800]=400

[2133, 3133, 400, 3375, 1800]=3375

((4*2133)+(4*3133)+((1*400)+(6*3375)+(2*1800))/(4+4+1+6+2)
=2665.53

√(4(2133-2665.53)²+4(3133-2665.53)²+1(400-2665.53)²+6(3375-2665.53)²+2(1800-2665.53)²)/((5-1)/5(4+4+1+6+2))
=925.91

Related topics

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

• Aggregate Points
• Summarize Within
• Summarize Nearby
• Join Features
• Dissolve Boundaries