Summary statistics

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: 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

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 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.

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	280+408+356+361+450+713 =2568
Minimum	Minimum of: [280, 408, 356, 361, 450, 713] =280
Maximum	Maximum of: [280, 408, 356, 361, 450, 713] =713
Mean	2568/6 =428
Standard Deviation	√((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.

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 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.

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	4000+5000=9000
Minimum	Minimum of: [4000, 5000]=4000
Maximum	Maximum of: [4000, 5000]=5000
Mean	((2*4000)+(3*5000))/(2+3) =(8000+15000)/5 =4600
Standard Deviation	√(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.

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 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.

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	2133+3133+400+3375+1800=10841
Minimum	Minimum of: [2133, 3133, 400, 3375, 1800]=400
Maximum	Maximum of: [2133, 3133, 400, 3375, 1800]=3375
Mean	((4*2133)+(4*3133)+((1*400)+(6*3375)+(2*1800))/(4+4+1+6+2) =2665.53
Standard Deviation	√(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