#### Table of Contents

### Introduction

The *Standard Distance, *also know as the *Standard Distance Deviation*, is the average distance all features vary from the *Mean Center *and measures the compactness of a distribution. The *Standard Distance* is a value representing the distance in units from the *Mean Center* and is usually plotted on a map as a circle for a visual indication of dispersion, the *Standard Distance* is the radius.

The *Standard Distance* works best in the absence of a strong directional trend. According to Andy Mitchell, if a directional trend is present you are better off using the *Standard Deviational Ellipse*.

You can use the *Standard Distance* to compare territories between species, which has the wider/broader territory, or to compare changes over time such as the dispersion of burglaries for each calendar month.

In a* Normal Distribution* you would expect around 68% of all points to fall within the *Standard Distance*.

Sources:

The Esri Guide to GIS Analysis, Volume 2: Spatial Measurements and Statistics.

An Introduction to Statistical Problem Solving in Geography

This course is designed to instill the basics of Python Programming by incrementally increasing your knowledge session-upon-session. In each section you will be given new material for a workbook to fill out and by the end of this course you will have your very own Python reference handbook. So how does this course have a GIS focus? Simple, most elements of the course have GIS and geospatial data in mind. Instead of using non-descript variables and values, we will use terms such as population, city, x_coord, y_coord, and so on. This will aid participants with pinpointing how they can relate geospatial data to Python.

### The Formula

The mean x-coordinate is subtracted from the x-coordinate value for each point and the difference is squared. The sum of all the squared values for x minus the x-mean is divided by the number of points. This is also performed for y-coordinates. The resulting values for x and y are summed and then we take the square root of this value to return the value to original distance units.

First we get the mean X and Y…

…and then the *Standard Distance.*

For *Point* features the X and Y coordinates of each feature is used, for *Polygons* the centroid of each feature represents the X and Y coordinate to use, and for *Linear* features the mid-point of each line is used for the X and Y coordinate.

### Using GeoPandas to Calculate the Standard Distance

The code below represents previous endeavours from when we used GeoPandas and Shapely to find the mean center for a dataset. We require the mean center for the standard distance calculations. In our example we will use a Shapefile, but you can use any input and output filetypes that you have available with your GeoPandas setup.

The code is heavily commented for ease of understanding the workflow. For a Point, we now we need get the mean value for all x values and y values. For a Polyline, we need to first get the midpoint of each line, and then get the mean value for all x values and y values for the midpoints. For a Polygon, we need to the the centroid value for each polygon, and then get the mean value for all x values and y values for the centroids.

` ````
```import geopandas as gpd
from shapely.geometry import Point
import math
## input shapefile path
in_shp = r"path\to\input\shapefile\input.shp"
## the output shapefile path for the standard distance polygon
out_shp = r"path\to\output\shapefile\output.shp"
## read in the shapefile to a GeoDataFrame
gdf = gpd.read_file(in_shp)
## get the geometry type from the first record
geom_type = gdf.geom_type[0]
## get the EPSG code
crs = gdf.crs
## required for the formula
num_features = len(gdf)
## for Point geometry
if geom_type == "Point":
## get all x and y values
gdf["x"] = gdf.geometry.x
gdf["y"] = gdf.geometry.y
## for LineString geometry
elif geom_type == "LineString":
## get all x and y values for the midpoints
gdf["midpoint"] = gdf.geometry.interpolate(0.5, normalized=True)
gdf["x"] = gdf["midpoint"].x
gdf["y"] = gdf["midpoint"].y
## for Polygon geometry
elif geom_type == "Polygon":
## get all x and y values for the centroids
gdf["centroid"] = gdf.geometry.centroid
gdf["x"] = gdf["centroid"].x
gdf["y"] = gdf["centroid"].y
## get the mean of the x and y values
mean_x = gdf['x'].mean()
mean_y = gdf['y'].mean()
## calculate the sum of the squared differences for x and y
sum_of_sq_diff_x = sum([math.pow(x - mean_x, 2) for x in gdf["x"]])
sum_of_sq_diff_y = sum([math.pow(y - mean_y, 2) for y in gdf["y"]])
## get the sum of the results
sum_of_results = (sum_of_sq_diff_x/num_features) + (sum_of_sq_diff_y/num_features)
## get the square root of the sum of the results to find the standard distance
standard_distance = math.sqrt(sum_of_results)
## create a point geometry representing the mean center
mean_center = Point(mean_x, mean_y)
## buffer the mean center by the standard distance
standard_distance_geom = mean_center.buffer(standard_distance)
## create a GeoDataFrame with the polygon geometry
gdf_standard_distance = gpd.GeoDataFrame(geometry=[standard_distance_geom], crs=crs)
## add fields for the CenterX, CenterY, and StdDist
gdf_standard_distance["CenterX"] = mean_x
gdf_standard_distance["CenterY"] = mean_y
gdf_standard_distance["StdDist"] = standard_distance
## write the standard distance polygon to the output shapefile
gdf_standard_distance.to_file(out_shp, driver="ESRI Shapefile")

### Standard Distance in Action

I downloaded crime data from DATA.POLICE.CO.UK for all crimes in the West Midlands which were a series of CSV files. I further processed the data for burglaries only and you can downland the shapefile for the example below here.

Running the script produces a Shapefile that contains the Standard Distance polygon for all burglaries in 2023

Below is a comparison between our GeoPandas tool and the Standard Distance tool output from ArcGIS Pro. Spot on!

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### Also in this series

- Mean Center
- Central Feature
- Median Center
- Weighted Mean Center
- Weighted Central Feature
- Standard Distance by Case