Mapping groundwater well samples for a particular contaminant might indicate how the toxin is spreading and, consequently, may be useful in deploying mitigation strategies.Mapping the distributional trend for a set of crimes might identify a relationship to particular physical features (a string of bars or restaurants, a particular boulevard, and so on).You can also specify the number of standard deviations to represent (1, 2, or 3). The values for the angle, tilt and roll of the ellipsoid are Euler angles which describe the orientation of the ellipsoid in 3D space. The attribute values for these output ellipsoids include three standard distances (long, short and height axes) information regarding the angle, tilt, and roll of the ellipsoid and the case field, if specified. You can also specify the number of standard deviations to represent (1, 2, or 3).įor three dimensional point data (your data is z enabled and contains 3D attribute information such as elevation), this tool creates a new feature class containing an ellipsoid multipatch centered on the mean center for all features (or for all cases when a value is specified for Case Field). The orientation represents the rotation of the long axis measured clockwise from noon. The attribute values for these output ellipse polygons include two standard distances (long and short axes) the orientation of the ellipse and the case field, if specified. Similarly, for the three dimensional case (x, y and z), the 61-99-100 percentage rule will result.įor two dimensional data, the Directional Distribution (Standard Deviational Ellipse) tool creates a new feature class containing an elliptical polygon centered on the mean center for all features (or for all cases when a value is specified for Case Field).
A more appropriate rule-of-thumb derived from the Rayleigh distribution suggests that a one standard deviational ellipse will cover approximately 63 percent of the features two standard deviations will contain approximately 98 percent of the features and three standard deviations will cover approximately 99.9 percent of the features in two dimensions (x and y). However, when working with higher dimensional spatial data (x, y and z variables), this breakdown of percentages is rarely observed. In a normal distribution, this would mean 68%, 95% and 99.7% of the data values will fall within one, two and three standard deviations respectively. When working with one dimensional data, the three sigma rule is the common rule-of-thumb conveying the percentage of data values that will fall within one, two and three standard deviations of the mean. Standard deviations help you understand the dispersion or spread of your data. Visit the Additional resources if you would like to learn more about eigenvalues and eigenvectors. These adjustment factors are provided in the table below. The variances are scaled by an adjustment factor in order to produce an ellipse or ellipsoid containing the desired percentage of the data points. These equations can be extended to solutions for three dimensional data. The standard deviations for the x- and y-axis are then: The sample covariate matrix is factored into a standard form which results in the matrix being represented by its eigenvalues and eigenvectors. Where x and y are the coordinates for feature i, represent the Mean Center for the features and n is equal to the total number of features. The Standard Deviational Ellipse is given as: The latter is termed a weighted standard deviational ellipse. You can calculate the standard deviational ellipse using either the locations of the features or the locations influenced by an attribute value associated with the features. While you can get a sense of the orientation by drawing the features on a map, calculating the standard deviational ellipse makes the trend clear. The ellipse or ellipsoid allows you to see if the distribution of features is elongated and hence has a particular orientation. In 3D, the standard deviation of the z-coordinates from the mean center are also calculated and the result is referred to as a standard deviational ellipsoid. The ellipse is referred to as the standard deviational ellipse, since the method calculates the standard deviation of the x-coordinates and y-coordinates from the mean center to define the axes of the ellipse. These measures define the axes of an ellipse (or ellipsoid) encompassing the distribution of features. A common way of measuring the trend for a set of points or areas is to calculate the standard distance separately in the x-, y- and z-directions.