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Moran’s I is a statistic used to measure spatial autocorrelation, or the degree to which the values of a variable are similar to those of its neighboring locations. It is used to quantify the degree to which values in a dataset tend to cluster together or disperse away from each other. It is calculated as the ratio of the variance of a spatial autocorrelation statistic, divided by the variance of the variable.
Moran’s I is a way to measure spacial autocorrelation.
In simple terms, it’s a way to quantify how closely values are clustered together in a 2-D space. It’s often used in geography and to measure how closely clustered different features are on a map like household income, level of education, etc.
Moran’s I: The Formula
The formula to calculate Moran’s I is:
I = (N/W)*ΣΣwij(xi–x)(xj–x)/Σ(xi–x)2
where:
- N: The number of spacial units indexed by i and j
- W: The sum of all wij
- x: The variable of interest (household income, years of schooling, etc.)
- x: The mean of x
- wij: A matrix of spacial weights
You’ll likely never have to calculate this measure by hand since most statistical software can calculate it for you, but it’s useful to know the formula being used under the hood.
The value for Moran’s I can range from -1 to 1 where:
- -1: The variable of interest is perfectly dispersed
- 0: The variable of interest is randomly dispersed
- 1: The variable of interest is perfectly clustered together
Along with computing Moran’s I, most statistical software will compute a corresponding p-value that can be used to determine whether or not the data is randomly dispersed or not.
Moran’s Test uses the following null and alternative hypotheses:
Null Hypothesis (H0): The data is randomly dispersed.
Alternative Hypothesis (HA): The data is not randomly dispersed, i.e. it is clustered in noticeable patterns.
If the p-value that corresponds to Moran’s I is less than a certain significance level (i.e. α = .05), then we can reject the null hypothesis and conclude that the data is spatially clustered together in such a way that it is unlikely to have occurred by chance alone.
Moran’s I: A Few Examples
The following examples represent fake maps with different values for Moran’s I.
Moran’s I = 0: Average Household income is randomly dispersed (i.e. random clusters in random areas).
Moran’s I = -1: Average Household income is perfectly dispersed.
Moran’s I = 1: Average Household income is perfectly clustered.
Refer to for a real-world example of computing Moran’s I in the statistical software R.