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I have calculated histogram bins with the -hist option in gdalinfo.

Is there an option to calculate skewness and kurtosis, using this or any other GDAL tool?

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GDAL does not have the functionality to compute histogram moments such as skewness or kurtosis. Try looking at GIS software (e.g. QGIS, ArcGIS) or if you are comfortable with Python you can read your raster as an array and then compute a variety of statistics (e.g. https://stackoverflow.com/questions/26377272/calculating-kurtosis-from-a-numpy-array)

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