Just had another look at the help for "How Focal Statistics Works":
Note that it says "if the input is integer ...., if the input is float...."
Statistics type
The available Focal Statistics statistics are majority, maximum, mean, median, minimum, minority, range, standard deviation, and sum. The default statistics type is mean.
Majority
If the input raster is integer, the values on the output raster will be integer; if the values on the input are floating point, the values on the output will be floating point.
When there is more than one majority value within a neighborhood, the processing cell location will receive NoData on the output.
Maximum
If the input raster is integer, the values on the output raster will be integer; if the values on the input are floating point, the values on the output will be floating point.
Mean
The output raster will always be floating point.
The mean statistic can be used with the weight neighborhood type.
Median
If the input raster is integer, the values on the output raster will be integer; if the values on the input are floating point, the values on the output will be floating point.
When the number of valid cell values in the neighborhood is odd, the median value is calculated by ranking the values and selecting the middle value. If the number of values in a neighborhood is even, the values will be ranked and the middle two values will be averaged.
Minimum
If the input raster is integer, the values on the output raster will be integer; if the values on the input are floating point, the values on the output will be floating point.
Minority
If the input raster is integer, the values on the output raster will be integer; if the values on the input are floating point, the values on the output will be floating point.
When there is more than one minority value within a neighborhood, the processing cell location will receive NoData on the output.
Range
If the input raster is integer, the values on the output raster will be integer; if the values on the input are floating point, the values on the output will be floating point.
The values for each cell location on the output grid are determined on a cell-by-cell basis by applying this simple formula: Focal Range = Focal Maximum �?? Focal Minimum
STD
The output raster will always be floating point.
The STD statistic can be used with the weight neighborhood type.
Sum
If the input raster is integer, the values on the output raster will be integer; if the values on the input are floating point, the values on the output will be floating point.
The STD statistic can be used with the weight neighborhood type.
Variety
The output raster will always be integer.