Ffmpeg Gaussian Filter (gblur) kernel size

Actually, the kernel size is a discrete kernel truncated from the continuous Normal Distribution, which is for satisfying the requirements of using convolution.

It's correct that only the sigma is requested to input. I supposed you want to know this one: About 68% of values drawn from a normal distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations, and about 99.7% are within three standard deviations. This fact is known as the 68-95-99.7 (empirical) rule or the 3-sigma rule.

So that's why the kernel size is 3x3 when sigma is 0.5. It's simply 0.5 * 3 = 1.5, and for its symmetry, the final kernel size is 3 in the case of 1D Normal Distribution. For the 2D case, it's 3x3.