I think the unit of PDF of normal distribution should be unitless or dimensionless.

http://upload.wikimedia.org/math/d/6...9d037a126c.png
But, I found a sigma in the denominator of PDF of normal distribution. So, the sigma should be unitless.

But, the sigma is the standard deviation of the random variable. This means the sigma and the random variable are also unitless.

Is this correct?

If so, can we apply normal distribution to any random variable with unit, e.g. meter or kg?

If we can do this, the unit of PDF of normal distribution would be 1/meter or 1/kg due to the existence of sigma in the equation.

I think this is not a correct expression of unit of probability.

Could any one help explain?