# Effect of units on shape of normal distribution

TL;DR: Is the shape of a bell curve / normal distribution idependent of the units chosen to represent a certain variable?

More detail: I was playing around with some data from this paper in excel:

Transmission (R) Temperature (C) Humidity (%)
mean 1.831 6.017 78.456
stdev 0.522 6.549 9.898

When I plot a normal distribution for each of these variables, I see that the greater the stdev, the flatter the curve:

In this case, humidity has the greatest stdev, so it has the flattest curve. This follows what I have learned in my (limited) stats education, but intuitively it seems wrong to me. Sure, humidity has a greater stdev than temperature, but the units are completely different, so it seems like an unfair comparison. Furthermore, if I simply change the units of humidity from a percent to a decimal it now has the smallest stdev (mean=0.78456, stdev=0.09898). When I try this in excel this actually changes the shape of the curve, with humidity now being the steepest rather than the flattest. It seems wrong to me that changing the units should actually change the shape of the distribution.

• The main unfair comparison is that all normal density curves "enclose" a total area of $1.$ So there is something wrong with the scale of plotting your curves. Commented Apr 26, 2021 at 14:45

hdr="Densities of NORM(0,1) [red], NORM(0,3) [green], and NORM(0,5)"

Recall that the probability density function tells us about "probability per foot" and that it has the property that it integrates to one. What follows, the $$y$$-axis units, the probability densities will depend on the units on $$x$$-axis. Imagine a simplified example, the uniform density, if you plot it, it's a rectangle, if you increase the units on $$x$$-axis, to preserve the area equal to one, the $$y$$-axis units would need to decrease. The same happens with the Gaussian curve, making it wider will make it flatter. Nothing else changes about the Gaussian except being wider and flatter vs thinner and spikier.