To ask this question I needed to simulate two circuits as an example. Imagine we want to measure the voltage across a device output. Call the differential voltage across its terminals Vd. And imagine we couple a usual scope to its terminals to observe the voltage. Call AGND is the scope's own ground. And call the usual triangle reference symbol as the earth ground for this example.
Now on the left side circuit, if the scope is earth grounded and if there is a common mode interference as Vcm the simulation shows the green plot as the difference between Scope and AGND nodes. But if I cut the earth connection of the scope as on the right circuit the simulation shows the blue plot as the difference between Scope1 and AGND1 nodes.
In the simulation above, cutting the earth is hiding the common mode interference information. Is that what happens in reality? Can this be a way to check whether the noise is common mode or differential in nature?