Oscilloscope V_rms vs V_avg for effective voltage across an LED?
Neither - use the peak voltage to estimate the current during that portion of the cycle then multiply by the duty cycle to get the average current.

For example if the LED is on for one third of the time the average current will be one third of the current you measure when it is on.

You do have a series resistor don't you? You can't estimate the current accurately through an LED by measuring the voltage across it. LEDs vary too much with temperature and between units to power from a voltage source.

You should have a series resistor and measure the voltage across that.

Alternativey you could use a current source modulated by the PWM signal.