I think, in machine learning it makes more sense to speak about model variance than about noise within variables. We see the effects of noise in the training data as a factor influencing model performance, but in most cases it is not necessary to quantify the noise. I will explain why:
Noise is an expression from physics and describes (in general) a perturbation with a broad and unspecific spectrum. There are many different forms of noise based on its origin or its statistical behavior. For example, white noise has a constant power density spectrum over a defined spectral area. In that sense you can simply measure noise, if you know what you are looking for. To quantify noise you need to know what you are comparing it to. A way to find your signal in the data.
An example: Let's say you have measurements of people heights in
[m] and weights in
[kg]. You want to build a model to predicts the
[kg] from the
[m] data. If you only have the raw measurements you don't know what is the signal and what is noise. You can still build a model. It will be influenced by the noise in both the
[m] and the
[kg] set. With this model you can calculate the training and test error to quantify the variance of you model. Though, that does not tell you much about the noise in each of the data sets. It tells you though how well the variables explain each other.
How to quantify the noise in this example? If you repeat the experiment 100 times within a time frame where you can assume that the peoples heights and weights did not change significantly (maybe within a day). You find that the measurements may still be different from each other (even for a single person) due to various factors such as peoples postures, inexact read-offs etc. Now, you plot a histogram and find that the variation follows a normal distribution. We assume that the true height is close to the mean of all measurements. Now, you know what is your signal e.g.
1.81 for a person that is
1.81m tall. Finally, you calculate the variance or standard error. Now, you quantified the noise
height = 1.81 +/- 0.01 normalized by
[m]. Reflecting an error of
1cm on average. Yes, a higher deviation means higher noise.
Effectively, you build a noise filter using the mean values. In some cases model performance increases significantly using such a filter. Though with enough training data, the model may learn the about the noise factors implicitly. But that is a different discussion.