I know that when you have data with a view outliers it is advised to use Median over Mean. What I am struggling with now is how to determine when you have this situation. Is there any concrete approach to know?
For example can the value of the standard deviation be used to indicate when to use the Median?
Can the value of the Range indicate this?
What numeric relationship can be used to know when it is better to use the Median over the Mean?
I wouldn't use the standard deviation as a measure of this. There are instances when the mean and median are the same regardless of standard deviation (see the normal distribution as an example of this).
Skewness may be a better measure for this. I would recommend reading the wikipedia page on skewness.
I wouldn't say there is a straight up quantitative (numeric) indicator for when to use median over mean; rather I think it's a function of what you're trying to do. If all you want to do is describe the data, then you should visualize it and determine if there is heavy skew or extreme outliers (both of which affect the mean but not so much the median). If there is, use median and measures of position to describe the data. For instance, US household income (which is highly right-skewed and contains extreme outliers like professional athletes and Jeff Bezos) is often described using median for this reason.
If instead you want to do tests on the data, you may want to use both approaches, check your assumptions and compare results. Parametric tests (where the distribution is assumed to be normal or can be made normal under the Central Limit Theorem) are often very robust, but there are instances when non-parametric tests (which make no assumptions about the underlying distribution) would be preferred. There's no black-and-white answer here, but rather relies on your justification for using the approach you did.
