Correlations between continuous and categorical (nominal) variables I would like to find the correlation between a continuous (dependent variable) and a categorical (nominal: gender, independent variable) variable. Continuous data is not normally distributed. Before, I had computed it using the Spearman's $\rho$. However, I have been told that it is not right.
While searching on the internet, I found that the boxplot can provide an idea about how much they are associated; however, I was looking for a quantified value such as Pearson's product moment coefficient or Spearman's $\rho$. Can you please help me on how to do this? Or, inform on which method would be appropriate?
Would Point Biserial Coefficient be the right option?
 A: The reviewer should have told you why the Spearman $\rho$ is not appropriate.  Here is one version of that:  Let the data be $(Z_i, I_i)$ where $Z$ is the measured variable and $I$ is the gender indicator, say it is 0 (man), 1 (woman). Then Spearman's $\rho$ is calculated based on the ranks of $Z, I$ respectively. Since there are only two possible values for the indicator $I$, there will be a lot of ties, so this formula is not appropriate. If you replace rank with mean rank,  then you will get only two different values, one for men, another for women.  Then $\rho$ will become basically some rescaled version of the mean ranks between the two groups. It would be simpler (more interpretable) to simply compare the means!   Another approach is the following.
Let $X_1, \dots, X_n$ be the observations of the continuous variable among men, $Y_1, \dots, Y_m$ same among women. Now, if the distribution of $X$ and of $Y$ are the same, then $P(X>Y)$ will be 0.5 (let's assume the distribution is purely absolutely continuous, so there are no ties). In the general case, define
$$
   \theta = P(X>Y)
$$
where $X$ is a random draw among men, $Y$ among women. Can we estimate $\theta$ from our sample? Form all pairs $(X_i, Y_j)$ (assume no ties) and count for how many we have  "man is larger" ($X_i > Y_j$)($M$) and for how many "woman is larger"  ($ X_i < Y_j$) ($W$).  Then one sample estimate of $\theta$ is
$$
  \frac{M}{M+W}
$$
That is one reasonable measure of correlation!  (If there are only a few ties, just ignore them).  But I am not sure what that is called,  if it has a name. 
This one may be close:       https://en.wikipedia.org/wiki/Goodman_and_Kruskal%27s_gamma
A: I'm having the same issue now. I didn't see anyone reference this just yet, but I'm researching the Point-Biserial Correlation which is built off the Pearson correlation coefficient. It is mean for a continuous variable and a dichotomous variable.
Quick read: 
https://statistics.laerd.com/spss-tutorials/point-biserial-correlation-using-spss-statistics.php
I use R, but I find SPSS has great documentation.
A: It would seem that the most appropriate comparison would be to compare the medians (as it is non-normal) and distribution between the binary categories. I would suggest the non-parametric Mann-Whitney test...
A: For the specified problem, measuring the Area Under the Curve of a Receiver Operator Characteristic curve might help.
I am not an expert in this so I try to keep it simple. Please comment on any error or wrong interpretation so I can change it.
$x$ is your continuous variable. $y$ is your categorical.
See how many True Positives and False Positives do you get if you choose a value of $x$ as being the threshold between positives and negatives (or male and female) and you compare this to the real labels.
For e.g. you choose 7, then above $x$=7 are all female (1) and below $x$=7 all male (0). Compare this to the real labels and get the number of true positives and false positives of your prediction.
Repeating the procedure explained above, from min($x$) to max($x$) you will generate the true positive and the false positive rates and then you can plot them like in the figure below and you can calculate the Area Under the Curve. 
The idea is that if there is no correlation between the variables, you will get the same ratio of true positives and true negatives for all values of $x$, nevertheless, if there is good correlation (and the same stands for anti-correlation) the ratio of true positives to true negatives will strongly vary as $x$ varies.
The above statement is calulcated with the Area Under the Curve.

Example of good correlation (right) and fair anti-correlation (left). 
A: I like to think of it in more practical terms. A simple use case for continuous vs. categorical comparison is when you want to analyze treatment vs. control in an experiment. If you show statistical significance between treatment and control that implies that the categorical value (Treatment vs. Control) does indeed affect the continuous variable. You can do this same thing with ANOVA metric when you have multiple treatment groups. I think this is the most practical way of evaluating whether your categorical variable in any way affects the distribution of the continuous value.
