The method in the attached paper my be helpful to you. If you have questions please ask. Best wishes, David

You may try robust method of regression. If you are familiar with R, there is a function rlm() in MASS package which can help you in this issue.

I would also recommend switching to R. Generally outliers are not what you need to worry about but influential data points (points which are both extreme in leverage and have large residuals). In R you will be able to get measures of influence such as Cook's distance and run robust regressions.

In many research areas this sort of ad hoc data pruning is frowned upon, so I think it best avoided by using a pre-specified algorithm or by using a more robust modelling approach.

You have already got useful answers, but I'll add a few more suggestions:

Cross validation statistics are very useful in detecting outliers, especially if there aren't too many of them. Most regression tools give e..g. Sudentized deleted residuals that are approximately t-distributed. For example, in your case these residuals (the 3rd column below) for the first few observations of your data are

CVpred CVresid CV.Stud.resid

1 14.95255 13.1150606 8.30604837

2 15.65248 9.5267553 5.71994754

3 22.56602 0.1774386 0.10153745

4 23.15908 0.1239113 0.07093672

showing clearly that the first 2 are outliers. There is also a package called outliers in R which contains all most commonly used outlier tests. I too recommend R (which is free) or some other good statistics software instead of using Excel. To carry out outlier tests, or cross validation, in Excel requires either rather tedious calculations, or use of some statistics add in, e.g. xlstat.

There is no unique indicator (AFAIK) that shows that outliers exist. But one can make use of Mahalanobis distance, Cook's distance, leverage, DFBeta and DFit  values case-wise to understand if that observation is a candidate outlier.

Edit: Since question poster wants a quick solution, for the specific two variable X&Y case as asked in the question, there is an alternative. You can compare Spearman rho to Pearson r. And if Spearman rho is greater then this can mean that there are outliers (Warning: It is not a necessary condition, it can indicate the existence of outliers when the relationship between X&Y is linear)

I'm going to disagree with previous posters. The correct answer is that you should not avoid plotting your data. There is no definition of outliers that works consistently, so there can be no automated statistic to identify and downright them. An outlier is a point that doesn't belong with the rest of the data, but the determination that a point "doesn't belong" is one that can't be made without knowing more about the data than your computer knows; it requires a human to examine and think about it. An influential point is one that changes your model if it is set aside--but how much "change" is enough to matter? That depends on your situation and is again a judgment call.  So my advice is plot your data, look at it, identify points that might deserve special attention or correction, and deal with them based on your understanding of your data. Don't abdicate responsibility to the computer.

And, BTW, be sure to plot the residuals from your analysis to check that there isn't curvature in the relationship. Outliers can hide that, and none of the automated methods mentioned above deal with it. You must first re-express your data to make the underlying relationship as straight as possible before you go hunting for outliers. Of course "straight enough" is yet another judgment call that you can't pawn off on the computer; you must look at the plots and think about the data and model.

I actually agree about plotting the data. However, I do think that using a subjective judgement to handle 'outliers' identified by whatever method (plotting or descriptive statistics) is dangerous.

However, I've re-read my answer and wish I hadn't written "pre-specified algorithm" as it isn't really what I meant. I think you need to identify a strategy for handling potential outliers in advance based on the context of the research. In exploratory data analysis I would be more relaxed but that should in turn lead to a sensible strategy for future data analysis. I've seen published work where the handling of outliers varies from study to study without real explanation in a a multi experiment paper and that seems extremely dodgy to me.

@Paul. Gee Paul, you didn't look at the paper I sent. I suggested residual plots and many of my references advocate graphing as well. BTW I also do a transformation before dealing with outliers. So what are you disagreeing with? Best wishes, David

David, I have no particular objection to your paper; I've used those methods. But it seemed clear to me that the questioner was likely to grab the M-estimator idea and never read far enough to get to the discussion of residuals. I wanted to re-orient the entire discussion to be about looking at and thinking about the data and not about algorithms to downweight possibly outlying points.

Paul, Thanks for your reply. I understand your position much better now. Wish you the best, David

Thank you all for your answers. I agree that plotting the data is perhaps the best approach for detecting outliers. However, I think that when you have hundreds of different data sets for analysis, manual detection certainly becomes cumbersome. In those cases, computational methods are highly valuable, at least for performing a first screening. Thanks for the suggestions on functions and packages in R. I will definitely try some of them. 

If you are using SAS there is a command that can be used in proc glm called influence that will flag extreme outliers for your examination. I don't know what the command would be using R or SPSS.