How can I (or should I) test that observation A tends to be greater than observation B for each subject? I measured 2 sediment parameters in multiple lakes.  The organic matter content at the sediment/water interface surfOM and the mean organic matter content of the whole core meanOM.
surfOM <- c(34.4600, 42.1700, 48.6700, 18.4711, 19.5300, 18.4238, 20.5601, 28.2800, 19.1739)
meanOM <- c(31.23571, 37.57117, 43.47832, 18.47110, 17.18266, 17.42377, 19.96015, 25.09943, 19.37392)

The two measures are correlated > cor(surfOM, meanOM) [1] 0.9967266 and surfOM is generally greater than meanOM:
 > surfOM - meanOM
[1]  3.22429  4.59883  5.19168  0.00000  2.34734  1.00003  0.59995  3.18057
[9] -0.20002

When reporting the results I say:

"The surface percent organic matter and the mean percent organic matter of the same core were highly correlated with slightly greater surface percent organic matter in most lakes."

I did not initially feel like the difference needed a statistical test because it just is what it is but both reviewers asked for a test showing that surfOM tends to be greater than meanOM in a lake.  
My initial thought was to calculate the confidence interval of surfOM - meanOM and show that it does not contain 0 but I am not sure if this is the most appropriate approach.
Questions:


*

*Does the difference between the two variables need a statistical test?

*What is an appropriate way of testing this difference?


(Note: this is just a subset of the data.  The complete dataset has 23 lakes)
 A: You can do a paired t-test. 
In R, t.test(surfOM,meanOM,paired=TRUE)
That will give you a p-value and a confidence interval for the mean of the differences. These only really makes sense if the lakes are a sample of a larger population of lakes, not if you have data on all the lakes in your population (e.g. all the lakes in some geographical area, or all the lakes of a certain type of interest).
A: When I hear the statement "A tends to be greater than B" it sounds like A is greater than B a bunch of the time, say, more than 50% of the time.  This can happen when $\mu_{A}$ is greater than $\mu_{B}$, but it can also happen that $\mu_{A}$ is greater than $\mu_{B}$... yet $P(A > B) < 0.50$. 
(For a concrete example of what I'm getting at, let B be identically 1/2, and let A put probability $1 - p$ on zero with its remaining probability $p$ on some integer $n \geq 1$.  We can make $P(A > B)$ as small as we like by letting $p \to 0$, yet we can make $\mu_{A}$ as big as we like by letting $n \to \infty$.) 
On top of all this, it looks like your "B" is itself a "mean", which complicates the language.
So, my first thought is to try to figure out whether you'd be more interested in $\mu_{A} > \mu_{B}$ or if you'd rather be more interested in $P(A > B) > 0.50$.  If you're interested in the means then the t-test should be just fine.
If you're more interested in $P(X > Y)$, you might consider the Sign Test.  It's a nonparametric test, it's light on assumptions, and you don't need to be so concerned about outliers. If you could additionally say that the distribution of the differences of organic matter was symmetric, you could do one better and go for the Wilcoxon Signed Rank test. 
Anyway, suppose you can say nothing more about the distributions, and suppose they are appropriate for the Sign Test.  The data given above had 8 out of 9 pairs with A greater than B. For these data, in R, you would do
binom.test(8, 9, alt = "greater")

The output would give a p-value and a 95% one-sided Clopper-Pearson confidence interval (which generally is conservative).
