Interpretation of non-significant results as "trends" Recently, two different co-workers have used a kind of argument about differences between conditions that seems incorrect to me. Both of these co-workers use statistics, but they are not statisticians. I am a novice in statistics.
In both cases, I argued that, because there was no significant difference between two conditions in an experiment, it was incorrect to make a general claim about these groups with regard to the manipulation. Note that "making a general claim" means something like writing: "Group A used X more often than group B".
My co-workers retorted with: "even though there is no significant difference, the trend is still there" and "even though there is no significant difference, there is still a difference". To me, both of these sound like an equivocation, i.e., they changed the meaning of "difference" from: "a difference that is likely to be the result of something other than chance" (i.e., statistical significance), to "any non-zero difference in measurement between groups".
Was the response of my co-workers correct? I did not take it up with them because they outrank me.
 A: That's a tough question!
First things first, any threshold you may choose to determine statistical significance is arbitrary. The fact that most people use a $5\%$ $p$-value does not make it more correct than any other. So, in some sense, you should think of statistical significance as a "spectrum" rather than a black-or-white subject.
Let's assume we have a null hypothesis $H_0$ (for example, groups $A$ and $B$ show the same mean for variable $X$, or the population mean for variable $Y$ is below 5). You can think of the null hypothesis as the "no trend" hypothesis. We gather some data to check whether we can disprove $H_0$ (the null hypothesis is never "proved true"). With our sample, we make some statistics and eventually get a $p$-value. Put shortly, the $p$-value is the probability that pure chance would produce results equally (or more) extreme than those we got,  assuming of course $H_0$ to be true (i.e., no trend).
If we get a "low" $p$-value, we say that chance rarely produces results as those, therefore we reject $H_0$ (there's statistically significant evidence that $H_0$ could be false). If we get a "high" $p$-value, then the results are more likely to be a result of luck, rather than actual trend. We don't say $H_0$ is true, but rather, that further studying should take place in order to reject it.
WARNING: A $p$-value of $23\%$ does not mean that there is a $23\%$ chance of there not being any trend, but rather, that chance generates results as those $23\%$ of the time, which sounds similar, but is a completely different thing. For example, if I claim something ridiculous, like "I can predict results of rolling dice an hour before they take place," we make an experiment to check the null hypothesis $H_0:=$"I cannot do such thing" and get a $0.5\%$ $p-$value, you would still have good reason not to believe me, despite the statistical significance.
So, with these ideas in mind, let's go back to your main question. Let's say we want to check if increasing the dose of drug $X$ has an effect on the likelihood of patients that survive a certain disease. We perform an experiment, fit a logistic regression model (taking into account many other variables) and check for significance on the coefficient associated with the "dose" variable (calling that coefficient $\beta$, we'd test a null hypothesis $H_0:$ $\beta=0$ or maybe, $\beta \leq 0$. In English, "the drug has no effect" or "the drug has either no or negative effect."
The results of the experiment throw a positive beta, but the test $\beta=0$ stays at 0.79. Can we say there is a trend? Well, that would really diminish the meaning of "trend". If we accept that kind of thing, basically half of all experiments we make would show "trends," even when testing for the most ridiculous things. 
So, in conclusion, I think it is dishonest to claim that our drug makes any difference. What we should say, instead, is that our drug should not be put into production unless further testing is made. Indeed, my say would be that we should still be careful about the claims we make even when statistical significance is reached. Would you take that drug if chance had a $4\%$ of generating those results? This is why research replication and peer-reviewing is critical.
I hope this too-wordy explanation helps you sort your ideas. The summary is that you are absolutely right! We shouldn't fill our reports, whether it's for research, business, or whatever, with wild claims supported by little evidence. If you really think there is a trend, but you didn't reach statistical significance, then repeat the experiment with more data!
A: This is a great question; the answer depends a lot on context.
In general I would say you are right: making an unqualified general claim like "group A used X more often than group B" is misleading. It would be better to say something like

in our experiment group A used X more often than group B, but we're very uncertain how this will play out in the general population

or

although group A used X 13% more often than group B in our experiment, our estimate of the difference in the general population is not clear: the plausible values range from A using X 5% less often than group B to A using X 21% more often than group B

or

group A used X 13% more often than group B, but the difference was not statistically significant (95% CI -5% to 21%; p=0.75)

On the other hand: your co-workers are right that in this particular experiment, group A used X more often than group B. However, people rarely care about the participants in a particular experiment; they want to know how your results will generalize to a larger population, and in this case the general answer is that you can't say with confidence whether a randomly selected group A will use X more or less often than a randomly selected group B.
If you needed to make a choice today about whether to use treatment A or treatment B to increase the usage of X, in the absence of any other information or differences in costs etc., then choosing A would be your best bet. But if you wanted be comfortable that you were probably making the right choice, you would need more information.
Note that you should not say "there is no difference between group A and group B in their usage of X", or "group A and group B use X the same amount". This is true neither of the participants in your experiment (where A used X 13% more) or in the general population; in most real-world contexts, you know that there must really be some effect (no matter how slight) of A vs. B; you just don't know which direction it goes.
A: Significant effect just means that you measured an unlikely anomaly (unlikely if the null hypothesis, absence of effect, would be true). And as a consequence it must be doubted with high probability (although this probability is not equal to the p-value and also depends on prior believes). 
Depending on the quality of the experiment you could measure the same effect size, but it might not be an anomaly (not an unlikely result if the null hypothesis would be true).
When you observe an effect but it is not significant then indeed it (the effect) can still be there, but it is only not significant (the measurements do not indicate that the null hypothesis should be doubted/rejected with high probability). It means that you should improve your experiment, gather more data, to be more sure.
So instead of the dichotomy effect versus no-effect you should go for the following four categories:

Image from https://en.wikipedia.org/wiki/Equivalence_test explaining the two one sided t-tests procedure (TOST)
You seem to be in category D, the test is inconclusive. Your coworkers might be wrong to say that there is an effect. However, it is equally wrong to say that there is no effect!
A: It sounds like they're arguing p-value vs. the definition of "Trend".
If you plot the data out on a run chart, you may see a trend... a run of plot points that show a trend going up or down over time.
But, when you do the statistics on it.. the p-value suggests it's not significant.
For the p-value to show little significance, but for them to see a trend / run in the series of data ... that would have to be a very slight trend.
So, if that was the case, I would fall back on the p-value.. IE: ok, yes, there's a trend / run in the data.. but it's so slight and insignificant that the statistics suggest it's not worth pursuing further analysis of.
An insignificant trend is something that may be attributable to some kind of bias in the research.. maybe something very minor.. something that may just be a one time occurence in the experiment that happened to create a slight trend.
If I was the manager of the group, I would tell them to stop wasting time and money digging into insignificant trends, and to look for more significant ones.
A: It sounds like in this case they have little justification for their claim and are just abusing statistics to reach the conclusion they already had. But there are times when it's ok to not be so strict with p-val cutoffs. This (how to use statistical significance and pval cutoffs) is a debate that has been raging since Fisher, Neyman, and Pearson first laid the foundations of statistical testing.
Let's say you are building a model and you are deciding what variables in include.  You gather a little bit of data to do some preliminary investigation into potential variables.  Now there's this one variable that the business team really is interested in, but your preliminary investigation shows that the variable is not statistically significant.  However, the 'direction' of the variable comports to what the business team expected, and although it didn't meet the threshold for significance, it was close. Perhaps it was suspected to have positive correlation to the outcome and you got a beta coefficient that was positive but the pval was just a bit above the .05 cutoff.
In that case, you might go ahead and include it.  It's sort of an informal bayesian statistics -- there was a strong prior belief that it is a useful variable and the initial investigation into it showed some evidence in that direction (but not statistically significant evidence!) so you give it the benefit of the doubt and keep it in the model.  Perhaps with more data it will be more evident what relationship it has with the outcome of interest.
Another example might be where you are building a new model and you look at the variables that were used in the previous model -- you might continue to include a marginal variable (one that is on the cusp of significance) to maintain some continuity from model to model.
Basically, depending on what you are doing there are reasons to be more and less strict about these sorts of things. 
On the other hand, it's also important to keep in mind that statistical significance does not have to imply a practical significance!  Remember that at the heart of all this is sample size.  Collect enough data and the standard error of the estimate will shrink to 0. This will make any sort of difference, no matter how small, 'statistically significant' even if that difference might not amount to anything in the real world.  For example, suppose the probability of a particular coin landing on heads was .500000000000001. This means that theoretically you could design an experiment which concludes that the coin is not fair, but for all intents and purposes the coin could be treated as a fair coin.
