Your sample is rather large (df=156, so n=158). The larger a sample, the tinyer are the differences that become "statistically significant".

The plot does not show the sample size. If you don't know the sample size, it's impossible to judge if a difference might be statistically significant. The very same plot with a smaller underlying sample size could come with a much larger p-value, and the very same plot with an even bigger sample size behind could come with an even smaller p-value.

You must not confuse the whiskers of the boxplot with standard errors or condifence intervals that are often used in barplots* or catepillar plots (points and error bars). There, the error bars usually indicate the statistical error (or precision) of the estimate (a mean, typically), and the lengths of these error-bars become smaller as the sample size increases. This corresponds to smaller p-values one gets from the comparisons of the means between groups.

It's also a common and persisting misconception that a "significant result" would indicate that there is a true difference whereas a "non-significant result" would indicate that there is no true difference. The p-value is NOT calculated to distinguish a possible true state of "there is a difference" from another possible true state of "there is no difference". Instead, p-values are calculated just to check if the observed data are already sufficient to interpret the observed direction of the difference (= the sign of the difference). In your case, the observed direction is: Cornuta > Ricordii, and the p=0.018 tells you that there is so much data behind this, that one can confidently believe that this is the case for the sampled population as well (the whole of the observed data is too incompatible or would be too implausible under any expectation that Cornuta < Ricordii). If the p-value was large, you would not conclude that there is no difference. You would only conclude that the data is not sufficient to interpret the observed direction, as the data would still be relatively compatible or plausible with the expectation that the difference is just opposite to the one observed [short: the data is not conclusive regarding the direction].

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Never ever use barplots. They are evil. Honestly.

Only bit to add to Jochen Wilhelm 's great response (though I'd put barplot plots as less evil than pie charts or treating the noun data as singular), is boxplots show medians (and some other bits of quantiles and outliers and far outliers depending on package) not means, so you can get those two bars (the medians) being exactly the same and the t-test, which examines means, being significant.

Another addition to Jochen Wilhelm 's very relevant remarks. Many years ago we carried out a layer trial over a long period with a lot of replicates. We found that very tiny differences in shell thickness between the different treatment groups became significant, due to the setup.

We then discriminated between significant and practically relevant differences!

The lesson for me was: always use your common sense and do not rely on statistics only.

Your plot is box plot (showing quartiles). This errorbar is not suitable for visual check of significance. It should be 95% CI with pooled SEs to reach your aim.

You can use "ggerrorplot" command from package "ggpubr".

To add to all the excellent replies, it is worth reiterating:

"Statistically significant"

and

"Biologically significant"

are not the same thing.

The former tells you "yes, by whatever arbitrary criteria you choose to set your thresholds at, this difference is real", while the latter tells you whether this difference actually means anything.

And a lot of times, it doesn't.

Most statistical tests are intended to identify whether consistent changes are present, regardless of whether they are of consequence or not. There is nothing in biology to prevent small, consistent, but completely irrelevant biological changes occurring. You might be able to infer interesting regulatory data from this, but it does not change the fact that the consequences remain irrelevant. I could take a pay cut of 20p per month, and if you analysed my monthly wages from year 1 to year 2, you would get a statistically significant drop. Does that mean I actually feel any consequence of being short £2.40 over the course of a year?

No. That's like, half a pint of lager at London prices.

Conversely, massive and dramatic (but highly variable) changes may fall beneath the threshold of significance, while also being quite obvious biologically relevant changes. An enzyme found in blood at levels of "5-10 units" in healthy patients and at levels of "1,000-1,000,000" in affected patients may actually be quite challenging to assess statistically, because in the "affected" group it varies by three orders of magnitude. That doesn't change the fact that it is demonstrably "not really there" in healthy patients, and "really, really definitely there" in affected patients.

Always, always, put your data into biological context. Here, for example, you could say "treatment X results in small but statistically significant reduction in total egg counts. These reductions are unlikely to be of biological consequence as presented, but they nevertheless indicate an influence on egg production at level Y or Z"

Honestly, a decent (if statistically poor) rule of thumb is to look at the data and make a straight up value judgement. If you look at it and think "yeah, that's definitely a drop", then it probably is, even if the stats don't support it due to high variability. If you look at it and think "nah, that's basically the same" then it probably is, even if stats tell you there's a significant (if incredibly small) drop.

A couple of comments on John Hildyard 's comment. At the end, his advice to look at the data and if they present something obvious is sometimes called the IOT test. I'll include my teaching notes at the bottom for this. A difficulty is if it is obvious on a plot but data patterns like it could often occur with H_0 being true, making a conclusive statement about the pattern is problematic.

Second, I am confused by stating that statistical significance tells you if an effect is "real". Maybe I am just missing something, but I get how it can provide information about the direction of an effect, and how things like testing band hypothesis (and other approaches) can provide information about non-point hypotheses, but at least in the fields I work in point nulls (or nils as people sometimes call them) are false. Are you referring to testing non-point hypotheses?

Here are my IOT notes.

\begin{quote}

Joe Berkson, a statistician at the Mayo Clinic, developed his own criterion, which he termed the IOT Test, or Inter Ocular Trauma Test, requiring a graph that hit you between the eyes. \\

\phantom{.} \hfill Sam Savage (\citeyear{Savage2009}, p.~325)

\end{quote}

And it has to hit you hard enough to produce trauma! Savage stresses the importance of plotting data, and that if the study is well done and the visualization appropriate, the hope is that anything of importance will be obvious. Savage's father, Leonard Savage, wrote an influential book in the history of statistics (\citeyear{Savage1972}, first edition published 1954) that compared different approaches to inference (but not the IOT test). The younger Savage presumably grew up with mathematical equations on scraps of paper throughout the house, but with a wise father explaining these were just technical details that can help you make wise decisions (Figure~\ref{fig:datapipeline}). Sam Savage's book focuses on looking beyond summary statistical measures. The IOT test is one of these. If you can design a study and if nature is forthcoming, a good plot should make the results obvious. Other ways of describing data can also pass the IOT test. An example is Jane Goodall showing David Greybeard using tools. For everyone seeing these photographs, the result is clear.

Unfortunately nature is not always so forthcoming to make its patterns and processes obvious enough to hit you between the eyes. Some effects of interest are subtle and can only be teased apart with advanced techniques. Further, what causes pain to one person's inter ocular region may sail safely over another person's head. The IOT test has not caught on within science, but the importance of presenting results clearly has.

If you define a significant difference in means by a t-test with p

Dear: First, analyze the data using another program ...

Second, make sure of the level of significance assumed in the experiment ...

Third, sometimes the drawing does not give a clear picture of the data under study

The difference may be caused by

A very important note!!!

Not merely related to this question.

t.test command in R has 2 methods: (the vectors can be separated by either ~ or comma)

1. t.test(outcome ~ groups)

2. t.test(outcome in group1, outcome in group0)

The first one is more common and is like other software such as SPSS. The second one is an immediate form for small data like Graph Pad software.

If a researcher wrongly perform "t.test(outcome, groups)" instead of "t.test(outcome ~ groups)", she/he will run into an exaggerated result; because the software thinks that the amounts of one group are e.g. 100, 120, 130, 115, ... and the amounts of the other group are 0, 0, 1, 0, 1, ... !!!!!!!