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In some legal and financial contexts, an option is material. That would mean an event/action/adjustment that is likely to affect some important outcome. For instance "A tax rate increase of 10% would make a material difference to our profitability."

Merriam-Webster:

... having real importance or great consequences

For the accounting term: https://en.wikipedia.org/wiki/Materiality_(auditing)

A result is said to be meaningful when it has some real-world significance.

full of meaning, significance, purpose, or value; purposeful; significant:

Dictionary.com

Substantial as in "a substantial salary" or "a substantial amount" fits the bill.

substantial - (adjective) significantly great MW

• The party has just lost office and with it a substantial number of seats.¹
  


• That is a very substantial improvement in the present situation.¹
  


• She inherited a substantial fortune from her grandmother.²
  


• All the evidence points to a substantial rise in traffic over the next few years.²
  

"Significant"

The relevant Oxford English Dictionary meaning of the word is "Sufficiently great or important to be worthy of attention; noteworthy; consequential, influential."

Non-negligible works well in certain contexts when discussing a variable that is so not so insignificant that it can be neglected.

The definition of negligible is:

so small, trifling, or unimportant that it may safely be neglected or disregarded

Non-negligible is, of course, the opposite.

To directly match “Significant enough to make a difference” I’d suggest impactful:

Oxford Dictionaries

Impactful - Having a major impact or effect.

For the specific context you’ve provided, though, I think noteworthy fits the sentence much better:

Oxford Dictionaries

Noteworthy - Worth paying attention to; interesting or significant.

Because you mention statistical significance, I would like to discuss the usage in a technical context.

Statistical significance essentially means that an observed difference or change is unlikely to have occurred by chance, and hence that it very probably is indicative of some real phenomenon. As you have discovered, this does not necessarily mean that the change or difference is of any importance. Even extremely small changes may be statistically significant if the measurement process is very precise.

Sometimes, this creates the requirement of distinguishing between "differences that are statistically significant, but inconsequential" and "differences that are statistically significant, and are also large enough to have some practical impact." Unfortunately, there does not seem to be a single word or short phrase that is universally understood to have this technical meaning.

In my own field, we say "technically significant" to mean "of practical importance, whether or not it is also statistically significant." However I am aware of other fields that use the same phrase to mean almost the opposite: "statistically significant but unimportant."

So in a technical context, it seems that if there is any risk of confusion it is best to clearly state what you mean instead of trying to form a compact phrase.

I'd stick with 'significant'.

"We need to find results that are significant".

"Enough to make a difference" is exactly what "significant" MEANS. 'Significant enough to make a difference' is tautology.

I'd avoid 'statistically significant' outside a mathematical context. As people have mentioned, that means something special.

The accepted term in medicine and psychology is "clinically significant", as compared to statistical significance.

In medicine and psychology, clinical significance is the practical importance of a treatment effect—whether it has a real genuine, palpable, noticeable effect on daily life.

Note that this usage is endorsed by the American Psychological Association (APA) per Vacha-Haase et al. (2000). I do not know of any comparable standards in medicine (being a somewhat psychology-affiliated statistician), but these may be findable.

A simple word that conveys the meaning is to matter. Using your example:

We need to find results that matter.

According to dictionary.com the definition for the verb "to matter" is

to be of importance; signify.

Similary, if something is "no matter",

it is unimportant; it makes no difference.

Notable comes to mind. From the OED:

1. Worthy or deserving of attention, esp. on account of excellence, value, or importance; significant in size or amount; noteworthy, remarkable, striking, signal, eminent.

While this is similar to "Noteworthy" (as talked about in this answer), notable tends to emphasize the greatness of a thing. For more on the subtle but noteworthy differences, see this thread.

Substantial has already been suggested, but I would suggest that substantive would be a better choice. From Merriam-Webster:

: having substance : involving matters of major or practical importance to all concerned

As already noted on substantial, this depends on exactly what you want to say. An alternative would be impactful which seems to better fit what you want than meaningful. Meaningful would more normally be used when there are a variety of ways of measuring meaning. Impactful is more singular. It is full of one impact, not a variety of meanings.

I would find it better than substantial, because it relates more to the importance rather than the size in normal usage. It's also more of a binary choice. Something is either substantive or not. Substantial is more relative and depends more on context. Substantive is generally used in ways that are not modified by words like rather or very.

Substantive has fewer meanings than material, which is a reasonable synonym. Thus, I would prefer it as being more obvious about what was meant.

Salient. In tech it is common to specify salient characteristics when requesting quotes from vendors.

Considerable

Considerable

adjective

notably large in size, amount, or extent.
"a position of considerable influence"
synonyms: sizable, substantial, appreciable, significant; More

I think in your example it means that the results are worth considering because of their significance.

Most of the other answers provide a word that is much more than just significant enough to make a difference. I'd like to suggest a word that quite literally means "[just] significant enough to make a difference."

Adequate

sufficient for a specific need or requirement

good enough

You didn't note speicifally how much of a difference had to be made, so I chose a word that covered the least amount possible to meet the needs of your test case.

From comments it looks like this is being used in communications with clients. Without knowing more about those clients and how they might interpret terminology it's difficult to come up with an ideal term, but here are a few possible scenarios.

If you are talking about features or attributes of something which are important enough that they should be paid attention to in a particular environment or context, relevant can be used.

Relevant in this context means a thing is of enough significance that it should not be disregarded, but it does not convey extreme importance. If something is relevant, it needs to be taken into account, but may not be the most important aspect.

Example: "Pages two and three of the instructions are relevant to
customers in Hawaii only."

Noticeable is similar in that it indicates a difference worthy of paying attention to, but not in a dramatic or extreme manner, and is often used to describe a change in a measurement or quantity.

Example: "There was a noticeable improvement in gas mileage after I
replaced my tires."

This indicates the improvement was enough to be worth considering, but it does not convey an extreme or surprising amount of improvement.

If you want to convey that the significance is just enough to achieve a specific threshold and no more, then sufficient will work.

Example: "My exam score was sufficient to ensure I got a B in the
class." The implication is that the score was just high enough, not
exceptional or perfect.

Not only do you want statistically significant results, you want results that are effectual.

Random House dictionary

effectual: producing or capable of producing an intended effect; adequate

With a sufficiently large sample size, even trivially small effects can be measured with statistical significance. But it sounds like you want the ones that result in a larger effect—that is, the most effectual.

One that's used in British English (esp. sports reporting) is telling. Idiomatically, a player whose impact on a match is significant enough to make a difference may be said to make a telling contribution.

See here for example usage.

My suggestion is decisive.

Oxford Dictionaries:

Decisive

Settling an issue; producing a definite result.

Statisticians distinguish between significance and effect size. While these two things tend to run together, results can be highly significant but show a small effect, while other results can show a big effect yet still be insignificant. Typically a very big sample can yield the former, while a very small sample often results in the latter.

Thus your proposed sentence might well be worded as

"We need to find results that show a significant large effect."

You need significance because insignificant results are of dubious value, even if they look impressive. I do not know a one-word adjective that means exactly "showing a large effect". The nearest I can think of is the word just used—impressive—but it is somewhat imprecise as to just what it describes.

"Important".

A current dictionary defines that as, "of great significance or value", but its earlier meaning (and the way I understand it) is "being of consequence" -- see also one of the meanings of the noun "import", which is given as, "the implicit meaning or significance of something".

Altohugh you cite statistics, it is not completely clear if the context in which you would put the word you seek is some scientific or technical paper.

Assuming the context is not so specific, I think a common word that may be interpreted, in general contexts, as "something more than significant", is remarkable.

We need to find results that are remarkable.

overwhelming

(adjective) very great in amount.

Or in your specific instance, maybe

compelling

is a better choice. As in "we need to find results which are compelling."

compelling

(adjective) forceful, demanding attention, convincing.

crucial might work, it encompasses both the significance and the decisiveness:

2 a : important or essential as resolving a crisis : decisive

// She played a crucial role in the negotiations.

b : marked by final determination of a doubtful issue

// the crucial game of a series

c : important, significant

// … what use we make of them will be the crucial question.

— Stanley Kubrick

// Vitamins are crucial for maintaining good health.

_{(source: Merriam-Webster)}

Another word which may work here is worthwhile (in the sense of worth considering).

"We need to find results that are worthwhile".

ODO:

worthwhile

(also worth one's while, worth while)
ADJECTIVE

Worth the time, money, or effort spent; of value or importance.

‘Questions were thrown at me on a subject until I could give no more
worthwhile answers.’