The phrase "significant degree" means a considerable or noticeable amount or level of something. It implies that there is a substantial impact or importance associated with the degree being referred to. Full definition
Although the students demonstrated a moderate to significant degree of authentic intellectual work in creating their films, we noted that the documentary medium is ill - defined for the K - 12 classroom. (citejournal.org)
I don't hate anybody, just the ideas of the religious people that at significant degree insult, chase, and kill minorities that don't fall in their pot. (religion.blogs.cnn.com)
Then more research followed, which showed that roughly 2 % of adults have this to a clinically significant degree. (protomag.com)