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The phrase "a much higher degree" is correct and usable in written English.
It can be used to emphasize a significant increase or level of something compared to another.
Example: "The new software offers a much higher degree of efficiency than the previous version."
Alternatives: "a significantly greater extent" or "a considerably higher level".
Exact(54)
But the trade center site plan involves a much higher degree of security.
"We have a much higher degree of accountability," said the Dutchman, whose team won 19 medals in Barcelona.
"This implies a much higher degree of social organization and behavioral complexity than is frequently attributed to the Neanderthals".
Other scholars have developed the ethnographic classification of religion to a much higher degree than did Müller.
A much higher degree of orientation of PP was found in the composites compared with the pure PP.
Enriched uranium can be used to fuel power plants, which is Iran's stated purpose, or to provide material for bombs, if refined to a much higher degree.
"This isn't 1970," she said, "and in the year 2000 we have a much higher degree of public awareness of the significance of the pollution problem.
We will need both a much higher degree of political mobilisation and a greater degree of personal action and behaviour change in order to succeed.
It demands of us a much higher degree of care when we talk about the meaning or the effect of a piece of writing.
Ceda's chief executive, Stephen Martin, says the vast majority of Australians will need to have a much higher degree of skill that particularly embraces computerisation.
Similar(1)
In contrast, a much higher degree of colocalization was observed for pMRLC and Stbm-Venus.
More suggestions(17)
a significantly higher degree
a much higher grade
a considerably higher degree
a much broader degree
a much larger degree
a much greater degree
a far higher degree
a much higher magnitude
a substantially higher degree
a much higher value
a much higher incidence
a much higher likelihood
a much higher profit
a much higher concentration
a much higher density
a much higher prevalence
a much higher dose
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com