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The phrase "achieved a higher" is correct and usable in written English.
It can be used when discussing accomplishments or improvements in comparison to a previous state or standard.
Example: "After months of hard work, she achieved a higher level of proficiency in her language skills."
Alternatives: "reached a greater" or "attained a superior".
Exact(60)
They've both achieved a higher utility.
Smad3 achieved a higher rank for Oct4, compared to the other genes.
Now that yoga has achieved a higher plane of style consciousness, saffron robes are just too bland.
CiF actually achieved a higher awareness score amongst the Guardian audience than it's digital counterpart, 75% versus 73%.
OECD statistics show that 38% of UK adults (aged 25-64) hachievedeved a higher education qualification of some kind.
Yet, for Dr. von Klemperer, their efforts achieved a higher goal.
However, the TOMM achieved a higher overall classification rate than the Dot Counting Test.
The resulting extra NADPH-consuming mutant grew much faster and achieved a higher biomass concentration.
Moreover, first year students achieved a higher average success rate than third year students.
The use of 3 indicators of post-index adherence achieved a higher c-statistic than the best-performing model using pre-index information (0.74 vs. 0.72).
That suggested its designers had used more atomic fuel, had achieved a higher rate of burning, or had engaged in thermonuclear boosting.
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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