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The phrase "approximately the same way" is correct and usable in written English.
It can be used when comparing methods, processes, or actions that are similar but not identical.
Example: "The two experiments were conducted approximately the same way, leading to similar results."
Alternatives: "roughly the same manner" or "about the same method".
Exact(11)
The clock is not exactly obsolete -- time is still measured approximately the same way, and Mars is enjoying a resurgence in popularity.
In soil, the beads performed in approximately the same way as the free cells.
This resulted in images of uniform size containing faces oriented approximately the same way.
Even when censoring data occur, the two models fit the data beyond detection limit approximately the same way.
In the opposite case the amplitude decreases with distance in approximately the same way as on the surface of a semi-infinite medium.
Both are open source frameworks and work approximately the same way.
Similar(48)
The main hypothesis motivating the procedure described here is that the partial frequencies of a given instrument or vocal vary approximately in the same way with time.
Concordance is a form of linking to establish score comparability from different tests that measure ideally similar, but not necessarily equivalent constructs, to be used approximately in the same way and given similar interpretations [ 16].
Participants are informed that they will be questioned about the story again later, and delayed recall is recorded in the same way after approximately 40 minutes.
Get to know your area and find corridors where people are going the same way at approximately the same time.
Spread a thin layer of spreadable avocado (approximately 1 tbsp - 15 mL) the same way as you did the mayonnaise.
More suggestions(16)
approximately the same manner
roughly the same way
approximately the same data
approximately the same resources
order the same way
about the same way
around the same way
approximately the same means
approximately the same number
approximately the same output
approximately the same height
approximately the same weight
approximately the same amount
approximately the same valence
approximately the same rate
approximately the same codon
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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