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The phrase "assigned at random" is correct and usable in written English.
It can be used when referring to something that has been allocated or designated without any specific criteria or pattern.
Example: "The participants in the study were assigned at random to either the control group or the experimental group."
Alternatives: "allocated randomly" or "designated at random".
Exact(60)
In completely randomized designs, patients are assigned at random to one of the options, requiring that patients and clinicians find each of the options acceptable.
Previously, cases were assigned at random.
The consumers were assigned at random to three groups.
Mice were assigned at random for xenograft assay.
Seventy patients were assigned at random to an antibiotic protocol.
Ms. Ogden was assigned at random to the group taking the standard form of Rebetron.
Myles would be assigned at random to one of four schools.
Assigned at random, 174 women received hormones and 171 did not.
After the letter and the investigation that followed, cases were assigned at random.
In the study, 14 volunteers were assigned at random to watch either a stressful or a humorous 20-minute video.
They were then assigned at random to one of three groups.
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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