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The phrase "a medium effect on" is correct and usable in written English.
It can be used when discussing the impact or influence of one variable on another, particularly in research or analytical contexts.
Example: "The study found that the new teaching method had a medium effect on student engagement levels."
Alternatives: "a moderate impact on" or "an average effect on".
Exact(6)
A 2014 Pew Research Center survey found that party identification and ideology had only a weak effect on whether people believed that vaccines were safe (though there was a "medium" effect on whether they should be required).
Experiments aimed at identifying the mechanism of influence of the SIP moieties on hydrolytic stability indicated that hydrolysis was enhanced by the presence of ionic multiplets which increase amorphous content, imbibe water, and perhaps exert a medium effect on the hydrolysis of esters associated with the ionic groups.
The PSQI reveals a medium effect on sleep parameters in subscales sleep quality and sleep efficiency.
Using Cohen's classification of effect sizes, 31 effect size was small for smoking behaviour and attitudes, while there was a medium effect on smoking-related knowledge.
Very low-graded evidence from another study suggested mutual support had a medium effect on symptoms of depression and anxiety (SMD = -0.42, -0.66 to -0.18) but no effect on recovery (SMD = -0.11, -0.35 to 0.13).
We found moderate evidence of a medium-to-large effect on pain and TPC for BT with mineral/thermal water (five studies, 177 participants; including three high-risk and two unclear-risk studies), a medium effect on HRQOL, and no significant effect on depressive symptoms (BDI).
Similar(54)
They detected a medium effect size on kinetic changes during stance at a p = 0.05 and power = 80%, and estimated sample size was 5 children per group.
The study was powered to detect a medium effect size on the primary study outcome measures of 0.5 units of standard deviation with 79% probability if the intraclass correlation coefficient within schools was 0.10 and with 90% probability for a coefficient of 0.06 for each primary outcome measure.
However, there was evidence of a small overall intervention effect on marijuana-related consequences and a medium effect in stratified analyses in the on-site condition.
There was evidence of a medium effect (SMD 0.62 (0.58 to 0.67)) on zinc concentration.
The sample size was calculated, based on a medium effect size, to achieve a power level of over 85% [ 8].
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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