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"show a positive correlation" is correct and can be used in written English.
You would use this phrase when you are discussing two separate things that have a direct connection to one another. For example, "The data showed a positive correlation between taking math classes and higher test scores."
Exact(60)
RNFL measurements did not correlate with age but did show a positive correlation with SE.
Half of the peer-reviewed studies on the effect of the drinking age on fatalities show a positive correlation.
The values of the indices show a positive correlation.
Descriptive results show a positive correlation between naturalisation and OJT participation.
However, in present study, LMA didn't show a positive correlation with leaf thickness.
Approximants-Both /r/ and /w/ show a positive correlation with Psychomotor Retardation.
The distribution of the samples show a positive correlation between two dimensions.
The results also show a positive correlation with optical emission from the plasma and SRIM simulation.
These quantities show a positive correlation with the increase of the geomagnetic activity.
Coreflooding recovery factor results show a positive correlation with dynamic interfacial properties, though not a straightforward one.
Our results show a positive correlation between the shear strength and the surface energy of the investigated metals.
More suggestions(14)
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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