Is there any difference between there was an increase of 2% and “there was an increase by 2%”?
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The former sounds better to me. The preposition "by" would be more appropriate if you had used "increase" as a verb instead of a noun. (i.e., "It increased by 2%" as opposed to "There was an increase of 2%.")
"Increase" is being used as a verb in the phrase "increase by 2%". This makes the second sentence incorrect (because it already has the verb "is").
"Increase" is being used as a noun in the phrase "increase of 2%", and this makes sense because it is being used as a direct object in the first sentence.
In your example, "increase" is a noun. So here are your options (in bold):
There can be an increase:
(i) of a certain amount
(ii) of something that has grown larger
(iii) in something used a means of measurement, e.g. size, temperature, etc.
(iv) in something being measured statistically
(v) from a lower amount to a higher amount
(vi) over or on an earlier amount or the period when that earlier amount was calculated
(c) A Valency Dictionary of English (2004)
As you can see, *an increase by is not idiomatic English ("unnatural for a normal Englishman to say or write").