Somehow and sometimes. The question is hard to answer without deeper explanation. Antioxidant activities can be measured with different methods - that is one problem and another is that there are a lot of compounds with antioxidant activities which are not phenolic compounds eg. vit. C. The most common and simple method for estimation phenolic compounds in total is colorimetric method with Folin-Ciocalteu reagent.
Though it is true that in most of the cases the antioxidant activity of the plant extract is correlated with higher phenolic contents. The free radicals get stabilizes by gaining an electron or by accepting the hydrogen. In plant, phenolic compounds are well known hydrogen donors. But in plants there are variety of phytochemicals which can donate hydrogen to free radicals to stabilize it. Hence sometimes, we come to know that the plant having high amount of phenolic compounds do not exhibits maximum antioxidant activity and in other cases the some plants having low phenols show maximum antioxidant activity. This is due to the fact that the other non phenolic compounds are also contributing to the antioxidant activity in addition to antioxidant activity by Phenolic compounds by donating hydrogen to free radicals. Thus, the antioxidant activity of plant extract always not correlates with amount of Phenols in plants. The Activity is affected by other factors such as glycosylation of aglycones, other hydrogen donating groups (-NH and –SH), flavanol aglycones, quercetin, myricetin and kaemperol containing multiple hydroxyl groups had higher antioxidant activity than their glycosides such as rutin, myricitrin, astragalin. The glycosylation of the flavonoids reduces the antioxidant activity of the compounds. It is from me.
Rice-Evans, C., Miller, N. J. and Paganga, C. (1996). Structure antioxidant activity relationship of flavonoids and phenolic acids. Free Radical Biological and Medicine, 20(7):933-956.