This really bears more examination. If one looks at the actual tables the XP per day (page 84 DMG) would translate to 3 deadly encounters.
A deadly encounter (by the same logic) is 2 medium encounters or one hard and one easy encounter.
8 hard encounters at level 1 is 600XP (for one adventurer) this is twice the daily XP budget. The party size does not matter for this consideration since the numbers scale with party size.
That is a level 1 hard encounter for a party of 4 is 300XP and 8 of them is 2,400XP as compared to the daily XP budget for a party of 4 is 1,200XP.
6 hard encounters for a party of 4 at level 1 is 1,800XP which still exceeds the daily budget.
The daily XP budget can be meet in 4 hard encounters (or 3 deadly - which matches my experience)
To get 6 to 8 encounters, withing the daily expected budget, one has to go for easier encounters.
1 hard encounter, 2 medium and 5 easy encounters will net the expected daily XP budget and I think that the statement on page 84 of the DMG about 6 to 8 medium to hard encounters needs to be considered in conjunction with the math implied on the Adventuring Day XP table on the same page and the encounter budget implied by the XP Thresholds by Character Level table on the previous page.