The Carrying Out of the Purifying, Washing of Clothes And Making of Atonement for the Levites Is Summarised (Numbers 8:21).
‘And the Levites purified themselves from sin, and they washed their clothes, and Aaron offered them for a waveoffering before Yahweh; and Aaron made atonement for them to cleanse them.'
Compare verses 7-8. In accordance with Yahweh's requirements, the Levites purified themselves from sin by means of the water of the purification for sin offering, shaved their bodies and washed their clothes. Then Aaron offered the Levites as a ‘waveoffering' before Yahweh, presenting them to Him, and through the offerings made atonement for them to cleanse them.