Just a guess...

You probably have to do this ..

z = ThisWorkbook.Sheets("USRMResultsAnalysis").Cells(Rows.count, 1).End(xlUp).Offset(1, 0).row

..before each row is copied to get the next (correct) destination row

or add an increment (1, 2 or 3) to z, maybe something like

ThisWorkbook.Sheets("SampleResult").Select   ' <<<<<<<<<<<<<<<<<<<<<<< don't need to select something to act on or with it. also Thisworkbook is probably unnecessary
  z = ThisWorkbook.Sheets("USRMResultsAnalysis").Cells(Rows.count, 1).End(xlUp).Offset(1, 0).row

Sheets("SampleResult").Range("av2").Copy 'A3 
Sheets("USRMResultsAnalysis").Cells(z, 1).PasteSpecial Paste:=xlPasteValuesAndNumberFormats

Sheets("SampleResult").Range("av4").Copy 'B3
Sheets("USRMResultsAnalysis").Cells(z, 2).PasteSpecial Paste:=xlPasteValuesAndNumberFormats

Sheets("SampleResult").Range("av2").Copy 'A3
Sheets("USRMResultsAnalysis").Cells(z+1, 1).PasteSpecial Paste:=xlPasteValuesAndNumberFormats

Sheets("SampleResult").Range("av4").Copy 'B3
Sheets("USRMResultsAnalysis").Cells(z+1, 2).PasteSpecial Paste:=xlPasteValuesAndNumberFormats

Sheets("SampleResult").Range("av2").Copy 'A3
Sheets("USRMResultsAnalysis").Cells(z+2, 1).PasteSpecial Paste:=xlPasteValuesAndNumberFormats

Sheets("SampleResult").Range("av4").Copy 'B3
Sheets("USRMResultsAnalysis").Cells(z+2, 2).PasteSpecial Paste:=xlPasteValuesAndNumberFormats

Sheets("SampleResult").Range("av2").Copy 'A3
Sheets("USRMResultsAnalysis").Cells(z+3, 1).PasteSpecial Paste:=xlPasteValuesAndNumberFormats

Sheets("SampleResult").Range("av4").Copy 'B3
Sheets("USRMResultsAnalysis").Cells(z+3, 2).PasteSpecial Paste:=xlPasteValuesAndNumberFormats