If your new employee has been paid monthly to date, most payroll systems calculate the tax free allowance of just under £540 per month, totalling £6475 over the course of the tax year, and not use up the whole allowance over a few months, whereas your calculation is based on deducting 20% from the gross pay.
If your new employee has another source of income, i.e. a pension which uses up the tax allowance, then you should be informed by HMRC to use a tax code of 'BR' (Basic Rate) for which no tax free allowance is given, and your calculation is correct according to the HMRC Tax Calculator.