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Stock-Based Compensation - Schedule of Stock-Based Compensation Expense (Income) (Details) - USD ($) $ in Millions | 12 Months Ended
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Sep. 30, 2020 | Sep. 30, 2019 | Sep. 30, 2018
Equity-based compensation plans |  |  |
Stock-based compensation expense | $ 30.2 | $ 30.0 | $ 26.9
Income tax benefit recognized for stock-based compensation | (3.6) | (4.9) | (5.8)
Stock-based compensation cost, net of tax | 26.6 | 25.1 | 21.1
Stock options |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | 6.8 | 6.9 | 6.6
Nonvested Stock Awards [Member] |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | 15.8 | 15.0 | 13.7
Income tax benefit recognized for stock-based compensation | (3.1) | (2.1) | (3.0)
Performance Shares [Member] |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | 6.7 | 7.1 | 6.4
Stock Appreciation Rights (SARs) |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | 0.2 | 0.4 | (0.2)
Restricted Stock Units (RSUs) |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | $ 0.7 | $ 0.6 | $ 0.4

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