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If necessary, sign in with your username and password. Click "Skype" in the menu bar at the top of the screen. It is in the top left next to the apple icon. Select "Preferences" from the drop-down menu. Alternatively, you can hit Command+, on the keyboard. Click the button at the top of the window that says "Privacy." It is marked with a picture of a "do not disturb" sign. " This is directly under the "Save chat history for:" option. You will be asked to confirm your choice. Click "Delete All" to finish. At this point, you can exit the window you don't need to save your changes like on Windows.
Open Skype. Open the Preferences menu. Select the Privacy tab. Click "Delete All Chat History.