With the release of the new Analysis tools in the ArcGIS Online web map, you will need to create at least one feature service to enable the appearance of the Perform Analysis option when you want to perform analysis on any feature layer. Without a previously created feature service the Perform Analysis option will not appear. This is a glitch that will be resolved in a future release.
An easy way to create a simple feature service to "seed" and enable the Perform Analysis choice is to add a spreadsheet with a single location to My Contents in your org subscription with Administrator or Publisher privileges. When you add the spreadsheet with these privileges you can select the option to spin it up as a feature service, thereby seeding the feature service capability needed to enable Perform Analysis as an option.
If you've already created a hosted feature service in your organization, this is not necessary.