You can make an activity completely private on Strava, either from your account on the web or from the Strava mobile App. You will be the only person who can see this activity on Strava.
1. On the website - When you "Edit" the activity with the pencil icon, click the "Private" checkbox. You can also make your ride private by toggling the 'Private' slider at the 'Save and View' page.
2. On the iPhone App - After pressing "Finish Ride/Run" there is a 'lock' icon for the Private activity feature that can be turned on or off.
Additionally, there are other privacy settings which you can familiarize yourself with under the "Privacy" pulldown under your name when you are logged in.