Why is my WIFI always on? I have turned off WIFI in the settings, but it's still always running. Doesn't this graph indicate that WIFI is really on all the time? How can this be when it's definitely "off" in Settings?

I'm worried that this constant WIFI use drains my battery and I want to avoid this.

This is a Nexus 4 running Kitkat 4.4.4. 

I have also installed the Llama automation app which can optionally turn on WIFI every x minutes, but I have made sure that option is definitely turned off. This behavior continues even when Llama is not running. 
I don't think there's any other app on my phone that would have the capability to turn on WIFI on its own.

Is there some way I could find out whether one of the installed apps is responsible for this?
Android is automatically keeping Wi-Fi on while the phone is sleeping (Keep Wi-Fi on during sleep) and always scans for available networks (Scanning always available). They must be keeping it on. Try disabling them from Settings &gt; Wi-Fi &gt; Advanced.