The sensors output is sent to the local server, that the Arduino is hosting
without refreshing the whole page?
With AJAX, you can make requests for data, where the Arduino can simply respond with the data, say as json formatted within HTTP.
The Arduino doesn't need to concern itself with the client side application, just send the data when asked for it, which can be adjusted to go pretty fast.
For even faster data updates between an Arduino system and web based application, then using TCP sockets might the way to go, but you need a good understanding of all things 'Ethernet and TCP'.
Anyhow, you provide very little information in terms of the code on your Arduino and the actual data path to any connected client.
I'm guessing the analogue sensor value is stored and used by the web server code you have on your Arduino, and it is this value that is sent to the client when a HTTP connection is made, is that correct ?
Paul - VK7KPA