How do I get the below information from Exchange Server 2013?

            Traffic Patterns
            Which departments are the highest users of email  (inter departmental)
            Which departments talk the most over email to each other (cross department)
            What is the average message size
            What is the average attachment size
            How many times has a messaged been replied to
            How quickly does it take for the message to be replied to

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Based on your query, Message tracking log should be your easiest approach to find out such points.

You can also download Exchange Server Mailbox Statistics Tool from below given resource to collect the information : https://gallery.technet.microsoft.com/office/Exchange-Server-Mailbox-7dd53529

Here is another informative resource : http://community.spiceworks.com/scripts/show/2746-exchange-2013-daily-traffic-report

Moreover, to get more depth information into real time, you may also consider on our Lepide exchange reporter tool that provides comprehensive reports on the mailbox traffic of the entire network with details such as, all emails sent and received within a particular duration, emails sent outside and inside organization, interaction between users in a particular duration.

Exchange Server Mailbox Statistics Tool can help you out:

https://gallery.technet.microsoft.com/office/Exchange-Server-Mailbox-7dd53529

*Cheers!*

Aviral Rana