I have one diagram in google analytics:
My question is how Day 0 is calculated as 1.09?
Day 0 for a user is the day they first visited your site, and the Sessions Per User value for that day is the average number of sessions a user had on their first day (where the users being considered are only the users who were new during the Acquisition Date Range). The fact that your value is 1.09 means that some of your new users returned for a second session on the same day as their first session.
As you move to the right on the graph the average is all sessions from Day 0 through the day you are hovering over, divided by the number of new users acquired during the acquisition date range. It increases early, because most repeat visits happen within a few days of the first visit, and then levels off - very few people come back again and again over multiple weeks.
To get the details of this graph clear, let's look at a very small example: an acquisition date range that is some Monday-Friday, with just 3 users who visited the site for the first time during that week.
User A: first visit is Monday, comes back a second time on Monday and once on Tuesday
User B: first visit is Tuesday, comes back once on Friday
User C: first visit is Wednesday, no additional visits
So the week's sessions look like this:
Monday: A, A
Tuesday: A, B
Note that any user who visited but was not new during the week is excluded.
Lifetime Value is measured for each user's individual "lifetime", which starts on the day of their first session (Day 0 for that user).
User A's Day 0 is Monday; they visit twice then and again on their Day 1
User B's Day 0 is Tuesday; they visit again on their Day 3
User C's Day 0 is Wednesday
So to compute the numbers for the LTV graph, we reorganize the sessions to look like this:
Day 0: A, A, B, C
Day 1: A
Day 2: none
Day 3: B
Day 4 and later: none
We have 3 users acquired during the acquisition date range, and since we are looking at Lifetime Value we take all sessions starting from Day 0 and ending with the day we are computing LTV for (the cumulative sum of sessions).
The Day 0
Sessions Per User (LTV) value is
(4 sessions)/(3 users) = 1.33.
The Day 1 value is
(5 sessions)/(3 users) = 1.67.
The Day 2 value is also 1.67, since there are no additional sessions that day.
The Day 3 value is
(6 sessions)/(3 users) = 2.00.
Days 4 and beyond also have a value of 2.
In this example, the LTV graph could continue past the end of the week -- and if User B came back for a visit on Saturday, the
Sessions Per User (LTV) value would increase to 2.33 on Day 4. Sessions after the end of the set date range are included, but only sessions belonging to these 3 users, because they were the only new users in the Monday-Friday Acquisition Date Range.