MongoDB: calculate 90th percentile among all documents - mongodb

Question

I need to calculate the 90th percentile of the duration where the duration for each document is defined as finish_time - start_time.
My plan was to:
Create $project to calculate that duration in seconds for each document.
Calculate the index (in sorted documents list) that correspond to the 90th percentile: 90th_percentile_index = 0.9 * amount_of_documents.
Sort the documents by the $duration variable the was created.
Use the 90th_percentile_index to $limit the documents.
Choose the first document out of the limited subset of document.
I'm new to MongoDB so I guess the query can be improved. So, the query looks like:
db.getCollection('scans').aggregate([
{
$project: {
duration: {
$divide: [{$subtract: ["$finish_time", "$start_time"]}, 1000] // duration is in seconds
},
Percentile90Index: {
$multiply: [0.9, "$total_number_of_documents"] // I don't know how to get the total number of documents..
}
}
},
{
$sort : {"$duration": 1},
},
{
$limit: "$Percentile90Index"
},
{
$group: {
_id: "_id",
percentiles90 : { $max: "$duration" } // selecting the max, i.e, first document after the limit , should give the result.
}
}
])
The problem I have is that I don't know how to get the total_number_of_documents and therefore I can't calculate the index.
Example:
let's say I have only 3 documents:
{
"_id" : ObjectId("1"),
"start_time" : ISODate("2019-02-03T12:00:00.000Z"),
"finish_time" : ISODate("2019-02-03T12:01:00.000Z"),
}
{
"_id" : ObjectId("2"),
"start_time" : ISODate("2019-02-03T12:00:00.000Z"),
"finish_time" : ISODate("2019-02-03T12:03:00.000Z"),
}
{
"_id" : ObjectId("3"),
"start_time" : ISODate("2019-02-03T12:00:00.000Z"),
"finish_time" : ISODate("2019-02-03T12:08:00.000Z"),
}
So I would expect the result to be something like:
{
percentiles50 : 3 // in minutes, since percentiles50=3 is the minimum value that setisfies the request of atleast 50% of the documents have duration <= percentiles50
}
I used percentiles 50th in the example because I only gave 3 documents but it really doesn't matter, just show me please a query for the i-th percentiles and it will be fine :-)