First of all, according to the docs, the default method used for interpolation is linear:

linear: i + (j - i) * fraction, where fraction is the fractional part of the index surrounded by i and j.

We first need to find out how index is calculated (and then get its fractional part):

index = distance_between_first_and_last_element * q

So, in your example, we'd have a distance of 7 (because 2 and 19 are 7 positions apart). This leads to:

 index = 7 * 0.3 = 2.1

Of course, we need to keep only the fractional part of it, which is simply 0.1, so fraction is equal to 0.1.

Moreover, the 0.3 quantile that you've asked for is between 5 and 8 (which are numbers in your list), so these will stand for i and j respectively.

Why? Because your index is 2.1, which is between 2 and 3, so:

i = s[2] # which is equal to 5 in your Series
j = s[3] # which is equal to 8 in your Series

All these lead to:

5 + (8-5) * 0.1

which equals to:

5.3

According to the same logic, we can calculate the 0.25 percentile:

index = 7 * 0.25 # 1.75
fraction = 0.75 # the fractional part of index
i = s[1] # 4
j = s[2] # 5
answer = 4 + (5-4) * 0.75 -> 4.75