Very interesting. There is clearly a bug at least in my version of Pandas (0.25.1) with df.groupby(...).quantile(<array-like>). That codepath is different and seems to be broken even on very simple examples like:

df = pd.DataFrame(
    {"A": [0., 0., 0.], "B": ["X", "Y", "Z"]}
)
result = df.groupby("B").quantile([0.5, 0.9])

While it would work on a 2-element version one:

df = pd.DataFrame(
    {"A": [0., 0.], "B": ["X", "Y"]}
)
result = df.groupby("B").quantile([0.5, 0.9])

I would avoid using groupby with quantile on array-like objects until the code is fixed even in cases it works now since it is likely error-prone.

Blame also shows a lot of fairly fresh updates there (10, 16 months) dealing exactly with these pieces of code too.

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