BUG: Series.describe treating pyarrow timestamps/timedeltas as categorical
State: closed

- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [x] Added an entry in the latest `doc/source/whatsnew/v2.0.2.rst` file if fixing a bug or adding a new feature.


Fixes the following such that pyarrow-backed timestamps and timedeltas are described in the same way numpy-backed timestamps and timedeltas are described. 

Current behavior:

```
In [1]: import pandas as pd

In [2]: df_numpy = pd.DataFrame({
   ...:     "datetime": pd.to_datetime(range(10)),
   ...:     "timedelta": pd.to_timedelta(range(10)),
   ...:     "numeric": range(10),
   ...: })

In [3]: df_pyarrow = df_numpy.convert_dtypes(dtype_backend="pyarrow")

In [4]: df_numpy.describe()
Out[4]: 
                            datetime                  timedelta   numeric
count                             10                         10  10.00000
mean   1970-01-01 00:00:00.000000004  0 days 00:00:00.000000004   4.50000
min              1970-01-01 00:00:00            0 days 00:00:00   0.00000
25%    1970-01-01 00:00:00.000000002  0 days 00:00:00.000000002   2.25000
50%    1970-01-01 00:00:00.000000004  0 days 00:00:00.000000004   4.50000
75%    1970-01-01 00:00:00.000000006  0 days 00:00:00.000000006   6.75000
max    1970-01-01 00:00:00.000000009  0 days 00:00:00.000000009   9.00000
std                              NaN  0 days 00:00:00.000000003   3.02765

In [5]: df_pyarrow.describe()
Out[5]: 
                   datetime        timedelta  numeric
count                    10               10     10.0
unique                   10               10     <NA>
top     1970-01-01 00:00:00  0 days 00:00:00     <NA>
freq                      1                1     <NA>
mean                    NaN              NaN      4.5
std                     NaN              NaN  3.02765
min                     NaN              NaN      0.0
25%                     NaN              NaN     2.25
50%                     NaN              NaN      4.5
75%                     NaN              NaN     6.75
max                     NaN              NaN      9.0
```

New behavior:

```
In [4]: df_pyarrow.describe()
Out[4]: 
                            datetime                  timedelta   numeric
count                             10                         10  10.00000
mean   1970-01-01 00:00:00.000000004  0 days 00:00:00.000000004   4.50000
min              1970-01-01 00:00:00            0 days 00:00:00   0.00000
25%    1970-01-01 00:00:00.000000002  0 days 00:00:00.000000002   2.25000
50%    1970-01-01 00:00:00.000000004  0 days 00:00:00.000000004   4.50000
75%    1970-01-01 00:00:00.000000006  0 days 00:00:00.000000006   6.75000
max    1970-01-01 00:00:00.000000009  0 days 00:00:00.000000009   9.00000
std                              NaN  0 days 00:00:00.000000003   3.02765
```

Labels:
name: Bug
name: Arrow
description: pyarrow functionality

Comments:
Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

```
git checkout 2.0.x
git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
git cherry-pick -x -m1 9442dfcc4f87c68632eb3ffbaf8dae6da4b532bd
```

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

```
git commit -am 'Backport PR #53001: BUG: Series.describe treating pyarrow timestamps/timedeltas as categorical'
```

4. Push to a named branch:

```
git push YOURFORK 2.0.x:auto-backport-of-pr-53001-on-2.0.x
```

5. Create a PR against branch 2.0.x, I would have named this PR:

> "Backport PR #53001 on branch 2.0.x (BUG: Series.describe treating pyarrow timestamps/timedeltas as categorical)"

And apply the correct labels and milestones.

Congratulations — you did some good work! Hopefully your backport PR will be tested by the continuous integration and merged soon!

Remember to remove the `Still Needs Manual Backport` label once the PR gets merged.

If these instructions are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                
Mind backporting @lukemanley?
> Mind backporting @lukemanley?

just backported in https://github.com/pandas-dev/pandas/pull/53031