Pandas version checks
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[X] I have checked that this issue has not already been reported.
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[X] I have confirmed this bug exists on the latest version of pandas.
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[ ] I have confirmed this bug exists on the main branch of pandas.
Reproducible Example
>>> a = pd.Series([100.0, 200.0], dtype=pd.ArrowDtype(pa.float64()))
>>> b = a.describe()
>>> b.dtype
Float64Dtype()
>>>
Issue Description
Describe with arrow backend returns an extension array datatype instead of pyarrow datatype.
Expected Behavior
I expect the returned series to be of dtype double[pyarrow].
Installed Versions
>>> pd.show_versions()
C:\pandas2_ps_04323\lib\site-packages\_distutils_hack\__init__.py:33: UserWarning: Setuptools is replacing distutils.
warnings.warn("Setuptools is replacing distutils.")
INSTALLED VERSIONS
------------------
commit : 478d340667831908b5b4bf09a2787a11a14560c9
python : 3.8.10.final.0
python-bits : 64
OS : Windows
OS-release : 10
Version : 10.0.22621
machine : AMD64
processor : Intel64 Family 6 Model 85 Stepping 7, GenuineIntel
byteorder : little
LC_ALL : None
LANG : None
LOCALE : English_United States.1252
pandas : 2.0.0
numpy : 1.24.2
pytz : 2023.3
dateutil : 2.8.2
setuptools : 67.5.1
pip : 23.0.1
Cython : None
pytest : 7.0.1
hypothesis : None
sphinx : 4.2.0
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : 3.1.2
IPython : 8.12.0
pandas_datareader: None
bs4 : 4.12.0
bottleneck : None
brotli : None
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : None
numba : None
numexpr : None
odfpy : None
openpyxl : None
pandas_gbq : None
pyarrow : 11.0.0
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : None
snappy : None
sqlalchemy : None
tables : None
tabulate : 0.8.9
xarray : None
xlrd : None
zstandard : None
tzdata : 2023.3
qtpy : None
pyqt5 : None
Comment From: mroeschke
Thanks for the report again!