BUG: groupby.var() does not return arrow types with arrow backed series as input.

[wirable23]

Pandas version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

>>> df = pd.DataFrame({"A": pd.Series([True, True], dtype="bool[pyarrow]"), "B": pd.Series([decimal.Decimal(123), decimal.Decimal(12)], dtype=pd.ArrowDtype(pa.decimal128(6,3)))})
>>> df
      A        B
0  True  123.000
1  True   12.000
>>> df.groupby("A").var().dtypes
B    float64
dtype: object
>>>

Issue Description

The input is a series with type pd.ArrowDtype(pa.decimal128(6,3)), yet return type of var operation is numpy float64.

Expected Behavior

I expect return type to be double[pyarrow], since pyarrow type was given as input (based on #53831).

Installed Versions

INSTALLED VERSIONS

commit : 6e194cf
python : 3.11.2.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 : 2.1.0rc0
numpy : 1.25.2
pytz : 2023.3
dateutil : 2.8.2
setuptools : 68.0.0
pip : 23.2.1
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 : 8.14.0
pandas_datareader : None
bs4 : 4.12.2
bottleneck : None
brotli : None
dataframe-api-compat: None
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : None
numba : None
numexpr : None
odfpy : None
openpyxl : None
pandas_gbq : None
pyarrow : 12.0.0
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

[wirable23]

Also, return type of sum seems to be data dependent:

>>> df = pd.DataFrame({"A": pd.Series([True, True], dtype="bool[pyarrow]"), "B": pd.Series([1,2], dtype="int8[pyarrow]")})
>>> df.groupby("A").sum()
      B
A
True  3
>>> df.groupby("A").sum().dtypes
B    int8[pyarrow]
dtype: object
>>> df = pd.DataFrame({"A": pd.Series([True, True], dtype="bool[pyarrow]"), "B": pd.Series([125,125], dtype="int8[pyarrow]")})
>>> df.groupby("A").sum().dtypes
B    int64[pyarrow]
dtype: object
>>>

[wirable23]

Hi @mroeschke

It looks like in groupby.var/groupby.std, for "float[pyarrow]", dtype is returned as "float[pyarrow]", not "double[pyarrow]". Is that expected? The mean and variance/stddev kernels in arrow for float32 return float64, and it would mean if you calculate mean in describe vs groupby.mean with 1 group for float32, results would be different.

[mroeschke]

Today these operations are implemented by converting to masked types (Int64, Float64, etc), computing the op, and converting back to arrow, and the conversion back to arrow may return a less precise type. So yes, these ideally should mirror the arrow kernels eventually.