BUG: Wrong skewness and kurtosis in `Rolling.{skew,kurt}` for degenerate distributions

[Alvaro-Kothe]

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

import pandas as pd

data = [42] * 10
window = pd.Series(data).rolling(5)

print(window.skew())
# 0    NaN
# 1    NaN
# 2    NaN
# 3    NaN
# 4    0.0
# 5    0.0
# 6    0.0
# 7    0.0
# 8    0.0
# 9    0.0
# dtype: float64

print(window.kurt())
# 0    NaN
# 1    NaN
# 2    NaN
# 3    NaN
# 4   -3.0
# 5   -3.0
# 6   -3.0
# 7   -3.0
# 8   -3.0
# 9   -3.0
# dtype: float64

Issue Description

When applying rolling.skew() and rolling.kurt() to a series of constant values, the results are 0.0 for skew and -3.0 for kurtosis once the window is full. However, since the data is constant, the variance is zero, making both skewness and kurtosis mathematically undefined.

Expected Behavior

The expected result should be NaN for these cases, as division by zero (variance) occurs in the formulas.

Installed Versions

INSTALLED VERSIONS

commit : ea75dd7
python : 3.13.9
python-bits : 64
OS : Linux
OS-release : 6.17.4-200.fc42.x86_64
Version : #1 SMP PREEMPT_DYNAMIC Sun Oct 19 18:47:49 UTC 2025
machine : x86_64
processor :
byteorder : little
LC_ALL : None
LANG : pt_BR.UTF-8
LOCALE : pt_BR.UTF-8

pandas : 3.0.0.dev0+2592.gea75dd7b3b
numpy : 2.3.4
dateutil : 2.9.0.post0
pip : 25.2
Cython : 3.2.0b1
sphinx : 8.2.3
IPython : 9.6.0
adbc-driver-postgresql: None
adbc-driver-sqlite : None
bs4 : 4.14.2
bottleneck : 1.6.0
fastparquet : 2024.11.0
fsspec : 2025.9.0
html5lib : 1.1
hypothesis : 6.142.3
gcsfs : 2025.9.0
jinja2 : 3.1.6
lxml.etree : 6.0.2
matplotlib : 3.10.7
numba : 0.62.1
numexpr : 2.14.1
odfpy : None
openpyxl : 3.1.5
psycopg2 : 2.9.11
pymysql : 1.4.6
pyarrow : 21.0.0
pyiceberg : 0.10.0
pyreadstat : 1.3.1
pytest : 8.4.2
python-calamine : None
pytz : 2025.2
pyxlsb : 1.0.10
s3fs : 2025.9.0
scipy : 1.16.2
sqlalchemy : 2.0.44
tables : 3.10.2
tabulate : 0.9.0
xarray : 2025.10.1
xlrd : 2.0.2
xlsxwriter : 3.2.9
zstandard : 0.25.0
qtpy : None
pyqt5 : None

[Alvaro-Kothe]

Another place that I think should be addressed too is in the non-window version.

import pandas as pd

data = pd.Series([42] * 10)

print(data.skew())
# 0.0

print(data.kurt())
# 0.0

[rhshadrach]

scipy.stats does indeed return NaN for these cases, I see the value in agreeing. However I'd be concerned about the performance hit we'd experience in the usual case for these edge cases, adjustments for this should be carefully benchmarked.