def kurtosis(data, m=None, s=None):
"""kurtosis(data [,m [,s]]) -> sample excess kurtosis of data.
The kurtosis of a distribution is a measure of its shape. This function
returns an estimate of the sample excess kurtosis usually known as g₂
"g\\N{SUBSCRIPT TWO}". For the population kurtosis, see ``pkurtosis``.
WARNING: The mathematical terminology and notation related to
kurtosis is often inconsistent and contradictory. See Wolfram
Mathworld for further details:
http://mathworld.wolfram.com/Kurtosis.html
>>> kurtosis([1.25, 1.5, 1.5, 1.75, 1.75, 2.5, 2.75, 4.5])
... #doctest: +ELLIPSIS
3.03678892733564...
If you already know one or both of the population mean and standard
deviation, you can pass the mean as optional argument m and/or the
standard deviation as s:
>>> kurtosis([1.25, 1.5, 1.5, 1.75, 1.75, 2.5, 2.75, 4.5], m=2.25, s=1)
2.3064453125
CAUTION: "Garbage in, garbage out" applies here. You can pass
any values you like as ``m`` or ``s``, but if they are not
sensible estimates for the mean and standard deviation, the
result returned as the kurtosis will likewise not be sensible.
If you give either m or s, and the calculated kurtosis is out
of range, a warning is raised.
If m or s are not given, or are None, they are estimated from the data.
If data is an iterable of sequences, each inner sequence represents a
row of data, and ``kurtosis`` operates on each column. Every row must
have the same number of columns, or ValueError is raised.
>>> data = [[0, 1],
... [1, 5],
... [2, 6],
... [5, 7]]
...
>>> kurtosis(data) #doctest: +ELLIPSIS
[1.50000000000000..., 2.23486717956161...]
Similarly, if either m or s are given, they must be either a single
number or have the same number of items:
>>> kurtosis(data, m=[3, 5], s=2) #doctest: +ELLIPSIS
[-0.140625, 18.4921875]
The kurtosis of a population is a measure of the peakedness and weight
of the tails. The normal distribution has kurtosis of zero; positive
kurtosis generally has heavier tails and a sharper peak than normal;
negative kurtosis generally has lighter tails and a flatter peak.
There is no upper limit for kurtosis, and a lower limit of -2. Higher
kurtosis means more of the variance is the result of infrequent extreme
deviations, as opposed to frequent modestly sized deviations.
CAUTION: As a rule of thumb, a non-zero value for kurtosis
should only be treated as meaningful if its absolute value is
larger than approximately twice its standard error. See also
``stderrkurtosis``.
"""
n, total = stats._std_moment(data, m, s, 4)
assert n >= 0
v.assert_(lambda x: x >= 1, total)
if n < 4:
raise StatsError('sample kurtosis requires at least 4 data points')
q = (n-1)/((n-2)*(n-3))
gamma2 = v.div(total, n)
# Don't do this:-
# kurt = v.mul((n+1)*q, gamma2)
# kurt = v.sub(kurt, 3*(n-1)*q)
# Even though the above two commented out lines are mathematically
# equivalent to the next two, and cheaper, they appear to be
# slightly less accurate.
kurt = v.sub(v.mul(n+1, gamma2), 3*(n-1))
kurt = v.mul(q, kurt)
if v.isiterable(kurt): out_of_range = any(x < -2 for x in kurt)
else: out_of_range = kurt < -2
if m is s is None:
assert not out_of_range, 'kurtosis failed: %r' % kurt
# This is a "should never happen" condition, hence an assertion.
else:
# This, on the other hand, can easily happen if the caller
# gives junk values for m or s. The difference between a junk
# value and a legitimate value can be surprisingly subtle!
if out_of_range:
import warnings
warnings.warn('calculated kurtosis out of range')
return kurt
def skewness(data, m=None, s=None):
"""skewness(data [,m [,s]]) -> sample skewness of data.
The skewness, or third standardised moment, of data is the degree to
which it is skewed to the left or right of the mean.
This returns g₁ "g\\N{SUBSCRIPT ONE}", the sample skewness. For the
population skewness, see function ``pskewness``.
WARNING: The mathematical terminology and notation related to
skewness is often inconsistent and contradictory. See Wolfram
Mathworld for further details:
http://mathworld.wolfram.com/Skewness.html
>>> skewness([1.25, 1.5, 1.5, 1.75, 1.75, 2.5, 2.75, 4.5])
... #doctest: +ELLIPSIS
1.71461013539878...
If you already know one or both of the population mean and standard
deviation, you can pass the mean as optional argument m and/or the
standard deviation as s:
>>> skewness([1.25, 1.5, 1.5, 1.75, 1.75, 2.5, 2.75, 4.5], m=2.25, s=1)
... #doctest: +ELLIPSIS
1.47132881615329...
CAUTION: "Garbage in, garbage out" applies here. You can pass
any values you like as ``m`` or ``s``, but if they are not
sensible estimates for the mean and standard deviation, the
result returned as the skewness will likewise not be sensible.
If m or s are not given, or are None, they are estimated from the data.
If data is an iterable of sequences, each inner sequence represents a
row of data, and ``skewness`` operates on each column. Every row must
have the same number of columns, or ValueError is raised.
>>> data = [[0, 1],
... [1, 5],
... [2, 6],
... [5, 7]]
...
>>> skewness(data) #doctest: +ELLIPSIS
[1.19034012827899..., -1.44305883553164...]
Similarly, if either m or s are given, they must be either a single
number or have the same number of items as the data:
>>> skewness(data, m=[2.5, 5.0], s=2) #doctest: +ELLIPSIS
[-0.189443057077845..., -2.97696232550900...]
A negative skewness indicates that the distribution's left-hand tail is
longer than the tail on the right-hand side, and that the majority of
the values (including the median) are to the right of the mean. A
positive skew indicates that the right-hand tail is longer, and that the
majority of values are to the left of the mean. A zero skew indicates
that the values are evenly distributed around the mean, often but not
necessarily implying the distribution is symmetric.
CAUTION: As a rule of thumb, a non-zero value for skewness
should only be treated as meaningful if its absolute value is
larger than approximately twice its standard error. See also
``stderrskewness``.
"""
n, total = stats._std_moment(data, m, s, 3)
assert n >= 0
if n < 3:
raise StatsError('sample skewness requires at least three items')
skew = v.div(total, n)
k = math.sqrt(n*(n-1))/(n-2)
return v.mul(k, skew)