Moments and Stats¶

The moments of a probability distribution encode its shape: the mean locates the center, the variance measures spread, skewness quantifies asymmetry, and kurtosis captures tail weight. Rather than estimating these from samples, scipy.stats distribution objects compute them analytically from the distribution's parameters, giving exact values with no sampling error. This page defines each moment and demonstrates the corresponding scipy.stats methods.

Mean and Variance¶

The mean (first moment about the origin) of a continuous random variable \(X\) with PDF \(f\) is:

\[ \mu = \mathbb{E}[X] = \int_{-\infty}^{\infty} x \, f(x) \, dx \]

The variance (second central moment) measures the expected squared deviation from the mean:

\[ \sigma^2 = \operatorname{Var}(X) = \mathbb{E}\!\left[(X - \mu)^2\right] = \int_{-\infty}^{\infty} (x - \mu)^2 \, f(x) \, dx \]

The standard deviation is \(\sigma = \sqrt{\sigma^2}\).

Computing Mean, Variance, and Standard Deviation¶

import scipy.stats as stats

# Normal distribution with μ=5, σ=2
dist = stats.norm(loc=5, scale=2)

print(f"Mean:     {dist.mean():.4f}")       # 5.0
print(f"Variance: {dist.var():.4f}")         # 4.0
print(f"Std dev:  {dist.std():.4f}")         # 2.0

These methods return exact analytical values. For the normal distribution, .mean() returns loc, .var() returns scale**2, and .std() returns scale.

Example: Gamma Distribution¶

The gamma distribution with shape \(a\) and scale \(\theta\) has mean \(a\theta\) and variance \(a\theta^2\):

dist = stats.gamma(a=3, scale=2)
print(f"Mean:     {dist.mean():.4f}")       # 6.0  (3 * 2)
print(f"Variance: {dist.var():.4f}")         # 12.0 (3 * 4)

Skewness and Kurtosis¶

The skewness measures asymmetry of the distribution about its mean:

\[ \gamma_1 = \mathbb{E}\!\left[\left(\frac{X - \mu}{\sigma}\right)^{\!3}\right] \]

A positive value indicates a right tail longer than the left; a negative value indicates the opposite. A symmetric distribution has \(\gamma_1 = 0\).

The excess kurtosis measures the heaviness of the tails relative to the normal distribution:

\[ \gamma_2 = \mathbb{E}\!\left[\left(\frac{X - \mu}{\sigma}\right)^{\!4}\right] - 3 \]

The subtraction of 3 makes the normal distribution the reference point with \(\gamma_2 = 0\). Positive excess kurtosis (leptokurtic) indicates heavier tails; negative (platykurtic) indicates lighter tails.

Excess vs raw kurtosis

SciPy reports excess kurtosis (subtracting 3). Some other software reports raw kurtosis without the subtraction. Always check the convention before comparing values.

The .stats() Method¶

The .stats() method returns multiple moments in a single call. The moments keyword controls which are computed, using a string of characters:

Character	Moment
`m`	Mean
`v`	Variance
`s`	Skewness
`k`	Excess kurtosis

dist = stats.norm(loc=0, scale=1)
mean, var, skew, kurt = dist.stats(moments='mvsk')
print(f"Mean={mean:.2f}, Var={var:.2f}, Skew={skew:.2f}, Kurt={kurt:.2f}")
# Mean=0.00, Var=1.00, Skew=0.00, Kurt=0.00

Example: Exponential Distribution¶

The exponential distribution with rate \(\lambda\) (scale \(= 1/\lambda\)) is right-skewed:

dist = stats.expon(scale=1.0)
mean, var, skew, kurt = dist.stats(moments='mvsk')
print(f"Mean={mean:.2f}, Var={var:.2f}, Skew={skew:.2f}, Kurt={kurt:.2f}")
# Mean=1.00, Var=1.00, Skew=2.00, Kurt=6.00

The skewness of 2 confirms the strong right skew, and the excess kurtosis of 6 indicates substantially heavier tails than the normal.

The .moment() Method¶

The .moment(n) method computes the \(n\)-th non-central moment \(\mathbb{E}[X^n]\):

dist = stats.norm(loc=0, scale=1)

# First non-central moment = mean
print(f"E[X]   = {dist.moment(1):.4f}")    # 0.0

# Second non-central moment = E[X^2] = Var + mean^2
print(f"E[X^2] = {dist.moment(2):.4f}")    # 1.0

# Fourth non-central moment
print(f"E[X^4] = {dist.moment(4):.4f}")    # 3.0

Non-central vs central moments

The .moment(n) method returns \(\mathbb{E}[X^n]\), not \(\mathbb{E}[(X-\mu)^n]\). To get the \(n\)-th central moment, compute it from non-central moments. For example, \(\operatorname{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2\).

The .entropy() Method¶

The differential entropy of a continuous distribution measures its uncertainty:

\[ h(X) = -\int_{-\infty}^{\infty} f(x) \ln f(x) \, dx \]

# Normal distribution: entropy = 0.5 * ln(2πeσ²)
dist = stats.norm(loc=0, scale=1)
print(f"Entropy: {dist.entropy():.4f}")     # 1.4189

Higher entropy indicates greater spread or uncertainty in the distribution.

Method Summary¶

Method	Returns	Notes
`.mean()`	\(\mathbb{E}[X]\)	First moment about the origin
`.var()`	\(\operatorname{Var}(X)\)	Second central moment
`.std()`	\(\sigma\)	Square root of variance
`.stats(moments='mvsk')`	Tuple of selected moments	Any subset of `m`, `v`, `s`, `k`
`.moment(n)`	\(\mathbb{E}[X^n]\)	\(n\)-th non-central moment
`.entropy()`	\(h(X)\)	Differential entropy (nats)

Summary¶

The moments of a distribution — mean, variance, skewness, and kurtosis — summarize its center, spread, asymmetry, and tail behavior. In scipy.stats, the .mean(), .var(), .std(), and .stats() methods compute these analytically from the distribution parameters, while .moment(n) provides arbitrary non-central moments. These exact values serve as benchmarks when comparing against sample estimates computed from data.