Normal PDF at a Point - Statistics for Data Analysis

Evaluate the Gaussian density f(x) = (1/(σ√(2π))) × exp(−½z²) where z=(x−μ)/σ. Four named pieces: z (the standardized value), coeff (1/(σ√(2π))), exp_term (exp(−½z²)), and their product pdf. At x=1, μ=0, σ=1 (standard normal) the density is ≈0.241971. Library: scipy.stats.norm.pdf(x, mu, sigma).

By hand

With the library

scipy.stats.norm.pdf(x, loc, scale) takes the point x first, then loc (mean μ) and scale (std σ). Returns the density value — the same formula evaluated at full floating-point precision.

naive.py

import math
x = 1
mu = 0
sigma = 1
z = (x - mu) / sigma
coeff = 1 / (sigma * math.sqrt(2 * math.pi))
exp_term = math.exp(-0.5 * z ** 2)
pdf = coeff * exp_term
print('RESULT:', round(pdf, 6))

library.py

from scipy.stats import norm
from dalib.display import set_display
set_display()

x = 1
mu = 0
sigma = 1
pdf = norm.pdf(x, mu, sigma)
print('x:', x, ' mu:', mu, ' sigma:', sigma)
print('RESULT:', round(pdf, 6))

x: 1  mu: 0  sigma: 1
RESULT: 0.241971

Implementation notes

The pdf gives a density, not a probability. f(x) can exceed 1 for narrow distributions (small σ). The probability of falling in an interval [a, b] is the integral of f, not f itself.
At x=μ the density is maximized: coeff = 1/(σ√(2π)), exp_term = 1 (since z=0). Density decreases symmetrically as |x−μ| grows.
Cross-reference: z-scores (ch02) for the standardization z=(x−μ)/σ that appears in the exponent; discrete-cdf (this chapter) for the analogous cumulative form.