The normal distribution

Heights, exam marks and measurement errors often follow a normal distribution: a symmetric bell curve described by its mean μ and standard deviation σ.

The 68-95-99.7 rule

About 68% of values lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.

z-scores

A z-score says how many standard deviations a value is from the mean:

z = (x − μ) ÷ σ

A mark of 82 in an exam with mean 70 and σ = 8 has z = 1.5: well above average. z-scores compare values on different scales.

Probabilities

P(X ≤ x) is the area under the curve to the left of x. Python's math.erf (the error function) computes it:

P(X ≤ x) = ½ × (1 + erf(z ÷ √2))

from math import erf, sqrt

def below(z):
    # Rounded: erf's last digits differ a little from one computer to another.
    return round(0.5 * (1 + erf(z / sqrt(2))), 6)

for k in [1, 2, 3]:
    print("within", k, "SD:", round(below(k) - below(-k), 4))

mean, sd, mark = 70, 8, 82
z = (mark - mean) / sd
print("z =", z, " P(X <= 82) =", round(below(z), 4))

Common mistakes

• Assuming every dataset is normal. Incomes, waiting times and anything that cannot go below zero are often skewed.

Mark lesson complete