The Normal Distribution and Z-Scores

The normal distribution is a symmetric, bell-shaped curve described by two numbers: the mean μ, which sets its center, and the standard deviation σ, which sets its width.

Many measurements, such as heights and test scores, approximate this shape, and many statistical methods assume it.

The 68-95-99.7 Rule

• About 68% of values fall within 1σ of the mean.
• About 95% fall within 2σ.
• About 99.7% fall within 3σ.

Z-Score

z = (x − μ) / σ

• z: the number of standard deviations a value sits above or below the mean. A negative z means below the mean.
• x: the value being checked.
• μ: the mean of the distribution.
• σ: the standard deviation of the distribution.

Converting values to z-scores puts them on a common scale, allowing you to compare scores measured in different units. Values with a z-score beyond about ±3 are often flagged as outliers.

Example: Exam scores are normal with mean 70 and standard deviation 10. Find the z-score for a mark of 85, the share of students scoring below 85, and the share scoring between 60 and 80.

1. z = (85 − 70) / 10 = 1.5
2. From the standard normal table, P(Z < 1.5) ≈ 0.9332, so about 93.3% score below 85.
3. Marks of 60 and 80 correspond to z = −1 and z = +1, so the share between them is about 68.3%.

from scipy.stats import norm

print(norm.cdf(1.5))
print(norm.cdf(1) - norm.cdf(-1))

Code Explanation:

• norm.cdf(1.5) returns the probability that a standard normal value is below 1.5, which is about 0.9332.
• norm.cdf(1) - norm.cdf(-1) subtracts the area below −1 from the area below +1, leaving the area between them, about 0.6827.
• The same idea is used when features are standardized before modelling: each value is replaced by its z-score.