A z-score, or standard score, expresses a value’s distance from a mean in standard-deviation units: z = (x − μ) / σ. Here x is the value, μ the mean, and σ the standard deviation. A score of 0 is at the mean; a positive score is above it, and a negative score is below it. Larger absolute scores indicate greater distance. Standardizing values this way makes observations measured on different scales easier to compare. section 3.1.1 section 14.1.11
A z-score is a distance, not a probability. When a normal distribution is an appropriate model, use its cumulative distribution function (CDF) or a z-score table to find the area below a score. Do not read the probability density function’s height at a score as the probability of observing that exact value. section 3.1.1
Find a probability with SciPy
For a standard normal distribution, `norm.cdf(1)` gives about 0.8413: roughly 84.13% of values fall at or below z = 1. The area between z = 1 and z = 1.5 is `norm.cdf(1.5) - norm.cdf(1)`, or about 9.18%. In Python: `from scipy.stats import norm; print(norm.cdf(1.5) - norm.cdf(1))`. section 3.1.1
Assess a project deadline
Suppose expected project duration is 29 weeks, its standard deviation is 1.41 weeks, and the deadline is 30 weeks. The deadline’s z-score is `(30 - 29) / 1.41`, about 0.709. Assuming normally distributed completion times, `scipy.stats.norm.cdf(0.709)` gives about a 76% chance of finishing by the deadline. This estimate depends on the normality assumption and on summing critical-path activity variances under an independence assumption. A reported z-table result of 78.52% disagrees with the CDF result; check the lookup rather than treating that gap as a rounding effect. section 13.4
In quality control, z-scores help assess how far measurements or defect-rate observations lie from their process mean and whether they warrant attention. A related z-statistic in a fitted time-series model describes an estimated coefficient’s distance from zero using its estimation uncertainty; it is not a raw observation’s distance from a data-set mean. section 14.1.11 section 7.3.10
