When the model is useful
Normal models describe many measurement errors and arise as approximations for sums or sample means under central-limit conditions. They are not justified simply because a dataset is large; check the data-generating process, skew, outliers, and dependence. See the central limit theorem, sampling distributions, and confidence intervals.
Convert a value into a probability statement
The normal distribution is continuous: probabilities are areas under its density, so P(X=x)=0 for any single exact value. For X~N(70,10²), a score of 85 has z=(85−70)/10=1.5. The probability of a score below 85 is the standard-normal area to the left of 1.5, about 0.933. A z-score tells distance from the model mean in standard-deviation units; it does not itself give the probability until you use the normal curve or a table.
A normal model is symmetric, so mean, median, and mode coincide at μ. The 68–95–99.7 rule describes approximate probability within one, two, and three σ for this model. It should not be applied to raw data just because a histogram looks vaguely bell-shaped or the sample is large. Extreme tails, hard bounds, mixtures of groups, and dependence can make the model unsuitable. A sampling distribution can be approximately normal even when individual data are not; see the central limit theorem.
Try this above
Change μ to shift the curve and σ to alter its spread. Compare the shaded areas within one and two standard deviations.
The normal model's density has the form proportional to exp(−(x−μ)²/(2σ²)); increasing σ spreads the curve while keeping total area one. A transformation to z units changes location and scale but preserves ordering. If parameters are estimated from data, the z-score is also estimated and inference should account for that uncertainty. Normal approximations are useful for calculations, not a substitute for examining residuals or checking whether the measurement process plausibly yields symmetric, unbounded errors.
When comparing measurements on different normal scales, standardisation removes units: z=1.5 means the value is 1.5 standard deviations above that model's mean whether the original unit is points, centimetres, or seconds. It does not make different populations identical or remove skew. Standard scores are comparable only when each distribution is a defensible model and the reference population is clear. A percentile based on the normal model can be wrong when the tails are heavier than normal, even if the centre looks bell-shaped.