DATA1001 · Foundations Of Data Science
The Normal Distribution
The Normal model — the bell curve N(μ, σ²) — is the bridge from describing data to reasoning about chance. Its master device is the standard unit, or z-score: z = (x − x̄)/s tells you how many SDs a value sits from its mean, putting any value on a common scale. From there the 68–95–99.7 empirical rule lets you split a roughly Normal distribution by eye, and the Normal table (pnorm / qnorm) lets you read an exact area below a value or find the value at a given percentile. The model also frames the idea of measurement error — observed = exact + chance error — which underpins the whole inference half of the course. Master the z-score and reading areas both ways (value → area with pnorm, area → value with qnorm) and you have the engine that every later test reuses.
What this chapter covers
- 01The Normal curve N(μ, σ²) and its shape
- 02Standard units / z-scores: z = (x − x̄)/s
- 03The 68–95–99.7 empirical rule
- 04Reading areas and percentiles off the Normal table (pnorm / qnorm)
- 05Measurement error: observed = exact + chance error
Worked example: z-scores, areas and percentiles
- +1(a) z-score: z = (78 − 70)/5 = 8/5 = 1.6.
- +1(a) Area below: pnorm(1.6) ≈ 0.945, so about 94.5% of students score below 78.
- +2(b) Empirical rule: the middle 95% lies within ±2 SD of the mean, i.e. 70 ± 2×5 = 60 to 80.
- +2(c) 90th percentile: qnorm(0.90) ≈ 1.28, so the score = 70 + 1.28×5 ≈ 76.4.
Key terms
- Normal distribution
- The symmetric bell-shaped curve N(μ, σ²), fully described by its mean and SD. Many measurements and (via the central limit theorem) most sample statistics are approximately Normal, which is why it is the reference curve for inference.
- Standard units (z-score)
- z = (x − x̄)/s, the number of SDs a value sits from its mean. It puts any value on a common, unitless scale and is the key that unlocks the Normal table — and the OV−EV part of the inference engine.
- 68–95–99.7 rule
- For a roughly Normal distribution, about 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. It lets you split a distribution and sanity-check Normal-table answers by eye.
- pnorm and qnorm
- The two directions of the Normal table. pnorm(z) returns the area (probability) below a z-score; qnorm(p) returns the z-score (or, scaled, the value) at a given cumulative probability. pnorm is value → area; qnorm is area → value.
- Measurement error
- The idea that an observed measurement = the exact value + chance error. Repeated measurements of the same quantity scatter around the truth with an SD that quantifies the instrument's precision — the seed of all later standard-error reasoning.
The Normal Distribution FAQ
What is a z-score and why standardise?
A z-score, z = (x − x̄)/s, counts how many SDs a value sits above or below its mean. Standardising puts values measured in different units on one common scale, so you can compare them and read probabilities straight off the single standard Normal table. It is also the OV−EV piece of the (OV−EV)/SE engine that runs the whole inference half of the course.
When can I use the 68-95-99.7 rule?
Only when the distribution is roughly Normal (symmetric, bell-shaped). Then about 68% of the data lie within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD. It is a fast way to give a range or sanity-check a table answer, but it is an approximation — for exact areas or non-round z-scores use pnorm/qnorm.
What's the difference between pnorm and qnorm?
They run in opposite directions. pnorm(z) takes a z-score and returns the area (probability) below it — value to area. qnorm(p) takes a probability and returns the z-score at that percentile — area to value. "What % score below 78?" uses pnorm; "what is the 90th-percentile score?" uses qnorm. Mixing them up is the classic Normal-model error.
How does measurement error fit in?
DATA1001 frames a measurement as observed = exact + chance error. If you weigh the same mass several times, the readings scatter around the true value, and the SD of those repeats measures the instrument's precision. This is the conceptual seed of the standard error: a statistic, like a single measurement, has a predictable spread around the quantity it estimates.
Exam move
Make the z-score automatic: standardise first (z = (x − x̄)/s), then decide which direction you need. For "how much is below / above / between", go value → area with pnorm and the empirical rule; for "what value sits at this percentile", go area → value with qnorm and rebuild raw units as x = mean + z×SD. Sanity-check every table answer against 68–95–99.7. Keep the measurement-error framing (observed = exact + chance error) in mind, because it is the bridge to the standard error and the inference engine in the back half of the course.
