STAT5002 Chap.3 The Normal Curve and Areas Under It
The Normal Curve and Areas Under It
This chapter turns a bell-shaped histogram into a model you can calculate with. The normal curve is fixed by two numbers, the mean μ for its centre and the SD σ for its spread, and the course writes it as N(μ, σ²), where the second entry is the variance. The total area under the curve is 1, so the area over any interval is the proportion of the data that falls in that interval.
Because that area has no closed form, you find it in R with pnorm: pnorm(x, mu, sigma) gives the lower tail, adding lower.tail = F gives the upper tail, and an interval is one lower tail minus another. Next comes rescaling. A value's standard unit, z = (x − μ) ÷ σ, says how many SDs it sits above or below the mean, and the area below a value on any normal curve equals the area below its z score on the standard curve N(0, 1).
Symmetry gives a second shortcut: a left tail equals the matching right tail. Together with the 68%, 95%, 99.7% rule, these let you estimate many areas with no software at all, splitting whatever lies outside the band equally between the two tails. The last part runs the question backwards.
When the percentage is given and the value is wanted, the answer is a quantile, found with qnorm(p, mu, sigma), and every normal quantile is the mean plus sigma times the matching standard quantile. Expect questions that hand you R output and ask which line answers which question, so the chapter keeps returning to the difference between a function that returns a proportion and one that returns a value.
What this chapter covers
- 01
Smoothing a density-scale histogram into a normal curve
- 02
General normal curve N(mean, variance) and the standard curve N(0, 1)
- 03
The normal curve formula and what mu and sigma control
- 04
Area under the curve as a proportion of the data
- 05
Lower tail, upper tail and interval areas with pnorm
- 06
Standard units and rescaling to the standard curve
- 07
Symmetry of the normal curve about its mean
- 08
The 68%, 95%, 99.7% rule and splitting the tails
- 09
Quantiles and percentiles with qnorm
- 10
Telling pnorm and qnorm apart in R output
Worked example · free
Share of bolts outside a length tolerance
- +1Standardise both limits: z = (39.2 − 40)/0.5 = -1.60 and z = (40.8 − 40)/0.5 = 1.60. The limits sit the same distance either side of the mean.
- +1Lower tail: pnorm(-1.6) = 0.0548, the share of bolts that are too short.
- +1Upper tail by symmetry: P(Z > 1.6) = P(Z < -1.6) = 0.0548, which pnorm(1.6, lower.tail = F) confirms.
- +1Add the two tails: 0.0548 + 0.0548 = 0.1096.
Key terms
- Normal curve
- A symmetric, bell-shaped curve fixed by a mean and an SD, written N(mean, variance) and used to model data whose histogram has that shape.
- Standard normal curve
- The normal curve with mean 0 and SD 1, written N(0, 1), onto which every general normal curve can be rescaled.
- Standard unit
- A value's distance from the mean measured in SDs, z = (x − mean)/SD; positive above the mean and negative below it.
- Lower tail area
- The area under the curve to the left of a value, equal to the proportion of data below it; pnorm returns it by default.
- Normal quantile
- The value Q with a stated proportion p of the data below it, found in R with qnorm(p, mu, sigma).
- Upper tail area
- The area under the curve to the right of a value, equal to 1 minus the lower tail area and returned by pnorm with lower.tail = F.
- Normal 68-95-99.7 rule
- A property of every normal curve: about 68%, 95% and 99.7% of the area lies within 1, 2 and 3 SDs of the mean.
The Normal Curve and Areas Under It FAQ
What is the difference between pnorm and qnorm in R?
pnorm takes a value on the data scale and returns the proportion of the curve below it. qnorm goes the other way: you give it a proportion between 0 and 1 and it returns the value with that share below. If the question gives a percentage and asks for a measurement, you need qnorm.
Why does N(10, 4) mean an SD of 2 and not 4?
The course writes a normal curve as N(mean, variance), and the variance is the SD squared. So the second number must be square-rooted before you pass it to pnorm or qnorm, which both expect the SD as their third argument.
How do I find the proportion above a value on a normal curve?
Use pnorm(x, mu, sigma, lower.tail = F), or equivalently 1 minus pnorm(x, mu, sigma). Both work because the whole area under the curve is 1, so the upper and lower tails always add to one.
When can I use the 68-95-99.7 rule instead of R?
When the cut-offs sit a whole number of SDs from the mean, such as one SD below or two SDs above. Find the share inside the band, halve what is left outside, and add or subtract tails. For other cut-offs you need pnorm.
What does a negative z score tell me?
A negative standard unit places the value below the mean, and its size counts the SDs between them. A z of -1.5 sits one and a half SDs under the centre, so by symmetry the share below it equals the share above +1.5, about 6.7%.
Exam move
Start by drawing five normal curves by hand and shading a lower tail, an upper tail, an interval, two outer tails and a central band; under each, write the single R expression that gives the shaded area. Then take one curve of your own, say mean 50 and SD 8, and compute the same area twice, once with pnorm on the original scale and once after converting to z scores, until the two answers agree without effort.
Drill the 68%, 95%, 99.7% rule by writing the percentiles it gives directly: the 2.5th, 16th, 50th, 84th and 97.5th. For quantiles, practise reading a percentage in a question and deciding whether the answer is a proportion or a value before touching R. Finally, collect three R outputs that mix pnorm and qnorm lines and label each line with the question it answers.
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