Theory
The curve with a measuring tape inside
CampusPulse heights: approximately normal, mean 168 cm, SD 6 cm.
Without measuring anyone new, statistics can now announce: about 68% of students stand between 162 and 174. About 95% between 156 and 180. A 186 cm student? Rarer than 1 in 40.
How? The normal curve is not just a shape: it ships with a built-in measuring tape, and today you learn to read it: first in percentages, then in the z-scores that make any two scales comparable.
Theory
The 68-95-99.7 dartboard
Picture the bell as a dartboard for nature's throws, rings drawn at 1, 2 and 3 SDs from the bullseye (the mean):
- Inner ring (±1σ): catches about 68% of throws.
- Middle ring (±2σ): about 95%.
- Outer ring (±3σ): about 99.7%: nearly everything.
Beyond the third ring lives the rare 0.3%: the flag for outliers. Three numbers, and they never change for any normal curve.
Theory
The empirical rule, worked
Heights: μ = 168, σ = 6.
- ±1σ: 162 to 174 → ~68% of students.
- ±2σ: 156 to 180 → ~95%.
- ±3σ: 150 to 186 → ~99.7%.
Tail questions use symmetry: if 95% sit inside 156..180, then 5% sit outside, split equally: 2.5% above 180, 2.5% below 156.
That halving step is where most marks are lost: outside-the-ring probability always splits between two tails.
Theory
The z-score: distance in SD units
For any value x:
z = (x - μ) / σ
z counts how many SDs x sits from the mean: positive above, negative below.
Height 180: z = (180 - 168)/6 = +2: two rings out.
Height 159: z = (159 - 168)/6 = -1.5.
z converts every normal scale to the standard normal (μ = 0, σ = 1): one universal curve, for which detailed tables (and R's pnorm, Unit 5) give areas for any z, not just whole rings.
Think first
The fairer topper
Riya scored 72 in Statistics (class: μ = 60, σ = 6). Aman scored 80 in Programming (class: μ = 75, σ = 10). Raw marks say Aman. Compute both z-scores before tapping: who actually performed more exceptionally?
Show the answer
Riya: z = (72 - 60)/6 = +2.0: two SDs above her class.
Aman: z = (80 - 75)/10 = +0.5: barely half an SD up.
Riya, decisively: only ~2.5% of her class reached where she did; Aman sits around the top third. Raw scores compare against different mountains; z-scores put everyone on the same one. This standardisation is exactly how boards scale marks across easy and hard papers.
Quiz
Heights are normal with μ = 168, σ = 6. Approximately what percentage of students are TALLER than 180 cm?
- About 2.5%: 180 is +2σ, 95% sit within ±2σ, and the remaining 5% splits between the two tails
- About 5%: everything outside ±2σ is above 180
- About 32%: 100 minus 68
- About 47.5%
Show the answer
About 2.5%: 180 is +2σ, 95% sit within ±2σ, and the remaining 5% splits between the two tails
180 = μ + 2σ. Inside ±2σ: 95%. Outside: 5%, but split by symmetry: 2.5% above 180, 2.5% below 156. Option B forgets the halving (the single most common bell-curve error); option C mixes up the rings (32% is what lies outside ±1σ, both tails together). Draw the bell, shade the tail, halve: three steps, full marks.
Watch out
Three bell-curve slips
Unhalved tails: outside-percentage always splits two ways: shade the picture before answering.
Wrong data: the rule is for (approximately) NORMAL data: applying 68-95-99.7 to skewed screen time misfires: check the shape (histogram, or last lesson's mean-vs-median test) first.
Sign carelessness: z = -1.5 means BELOW the mean; dropping the minus flips your answer to the wrong side of the curve.
Theory
One ruler, everything ahead
Everything measurable in this subject now reduces to "how many SDs out?": individual students (z with σ), survey results (z with the standard error, per the CLT), quality control limits (factories alarm at ±3σ), even IQ scores (defined as μ = 100, σ = 15). In Unit 5, R computes these areas exactly with pnorm() and draws the curve you have been imagining. The tape measure is now yours.
Summary
Key takeaways
- Empirical rule: ~68% within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ: for normal data only.
- Tail questions: outside-percentage halves by symmetry (95% inside → 2.5% per tail).
- z = (x - μ)/σ: distance from the mean in SD units; negative = below.
- z-scores standardise different scales: compare performances via z, not raw marks.
- The standard normal (μ = 0, σ = 1) is what z maps everything onto; tables/pnorm give exact areas.
- |z| > 3 is rare (0.3%): the outlier flag.
- Memory hook: the 68-95-99.7 dartboard.