Example: battery lifetimes
What is given?
Battery lifetimes follow a normal model. Mean = 100 hours. SD = 15 hours.
Question A: find a chance
“What is the chance a battery lasts less than 120 hours?”
120 hours is given in the question. Use normalcdf to find the chance below it.
Question B: find a lifetime
“What lifetime has 90% of batteries below it?”
Here 90% is given, but the lifetime is unknown. Use invNorm to find it, in hours.
Select Below 120: coral shows batteries lasting less than the given 120 hours.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
What are −1E99 and 1E99?
Calculator stand-ins for far left and far right.
Below 120: lower = −1E99; upper = 120.
Where do μ and σ go?
After the bounds in normalcdf.
Mean 100, SD 15: normalcdf(lower,upper,100,15).
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Find a probability
Read the question
Scores have mean 100, SD 15. Find the chance of a score below 120.
Enter the four inputs
normalcdf(−1E99,120,100,15)
Lower end · upper end · mean · SD. −1E99 means far left.
Read the result
0.909 ≈ 90.9%. Probability × 100 gives the percentage.
Below 120: coral is the area we want.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Stop wasting time on topics you know
Our AI identifies your weak areas and focuses your study time where it matters. No more overstudying easy topics.
Which part of the bell curve do I want?: Below → left. Above → right. Between → middle. Try each button below.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
| Question wording | Meaning | Shade |
|---|---|---|
| Below 120 | Less than 120 | Left of 120 |
| Above 120 | More than 120 | Right of 120 |
| Between 85 and 120 | From 85 to 120 | Middle |
| Lowest 8% | 8% below the cutoff | Left tail |
| Highest 2% | 2% above the cutoff | Right tail |
Highest 2%: where do I shade?
The highest values are on the right. Shade the small right-hand tail.
Lowest 8%: where do I shade?
The lowest values are on the left. Shade the small left-hand tail.
Find the height for the tallest 10%
Read the question
Mean height 170 cm, SD 6 cm. Where do the tallest 10% begin?
Use the area on the left
10% is above the cutoff. 100% − 10% = 90% below. Enter 0.90.
Find the cutoff
invNorm(0.90,170,6) = 177.689… cm.
Answer: 178 cm to 3 significant figures.
Set left area to 90%. The remaining 10% is the tallest group.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Fastest or slowest?
For times: fastest = smaller values; slowest = larger values.
Slowest 15%: enter 0.85 as the left area.
Top 2%: which calculator area?
2% above the cutoff → 98% below it.
invNorm needs the area below, so enter 0.98.
Bottom 8%: which calculator area?
8% below the cutoff is already the left area.
Enter 0.08 in invNorm.
See how examiners mark answers
Answer exam-style questions with model answers. Learn exactly what earns marks and what doesn't.
Find the middle 80%
Read the question
Diameters: mean 40 mm, SD 1.5 mm. Find the middle 80% interval.
Split the outside
20% outside → 10% per tail. Left areas at the two ends: 0.10 and 0.90.
Calculate both ends
invNorm(0.10,40,1.5) ≈ 38.1
invNorm(0.90,40,1.5) ≈ 41.9
Answer: 38.1–41.9 mm.
Coral = middle 80%. Each tail = 10%.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
An interval is not centred?
Subtract its area from the area below its known upper endpoint.
Example: mean 100, SD 15; P(k < X < 120) = 0.68.
P(X < 120) = 0.908788…
Left area at k = 0.908788… − 0.68 = 0.228788…
k = invNorm(0.228788…,100,15) ≈ 88.9.
Do not assume every 68% interval is μ ± σ.
Find the half-width m?
Find one endpoint, then its distance from the mean.
Central 60%, mean 4, SD 0.25: upper left area = 0.80. Upper endpoint = 4.210405…; m = 4.210405… − 4 = 0.210.
Chance → expected number
Read the question
200 batteries. Lifetimes have mean 100 hours, SD 15 hours. How many last more than 120 hours, on average?
Find the chance for one battery
normalcdf(120,1E99,100,15) = 0.0912112…
About 9.12% last beyond 120 hours.
Multiply by 200
200 × 0.0912112… ≈ 18.2 batteries on average. Use the unrounded probability.
Coral = batteries lasting more than 120 hours.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Can an expected number be a decimal?
Yes: it is a long-run average.
Example: 18.2 items is an expected count, even though an actual count is whole.
Add a transfer time?
Find the arrival cutoff, then add the transfer time.
Example: arrival mean 20 minutes late, SD 10; 99% cutoff = invNorm(0.99,20,10) = 43.263… minutes late. Add 10 minutes to change → 53.263… minutes after the scheduled arrival.
If asked for the earliest WHOLE minute meeting at least 99%, round UP to 54 minutes. A nearest-minute instruction is different.
Study smarter, not longer
Most students waste 40% of study time on topics they already know. Our AI tracks your progress and optimizes every minute.
A group count → a cutoff
Read the question
80 animals. Masses have mean 30 kg, SD 4 kg. The heaviest 12 animals are above a cutoff. Find it.
Turn 12 animals into an area
12 ÷ 80 = 0.15 above the cutoff.
Left area = 1 − 0.15 = 0.85.
Find the mass
invNorm(0.85,30,4) ≈ 34.1 kg.
85% below the cutoff; the heaviest 15% above it.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
One example, four exam tasks
Read the situation
Bag masses: mean 250 g, SD 12 g. Use the same mean and SD in each part.
Chance above 270 g
normalcdf(270,1E99,250,12) ≈ 0.0478.
Chance between 240 and 260 g
normalcdf(240,260,250,12) ≈ 0.595.
Expected bags below 235 g, out of 150
150 × normalcdf(−1E99,235,250,12) ≈ 15.8 bags.
Cutoff for the heaviest 8%
Heaviest = right-hand tail: 8% is above the cutoff.
invNorm needs the percentage below it: 100% − 8% = 92% → 0.92.
invNorm(0.92,250,12) ≈ 267 g (3 s.f.). Shade above the cutoff, to the right.
Above 270 is shaded. Try Between to compare regions.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
What if it says lightest 8%?
Lightest = left-hand tail. 8% is already below the cutoff.
Enter 0.08, not 0.92:
invNorm(0.08,250,12) ≈ 233 g (3 s.f.).
Heaviest: subtract from 1. Lightest: use the percentage directly.
Two independent measurements?
If one meets the condition with probability q, both do with probability q².
This uses the independence rule from 4.5.
Normal probability AND another event?
Use the conditional rate for that group.
P(heavy and lost) = P(heavy) × P(lost | heavy).
Example: 0.252492… × 1/12 ≈ 0.0210.
The question specifies distance from zero?
For 0 < D < 10, enter BOTH bounds.
normalcdf(0,10,26,7) ≈ 0.0110.
The next part counts qualifying items?
Carry the unrounded normal probability into the binomial method already taught in 4.8.2. Do not round it first.
Practice with exam-style questions
Answer exam-style questions and get AI feedback that shows you exactly what examiners want to see in a full-marks response.
Before you submit
Check the spread
N(100,15²) means SD 15, variance 225. Enter 15.
Check the region
Below → left. Above → right. invNorm always needs the left area.
Check the answer
Probability: 0 to 1. Measurement: include units. Expected number: group size × probability.
Check the shaded side matches your question.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Normal values have no ±1 boundary shift
This is a continuous model.
Example: P(X > 120) uses 120, not 119. Binomial count rules do not apply.
How much working?
Write the event and calculator inputs before the result.
Example: P(X > 120) = normalcdf(120,1E99,100,15) ≈ 0.0912.
Very small probability?
Keep significant figures.
Example: 0.000303 = 3.03 × 10⁻⁴, not 0.000.
Variance is not SD
N(100,15²) means SD = 15, variance = 225.
Enter 15 in normalcdf.
Probability or expected number?
A probability is between 0 and 1.
Among 200 items with probability 0.073, expected count = 200 × 0.073 = 14.6.
A percentage requested?
Multiply the probability by 100.
Example: 0.073 → 7.3%.
A sketch requested?
Include a recognisable bell, labelled mean and cutoff, and the correct shaded region.
Add independent normal variables before finding a probability
Cross-topic HL connection: Topic 4.15 teaches combinations of independent normal variables. For a sum, add means and variances; then take the square root to obtain the SD.
A two-stage journey
Independent travel times are A ~ N(8, 1.5²) and B ~ N(12, 2²), in minutes. Find P(A + B > 24).
Step by step
- Add means.
- Add variances, then find the SD.
- Use the new distribution on the GDC.
Final answer
About 0.0548. If a fixed three-minute break is added, the mean becomes 23 but the SD stays 2.5.
The sum has mean 20 and SD 2.5. The right-hand shaded tail is the probability above 24.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Feeling unprepared for exams?
Get a clear study plan, practice with real questions, and know exactly where you stand before exam day. No more guessing.
Given a measurement: which group did it come from?
Combine the normal model with conditional probability: First find the measurement probability within each group. Multiply by the chance of that group, add the paths, and only then divide.
An item from one of two production lines
60% of items come from line A and 40% from line B. Measurements are N(70, 5²) on A and N(60, 5²) on B. Given a measurement above 65, find the probability the item came from A.
Step by step
- Find the upper-tail probability within each group.
- Weight the two paths using the production shares.
- Restrict attention to the items above 65.
Final answer
About 0.888. Do not average the two means or divide by 0.6; the given condition is the high measurement.