What does a normal distribution show?: Most heights are near the average. Fewer are very short or very tall. The bell curve models this pattern.
Example: students’ heights
Given in this example
Heights follow a normal model with mean 170 cm and SD 10 cm. These numbers are supplied for this example; we are not calculating them.
What do they mean?
170 cm is the average height: the centre of the bell. 10 cm measures how spread out the heights are. Larger SD means more variation.
Try the sliders. They change this example, not the normal-distribution rule.
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Mean, median and mode?
All at the centre.
Example: this curve has mean = median = mode = 170.
What is probability?
Area under the curve. Total area = 1.
Example: half is below 170 → probability 0.5.
Is every data set normal?
No. Example: waiting for a bus.
Imagine most people wait 1–5 minutes, but a few wait 20–30 minutes. The data bunches near short waits, with a long tail towards longer waits.
That shape is not a symmetric bell, so a normal model would not fit this example.
In an exam, use a normal model when the question says the data is normally distributed.
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Example: test scores
Read what is given
Test scores follow a normal model. The question gives mean 70 marks and SD 8 marks.
Mean = average score
The bell is centred at 70 marks. A score of 70 is at the average.
SD = how spread out the scores are
One SD either side: 70 − 8 = 62 and 70 + 8 = 78 marks. These are reference points, not the lowest and highest possible scores.
Move SD: smaller = scores closer together; larger = scores more spread out.
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Variance given instead of SD?
Take the square root before entering the calculator.
Example: variance = 25 → SD = √25 = 5.
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Example: scoring above 70 marks
Read what is given
Test scores follow a normal model with mean 60 marks and SD 10 marks. Find the chance of scoring more than 70 marks.
Shade the scores we want
70 is above the average of 60. Mark 70 and shade everything to its right. Select Above 70 below.
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Half on each side
The mean splits the area in half: 50% below, 50% above.
Below the mean?
Exactly 0.5.
Example: P(X < 60) = 0.5.
Probability at one exact value?
For a continuous model, P(X = a) = 0.
Example: P(X = 60) = 0. So < and ≤ give the same normal probability.
Can every interval be treated as symmetric?
Only if its endpoints are equally far from the mean.
Example: 50–70 is symmetric around 60; 50–75 is not.
1. Imagine 100 students
Their scores follow a normal model with an average of 60 marks.
We are told 10 students score below 50 marks.
That is 10 out of 100 = 10% = 0.10.
2. Find the matching tail
Continue the example in card 1.
The average is 60 marks. A score of 50 is 10 marks lower.
The matching point is 60 + 10 = 70 marks.
Equal tails have equal areas, so 10% also score above 70.
Given heights → use the normal rule
Our height example
Given: mean 170 cm, SD 10 cm. One SD is a distance of 10 cm from the average.
The rule to remember
For any normal model: about 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD. These percentages come from the normal curve, not from 170 or 10.
Tap 1, 2 or 3 SD. Coral shows the percentage inside that height range.
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| Distance from average | Height range | Middle area | Outside total | Each tail |
|---|---|---|---|---|
| 1 SD: 170 ± 10 | 160–180 cm | About 68% | 32% | 16% |
| 2 SD: 170 ± 20 | 150–190 cm | About 95% | 5% | 2.5% |
| 3 SD: 170 ± 30 | 140–200 cm | About 99.7% | 0.3% | 0.15% |
Example: taller than 190 cm
Locate 190
190 = 170 + 2 × 10: two SD above the given mean.
Find the outside area
About 95% is in the middle. 5% is outside.
Keep one tail
Split the outside equally: 5% ÷ 2 = 2.5% above 190.
What does “within 1 SD” mean?
From one SD below to one SD above the mean.
Here: 170 − 10 = 160 to 170 + 10 = 180 cm. About 68 out of 100 students would fall in this range.
Estimate or calculate?
68 · 95 · 99.7 gives approximate percentages. Use the calculator method in the next micro for a calculated probability.
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Find the size of one SD
Read the question
Mean speed 64 km/h. A speed of 74 is two standard deviations above the mean. Find the SD.
Find the distance
74 − 64 = 10 km/h.
Split into two equal steps
10 ÷ 2 = 5 km/h. Each SD is 5.
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Below the mean?
Use the positive distance.
Example: mean 64; speed 49 is three SD below it → σ = (64 − 49)/3 = 5.
Find the value instead?
Add or subtract k × SD.
Example: mean 64, SD 5; two SD above = 64 + 2 × 5 = 74.
Read the middle and the tails
Start with the middle
The question says the middle 80% is between two endpoints.
Split the remaining 20%
100% − 80% = 20% outside. Symmetry gives 10% in each tail.
Area below the upper end
10% left tail + 80% middle = 90%. The remaining 10% is above the upper end.
The next micro shows how to calculate the endpoint values.
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Middle 60% instead?
40% outside → 20% in each tail.
Below the upper end: 20% + 60% = 80%.
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AA exam practice: compare normal curves without a calculator
Equal standard deviations mean equal spread. Shifting the mean shifts the curve without changing areas at matching distances from the mean. Use symmetry and supplied probabilities.
Worked example [5 marks]
X follows N(50, 5²) and Y follows N(80, 5²). You are given P(X < 45) = 0.1587. Without a calculator:
(a) Find P(Y > 85). [2]
(b) Find P(45 < X < 55). [2]
(c) Describe how the two curves compare. [1]
Step by step
- (a) Each cutoff is five units from its mean. Reflect the right tail to the left tail.
- (b) Remove the two equal tails.
- (c) The curves have equal spread and equal peak height. The Y curve is shifted 30 units to the right.
Final answer
(a) 0.1587. (b) 0.6826. (c) Same shape, shifted 30 units right.
Try the next question before opening its solution.
Your turn [4 marks]
W follows N(100, 8²). Without a calculator, use the 68–95–99.7 rule to estimate:
(a) P(W > 116). [2]
(b) P(92 < W < 108). [2]
Step by step
- (a) The cutoff is two standard deviations above the mean. Split the remaining 5% into equal tails.
- (b) The interval extends one standard deviation on either side of the mean.
Final answer
(a) Approximately 0.025. (b) Approximately 0.68.
Exam tip: These estimates and symmetry arguments need no normalcdf or z-transformation. Use ≈ when applying the empirical percentage rule.
Additional AA exam practice
IB-style question — use the properties
X ~ N(40, σ²).
Write down P(X < 40) and P(X > 40).
Step by step
- Symmetric about the mean 40.
- The other half.
Final answer
P(X < 40) = P(X > 40) = 0.5.
IB-style question — describe the shading
X ~ N(60, 5²).
Describe how to represent P(X > 68) on a sketch.
Step by step
- Mark the mean and the boundary.
- Shade to the right of 68.
Final answer
Draw the bell centred at 60, mark 68 to the right of centre, and shade the tail beyond 68.
IB-style question — compare
Two groups' marks are normal with the same mean but Group A has σ = 4 and Group B has σ = 9.
Which curve is taller and narrower, and what does that mean?
Step by step
- Smaller σ → taller, narrower.
- Interpret.
Final answer
Group A's curve is taller and narrower (σ = 4), meaning its marks are more clustered/consistent.
IB-style question — symmetry
X ~ N(50, σ²) and P(X < 44) = 0.18.
Find P(X > 56).
Step by step
- 44 and 56 are each 6 from the mean 50.
- Mirror points have equal tails (symmetry).
Final answer
P(X > 56) = 0.18.