Normal distributions | Year 12 | Mathematical Methods

01 / LEARN

The idea, step by step

A bell-shaped family

A normal distribution is symmetric about its mean. Mean, median and mode coincide. Increasing its standard deviation gives a wider, lower curve, but total area remains one. A normal variable can theoretically take any real value.

Here the notation N(mu, sigma squared) lists mean then variance. Some calculators ask for standard deviation instead; check the input label rather than copying the second parameter automatically. An implausibly large negative tail may make a normal model unsuitable for a nonnegative quantity.

Use symmetry and sensible approximations

Half the probability lies on each side of the mean. Approximately 68%, 95% and 99.7% lie within one, two and three standard deviations. These provide quick estimates and useful checks on calculated probabilities.

The empirical rule is approximate. Use a normal cumulative calculation when precision is requested. Draw the event first, and distinguish an individual measurement from a sample mean, whose spread is smaller under independent sampling.

A common trap

Reading variance as standard deviation.

Instead: With N(mean, variance), square-root the second parameter.

02 / WORKED EXAMPLES

See the thinking

Example 1

Read the parameters

$X \sim N (64, 25)$ . State its mean, standard deviation and P(X>64).

Mean is 64.
Standard deviation is $\sqrt{25} = 5$ .
Symmetry gives probability 0.5 above the mean.

64; 5; 0.5.

Example 2

Estimate a central region

A normal variable has mean 100 and standard deviation 8. Estimate P(84<X<116).

The endpoints are $100 \pm 2 (8)$ .
This is the central two-standard-deviation region.
The empirical rule gives about 95%; the calculated normal value is about 95.45%.

About 95%.

Check your understanding

$X \sim N (50, 16)$ . Find its standard deviation and median.

Show answer

4 and 50.

The square root of 16 is 4; normal symmetry puts the median at the mean.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

$X \sim N (12, 9)$ . State mean and standard deviation.
2 marks
A normal model has mean 40 and standard deviation 6. Give the central two-SD interval.
2 marks
Why is the probability at exactly the normal mean not 0.5?
2 marks
Does doubling the standard deviation change the total density area?
2 marks
A waiting-time distribution has a very long right tail. Is a symmetric normal model automatically suitable?
2 marks
$X \sim N (30, 4)$ . Estimate P(28<X<32) using the empirical rule.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

$X \sim N (12, 9)$ . State mean and standard deviation.
2 marks
The parameters are mean and variance.
12 and 3.
A normal model has mean 40 and standard deviation 6. Give the central two-SD interval.
2 marks
$40 \pm 2 (6)$ .
[28,52].
Why is the probability at exactly the normal mean not 0.5?
2 marks
A half-line has probability 0.5; one point has no area.
A singleton has probability zero.
Does doubling the standard deviation change the total density area?
2 marks
The curve widens and lowers.
No; it stays one.
A waiting-time distribution has a very long right tail. Is a symmetric normal model automatically suitable?
2 marks
The shape is inconsistent with a normal model; inspect actual behaviour.
No.
$X \sim N (30, 4)$ . Estimate P(28<X<32) using the empirical rule.
2 marks
The endpoints are one standard deviation from the mean.
About 0.68.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods: continuous random variables, normal distributions and sampling distributions of sample means. Original lessons and practice.

Official curriculum source ↗

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