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Module 5 · L13 of 15 ~40 min ⚡ +95 XP available

Normal Distribution Applications

You know the bell curve exists, now you'll use it. What percentage of bottles are underfilled? What mark lands a student in the top 8%? In this lesson you master two fundamental operations: finding probabilities given values (normal CDF) and finding values given probabilities (inverse normal). These skills appear in every field that touches statistics.

Today's hook, A machine fills bottles with mean 500 mL and standard deviation 8 mL. The company needs to know what percentage are underfilled (below 490 mL). Using the empirical rule, would you expect this to be closer to 10%, 1%, or 0.1%? Commit to a number before you start.

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Orient to normal applications

Work through the visible teaching and complete each embedded check.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question

Recall, your gut answer first

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A machine fills bottles with $\mu = 500$ mL and $\sigma = 8$ mL. The company wants to know what percentage of bottles are underfilled (less than 490 mL). Without calculating using the empirical rule, do you expect this percentage to be closer to $10\%$, $1\%$, or $0.1\%$? Explain your reasoning.

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The two operations

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Every normal distribution question is one of two types and only two. Lock this into muscle memory before touching any numbers.

Forward: value → probability

Given $x$, find $P(X < x)$. Standardise with $z = \frac{x-\mu}{\sigma}$, then use tables or calculator.

Inverse: probability → value

Given $P(X < x)$, find $x$. Look up $z$ first, then un-standardise: $x = \mu + z\sigma$.

Key z-values to memorise

$z_{90\%} \approx 1.28$ · $z_{95\%} \approx 1.645$ · $z_{97.5\%} \approx 1.96$ · $z_{99\%} \approx 2.33$

The fundamental identity that makes everything work: standardising converts any $X \sim N(\mu, \sigma^2)$ into $Z \sim N(0, 1)$. Every table and calculator in existence works with the standard normal.

Convert a raw normal score to a standard normal z-score.

$$z = \dfrac{x - \mu}{\sigma}$$

What you'll master

Know

Key facts

Standardise before using tables: $z = (x - \mu)/\sigma$
$P(X > x) = 1 - P(X < x)$ (complement rule)
Inverse normal: find $z$ first, then $x = \mu + z\sigma$

Understand

Concepts

Why we standardise, every normal curve has the same shape
The difference between "find the probability" and "find the cut-off value"
How quality control uses both forward and inverse operations

Can do

Skills

Calculate $P(X < x)$, $P(X > x)$, and $P(a < X < b)$
Find the value $x$ corresponding to a given probability
Solve contextual problems in manufacturing, assessment, and science

Build the application language

Work through the visible teaching and complete each embedded check.

Key terms

Normal CDFCumulative distribution function, gives the area (probability) to the left of a value.

StandardisationConverting $x$ to $z = (x - \mu)/\sigma$ to use standard normal tables.

z-scoreThe number of standard deviations a value is from the mean.

Inverse normalFinding the value $x$ given a probability, working backwards from the table.

Left-tail$P(X < x)$, cumulative probability; what tables give directly.

Right-tail$P(X > x) = 1 - P(X < x)$, use the complement rule.

Translate normal problems

Work through the visible teaching and complete each embedded check.

Using the Normal CDF, finding probabilities

core concept

To find a probability for any normal distribution, the strategy is always the same four steps:

Identify $\mu$ and $\sigma$ from the problem
Standardise the boundary value(s) using $z = \frac{x - \mu}{\sigma}$
Find the corresponding probability from a z-table or calculator
Interpret the result in context

Three probability types:

Three types of normal probability as a comparison table

The three fundamental normal probability types, all rely on the standard CDF.

Example: Exam scores are $N(72, 12^2)$. Find $P(X > 84)$.

$z = \frac{84 - 72}{12} = 1$. So $P(X > 84) = P(Z > 1) = 1 - P(Z < 1) \approx 1 - 0.8413 = 0.1587$. Approximately $15.9\%$ of students scored above 84.

Quality control context. A bolt must be between $49.5$ mm and $50.5$ mm to pass. If lengths are $N(50, 0.3^2)$: $z_{49.5} = -1.67$, $z_{50.5} = 1.67$, $P(-1.67 < Z < 1.67) \approx 0.905$. About $90.5\%$ pass inspection.

Left tail: $P(X < x) = P(Z < z)$ where $z = \frac{x-\mu}{\sigma}$; Right tail: $P(X > x) = 1 - P(Z < z)$, always use complement

Pause, copy the three normal CDF cases: left tail $P(X < x) = P(Z < z)$; right tail $P(X > x) = 1 - P(Z < z)$; middle $P(a < X < b) = P(Z < z_b) - P(Z < z_a)$, where $z = (x-\mu)/\sigma$, into your book.

Did you get this? True or false: to find $P(X > 84)$ when $P(Z < 1) = 0.8413$, the answer is $0.8413$.

Worked examples · 3 in a row, reveal as you go

Work normal-distribution examples

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PROBLEM 1 · CHOCOLATE BARS, FORWARD

Chocolate bar weights are $N(250, 5^2)$ g. Find $P(X < 242)$.

$z = \dfrac{242 - 250}{5} = -1.6$

Standardise the boundary value.

PROBLEM 2 · CHOCOLATE BARS, INVERSE

Same distribution $N(250, 5^2)$. Find the weight exceeded by $90\%$ of bars.

$P(X > x) = 0.90 \Rightarrow P(X < x) = 0.10$

Translate: "exceeded by 90%" means only 10% are above, so 10% are below.

PROBLEM 3 · COMMUTE TIMES, TWO-PART

Commute times are $N(35, 8^2)$ minutes. (a) What percentage take more than 45 min? (b) Find the 90th percentile.

$z = \dfrac{45 - 35}{8} = 1.25$; $P(X > 45) = 1 - P(Z < 1.25) \approx 1 - 0.8944 = 0.1056$

About $10.6\%$ take more than 45 minutes.

Quick check: IQ scores are $N(100, 15^2)$. What IQ score is at the 90th percentile?

Common errors · the 3 traps that cost marks

Avoid common normal-distribution traps

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Trap 01

Reading the wrong tail

$P(Z < 1) = 0.8413$ is the LEFT tail. For $P(Z > 1)$ you need $1 - 0.8413 = 0.1587$. Always check whether your table gives cumulative (left-tail) or upper-tail probabilities.

Trap 02

"Top 10%" vs. "bottom 10%"

"Top 10%" means $P(X > x) = 0.10$, so the cumulative left area is $0.90$, look up $z = 1.282$, NOT $z = -1.282$. Draw a sketch every time.

Trap 03

Forgetting to un-standardise

After finding $z$ in an inverse problem, students forget the final step: $x = \mu + z\sigma$. A bare z-score is not a weight, time, or mark, it has no units in context.

Fill in the blank: To find $P(X > x)$ using a left-tail table, you compute $1$ minus ___.

Quick-fire practice · 5 calculations

Practise and reflect

Work through the visible teaching and complete each embedded check.

For $X \sim N(100, 12^2)$, find $P(X < 112)$

For $X \sim N(100, 12^2)$, find $P(X > 88)$

Fish lengths are $N(35, 2^2)$ cm. What $\%$ are longer than 38 cm?

Exam marks $N(68, 10^2)$. Top $15\%$ get Distinction. Find minimum mark.

Components $N(50, 0.4^2)$ mm. Rejected if outside 49.2 to 50.8 mm. What $\%$ rejected?

Odd one out: Three of these are inverse normal problems and one is a forward (CDF) problem. Which is the odd one out?

Revisit your thinking

$490 = 500 - 10 = 500 - 1.25 \times 8 = \mu - 1.25\sigma$. This is between $1\sigma$ and $2\sigma$ below the mean. The left tail beyond $1\sigma$ is about $16\%$, and beyond $2\sigma$ is about $2.5\%$. So the answer lies between $2.5\%$ and $16\%$, closer to $10\%$ than to $1\%$ or $0.1\%$. The exact value (from tables) is approximately $\mathbf{10.6\%}$. The empirical rule gives a powerful first estimate before touching a calculator.

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Complete the short-answer practice

Apply the lesson methods, then compare each response with its comprehensive answer.

Focus-area checkpoint

checkpoint

Use the visible short-answer practice below during this lesson. After completing the focus area, use Checkpoint 1 for the checkpoint question bank.

Short answer

ApplyBand 43 marks

Q1. The diameters of ball bearings are $N(12.00, 0.05^2)$ mm. Bearings are acceptable if diameter is between $11.90$ mm and $12.10$ mm. (a) Find the z-scores for the lower and upper acceptance limits. (b) Find the percentage of acceptable bearings. (c) Find the percentage that are too large. (3 marks)

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ApplyBand 53 marks

Q2. Commute times are $N(35, 8^2)$ minutes. (a) What percentage take more than 45 minutes? (b) Find the 90th percentile commute time. (c) If 500 employees qualify for a work-from-home policy (commute exceeds the 90th percentile), how many employees does the company have in total? (3 marks)

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AnalyseBand 63 marks

Q3. A cereal manufacturer states boxes contain "at least 375 g." The filling machine is normally distributed. The company wants fewer than $0.5\%$ of boxes underweight. (a) If $\mu = 380$ g, find the maximum allowable $\sigma$ to 1 decimal place. (b) A proposal suggests $\mu = 385$ g, $\sigma = 4$ g. Calculate the percentage of underweight boxes. (c) Manufacturing cost increases $3\%$ per extra gram of mean fill. Evaluate which setting is more cost-effective while meeting the underweight requirement. (3 marks)

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Comprehensive answers (click to reveal)

Drill 1: $z=1$, $P \approx 0.8413$. 2: $z=-1$, $P(Z>-1)\approx 0.8413$. 3: $z=1.5$, $\approx 6.7\%$. 4: $z\approx1.036$, $x\approx78.4$. 5: $z=\pm2$, rejected $\approx4.6\%$.

Q1 (3 marks): (a) $z_{\text{lower}}=-2$, $z_{\text{upper}}=2$ [1]. (b) $P(-2<Z<2)\approx0.9545$, so $\approx95.5\%$ acceptable [1]. (c) $P(Z>2)\approx0.0228$, so $\approx2.3\%$ too large [1].

Q2 (3 marks): (a) $z=1.25$, $P(X>45)\approx10.6\%$ [1]. (b) $z\approx1.282$, $x=35+1.282(8)=45.26$ min [1]. (c) $10\%=500$ employees, total $=5000$ [1].

Q3 (3 marks): (a) $z=-2.576$, $\sigma=5/2.576\approx1.9$ g [1]. (b) $z=(375-385)/4=-2.5$, $P\approx0.62\%$, exceeds 0.5% so FAILS the requirement [1]. (c) Setting (a) at $\mu=380$ is cheaper ($\$380$ base) and meets the requirement; setting (b) at $\mu=385$ costs more AND fails, setting (a) wins [1].

Review and complete

Retrieve the central ideas, then mark the lesson complete or continue to the module quiz.

Retrieve the lesson before you leave

Translate less than, greater than and between into the correct CDF expressions.
State when a complement is needed for an upper-tail probability.
Outline the inverse-normal steps for finding a percentile cutoff.

Answer from memory first, then return to the matching Learn checkpoint to check and correct your response.

Take the full module quiz

quiz

A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.

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Mark lesson as complete

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