Normal Distribution Probability And Standardisation

Scope Label

Core 9758. This note covers the normal model, normal probabilities as areas, symmetry, and standardisation. It deliberately stops before inverse-normal parameter solving and linear combinations, which are handled in separate branches.

What This Note Assumes

You should already know:

• continuous probabilities are areas under density curves;
• $P(X=a)=0$ for a continuous random variable;
• $E(X)$ describes centre and $\mathrm{Var}(X)$ describes spread.

The normal distribution is a special continuous model with a particular symmetric bell shape.

The Normal Distribution as a Model

A normal random variable is written

$$X\sim N(\mu ,{\sigma }^2).$$

This notation says:

• $\mu$ is the mean and centre of the distribution;
• ${\sigma }^2$ is the variance;
• $\sigma$ is the standard deviation.

The second parameter is variance, not standard deviation.

For example, if a measurement is normally distributed with mean $50$ and standard deviation $2$, write

$$X\sim N(50,2^2),$$

or equivalently

$$X\sim N(50,4).$$

When standardising or using a calculator that asks for standard deviation, use $2$, not $4$.

Caption: A normal distribution is a continuous bell-shaped model controlled by its centre $\mu$ and spread $\sigma$.

Shape and Symmetry

The normal curve is:

• continuous;
• bell-shaped;
• symmetric about $x=\mu$;
• highest at the mean;
• more spread out when $\sigma$ is larger.

The total area under the curve is $1$.

Because the curve is symmetric about $\mu$,

$$P(X<\mu )=P(X>\mu )=\frac{1}{2}.$$

For any positive $a$,

$$P(X<\mu -a)=P(X>\mu +a).$$

Symmetry only applies when distances from the mean are equal. The interval

$$\mu -3<X<\mu +3$$

is symmetric about $\mu$, but

$$\mu -2<X<\mu +5$$

is not.

Standard-Deviation Landmarks

For a normal distribution, the standard deviation gives a useful scale for judging how far a value is from the centre.

Approximately:

• $68.3\%$ of values lie within one standard deviation of the mean;
• $95.5\%$ of values lie within two standard deviations of the mean;
• $99.7\%$ of values lie within three standard deviations of the mean.

Caption: The $68.3\%$, $95.5\%$, and $99.7\%$ landmarks describe spread around the mean for a normal distribution.

These percentages are useful for intuition and checking answers, but they are not a substitute for exact calculator or table work when exact probabilities are required.

Meaning of $\mu$ and $\sigma$

Changing $\mu$ shifts the curve horizontally:

• larger $\mu$ moves the curve right;
• smaller $\mu$ moves the curve left;
• the shape is unchanged if $\sigma$ stays fixed.

Changing $\sigma$ changes the spread:

• smaller $\sigma$ gives a narrower and taller curve;
• larger $\sigma$ gives a wider and flatter curve;
• the centre is unchanged if $\mu$ stays fixed.

Caption: The mean shifts the centre of the normal curve, while the standard deviation changes its spread.

Normal Probabilities as Areas

For a normal random variable, probabilities are areas under the normal curve.

The main region types are:

Region type	Probability form	Meaning
Left tail	$P(X<a)$	area to the left of $a$
Right tail	$P(X>a)$	area to the right of $a$
Central interval	$P(a<X<b)$	area between $a$ and $b$

Before calculating, identify the region. This prevents common errors such as treating $P(X>a)$ and $P(X<a)$ as interchangeable.

Endpoint inclusion does not matter:

$$P(a<X<b)=P(a\le X\le b).$$

Caption: Normal probability questions differ by the region being shaded: left tail, right tail, between two bounds, or outside an interval.

Standard Normal Distribution

The standard normal distribution is

$$Z\sim N(0,1).$$

It has:

• mean $0$;
• variance $1$;
• standard deviation $1$.

The standard normal distribution is not a separate topic from normal distribution. It is the reference scale used to compare normal variables.

Standardisation

If

$$X\sim N(\mu ,{\sigma }^2),$$

then

$$Z=\frac{X-\mu }{\sigma }\sim N(0,1).$$

Standardisation has two meanings:

• subtract $\mu$ to recentre the distribution at $0$;
• divide by $\sigma$ to measure distance in standard-deviation units.

A $z$-score answers:

How many standard deviations is this value from the mean?

Examples:

• $z=0$ means the value is at the mean;
• $z=1$ means one standard deviation above the mean;
• $z=-2$ means two standard deviations below the mean.

Caption: Standardisation recentres a normal variable at $0$ and rescales it so distances are measured in standard-deviation units.

Direct Probability Workflow

To find a direct normal probability:

1. Define the random variable.
2. State its distribution.
3. Identify $\mu$ and $\sigma$.
4. Identify the required region.
5. Standardise each boundary if working manually.
6. Use a table or calculator after the region is clear.
7. Interpret the probability in context.

For a boundary $x$,

$$z=\frac{x-\mu }{\sigma }.$$

Then rewrite the probability in terms of $Z$.

Core Example: Right-Tail Probability

Suppose

$$X\sim N(100,15^2).$$

Find

$$P(X>130).$$

The boundary is $130$. Standardising,

$$z=\frac{130-100}{15}=2.$$

So

$$P(X>130)=P(Z>2).$$

Using a table or calculator,

$$P(Z>2)\approx \mathrm{0.0228.}$$

The answer is small because $130$ is two standard deviations above the mean.

Calculator Discipline

Use the calculator only after the mathematical structure is clear.

Check:

• Did you enter the standard deviation rather than the variance?
• Did you identify whether the region is left-tail, right-tail, or central?
• Did you use the correct lower and upper bounds?
• Did you keep the context and units clear?

If

$$X\sim N(50,9),$$

then use standard deviation $3$, not variance $9$, in standardisation and calculator input.

Common Pitfalls

Pitfall	Better thinking
Using ${\sigma }^2$ as the standard deviation	The standard deviation is $\sigma$.
Forgetting that probabilities are areas	Sketch the region before calculating.
Applying symmetry to unequal distances	Check distances from $\mu$.
Treating $z$ as a meaningless number	$z$ is distance from mean in standard-deviation units.
Using the calculator first	Set up the distribution and region first.

Revision Checklist

• Can you read $N(\mu ,{\sigma }^2)$ correctly?
• Can you explain how $\mu$ and $\sigma$ affect the curve?
• Can you identify left-tail, right-tail, and central regions?
• Can you standardise a boundary value?
• Can you interpret a $z$-score?
• Can you avoid entering variance when the calculator needs standard deviation?