General Normal Distributions (HSC SSCE Mathematics Advanced): Revision Notes

General Normal Distributions

Introduction to general normal distributions

When we examine real-world data, we often find that it follows a bell-shaped curve pattern. For instance, when tossing 20 coins repeatedly and counting the number of heads, the resulting distribution looks very much like a normal curve. However, this curve differs from the standard normal distribution in two important ways: the mean is not zero, and the standard deviation is not equal to 1.

A general normal distribution is a bell-shaped probability distribution that can be centred anywhere on the number line and can have any degree of spread. We create these distributions by transforming the standard normal distribution through stretching and shifting operations.

Transforming the standard normal distribution

To create a general normal distribution with mean $\mu$ and standard deviation $\sigma$, we apply a sequence of transformations to the standard normal distribution. Think of this as reshaping and repositioning the standard bell curve to match our data.

Stretching to accommodate the standard deviation

The first transformation adjusts the spread of the distribution to match the desired standard deviation $\sigma$.

Step 1: Horizontal stretching

We begin by stretching the standard normal curve horizontally by a factor of $\sigma$. This is achieved mathematically by replacing $x$ with $\frac{x}{\sigma}$ in the standard normal formula.

The standard normal PDF is:

$y = \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$

After horizontal stretching, this becomes:

$y = \phi\left(\frac{x}{\sigma}\right) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2\sigma^2}}$

Step 2: Vertical adjustment

However, horizontal stretching increases the area under the curve by a factor of $\sigma$. Since we need the total area to remain 1 (the total probability must equal 1), we must compress the curve vertically by multiplying by $\frac{1}{\sigma}$:

$y = \frac{1}{\sigma}\phi\left(\frac{x}{\sigma}\right) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{x^2}{2\sigma^2}}$

Now we have a proper probability density function with area 1, spread controlled by $\sigma$, but still centred at 0.

Shifting to accommodate the mean

The final transformation moves the entire curve horizontally to centre it at the desired mean $\mu$.

Horizontal translation

We shift the curve $\mu$ units to the right by replacing $x$ with $(x - \mu)$:

$y = \frac{1}{\sigma}\phi\left(\frac{x - \mu}{\sigma}\right) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

This translation doesn't change the area under the curve or the positions of the inflection points relative to the centre. The inflection points are now located at $x = \mu - \sigma$ and $x = \mu + \sigma$, exactly one standard deviation on either side of the mean.

The general normal distribution formula

Complete formula

Let $f(x)$ be the probability density function for a general normal distribution with mean $\mu$ and standard deviation $\sigma$. The transformation process gives us:

$f(x) = \frac{1}{\sigma}\phi\left(\frac{x - \mu}{\sigma}\right)$

Alternatively, writing out the exponential form in full:

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Key properties

For a general normal distribution with mean $\mu$ and standard deviation $\sigma$:

• The curve is bell-shaped and symmetric about the vertical line $x = \mu$
• The total area under the curve equals 1 (it's a valid probability density function)
• The mean, median, and mode all coincide at $x = \mu$
• The points of inflection occur at $x = \mu - \sigma$ and $x = \mu + \sigma$ (one standard deviation from the mean)
• The curve approaches but never touches the horizontal axis as $x \to \pm\infty$

Connection to the standard normal

Comparing with real data

Let's return to our coin-tossing example. When we toss 20 fair coins repeatedly and count the heads, the mean number of heads is 10 and the standard deviation is $\sqrt{5} \approx 2.236$. We can overlay a normal distribution with $\mu = 10$ and $\sigma = \sqrt{5}$ on our experimental data:

Working with z-scores

What is a z-score?

When working with general normal distributions, we need a way to standardise values so we can use standard normal distribution tables and results. This is where z-scores come in.

Conversion formulas

There are two essential formulas for converting between raw scores ($x$ values) and z-scores:

From raw score to z-score:

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

From z-score to raw score:

$x = \mu + \sigma z$

For sample data (rather than population data), we use the sample mean $\bar{x}$ and sample standard deviation $s$ instead:

$z = \frac{x - \bar{x}}{s} \quad \text{and} \quad x = \bar{x} + sz$

Example conversion table

Here's how z-scores correspond to raw scores for a distribution with mean $\mu = 10$ and standard deviation $\sigma = 2$:

$x$	4	5	6	7	8	9	10	11	12	13	14	15	16
$z$	-3	-2.5	-2	-1.5	-1	-0.5	0	0.5	1	1.5	2	2.5	3

Notice how the mean ($x = 10$) corresponds to $z = 0$, and values that are 1, 2, or 3 standard deviations away from the mean have z-scores of $\pm 1$, $\pm 2$, or $\pm 3$ respectively.

Worked examples with z-scores

Using z-scores for probability calculations

The empirical rule for general normal distributions

The 68-95-99.7 rule

One of the most useful properties of the normal distribution is the empirical rule, also called the 68-95-99.7 rule. When a random variable $X$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$:

• Approximately 68% of values lie within one standard deviation of the mean: $\mu - \sigma \leq x \leq \mu + \sigma$
• Approximately 95% of values lie within two standard deviations of the mean: $\mu - 2\sigma \leq x \leq \mu + 2\sigma$
• Approximately 99.7% of values lie within three standard deviations of the mean: $\mu - 3\sigma \leq x \leq \mu + 3\sigma$

Quartiles for normal distributions

The quartiles divide the distribution into quarters. For any normal distribution:

First quartile (Q₁): The value below which 25% of the data falls

$Q_1 = \mu - 0.67\sigma$

Second quartile (Q₂): The median, which equals the mean for normal distributions

$Q_2 = \mu$

Third quartile (Q₃): The value below which 75% of the data falls

$Q_3 = \mu + 0.67\sigma$

Interquartile range (IQR): The range of the middle 50% of the data

$\text{IQR} = Q_3 - Q_1 = 1.35\sigma$

The IQR criterion for outliers

The IQR criterion is a common method for identifying outliers in a dataset. A value is considered an outlier if it lies outside the range:

$Q_1 - 1.5 \times \text{IQR} \text{ to } Q_3 + 1.5 \times \text{IQR}$

For a normal distribution, this translates to:

$\mu - 2.70\sigma \leq x \leq \mu + 2.70\sigma$

Applying the empirical rule in practice

Key takeaways