AOS4 Topic 2: Normal Distribution

The standard normal distribution

The simplest form of the normal distribution is a random variable with probability density function f given by:

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

The domain of \(f\) is \(\mathbb{R}\).

Because it is the simplest form of the normal distribution, it is given a special name: the standard normal distribution. The graph of the standard normal distribution is as shown.

The graph of the standard normal probability density function \( f \) is symmetric about \( x = 0 \), since \( f(-x) = f(x) \). That is, the function \( f \) is even.

The line \( y = 0 \) is an asymptote: as \( x \rightarrow \pm \infty \), \( y \rightarrow 0 \). Almost all of the area under the probability density function lies between \( x = -3 \) and \( x = 3 \).

The mean and standard deviation of the standard normal distribution:

It can be seen from the graph that the mean and median of this distribution are the same, and are equal to 0. While the probability density function for the standard normal distribution cannot be integrated exactly, the value of the mean can be verified by observing the symmetry of the two integrals formed below. One is just the negative of the other.

E(X) = \(\int_{-\infty}^{\infty} x f(x) dx = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} x e^{-\frac{1}{2} x^2} dx = \frac{1}{\sqrt{2\pi}} \left(\int_{\infty}^{0} x e^{-\frac{1}{2} x^2} dx + \int_{0}^{-\infty} x e^{-\frac{1}{2} x^2} dx\right)\)

Thus the mean, \( E(X) \), of the standard normal distribution is 0.

What can be said about the standard deviation of this distribution? It can be shown that \( E(X^2) = \int_{-\infty}^{\infty} x^2 f(x) dx = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} x^2 e^{-\frac{1}{2} x^2} dx = 1 \)

Therefore \( \text{Var}(X) = E(X^2) - E(X)^2 = 1 - 0 = 1 \) and \( \text{sd}(X) = \sqrt{\text{Var}(X)} = \sqrt{1} = 1 \).

Standard Normal Distribution

A random variable with the standard normal distribution has probability density function:

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

The standard normal distribution has mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \).

Henceforth, we will denote the random variable of the standard normal distribution by \( Z \).

The General Normal Distribution

The normal distribution does not apply just to the special circumstances where the mean is 0 and the standard deviation is 1.

Transformations of the Standard Normal Distribution

The graph of the probability density function for a normal distribution with mean \( \mu \) and standard deviation \( \sigma \) may be obtained from the graph of the probability density function for the standard normal distribution by the transformation with rule:

\((x, y) \to \left(\sigma x + \mu, \frac{y}{\sigma}\right)\)

This is a dilation of factor \( \sigma \) from the y-axis and a dilation of factor \( \frac{1}{\sigma} \) from the x-axis, followed by a translation of \( \mu \) units in the positive direction of the x-axis, for \( \mu > 0 \). (In Section 15D, this was discussed for probability density functions in general.)

Conversely, the transformation which maps the graph of a normal distribution with mean \( \mu \) and standard deviation \( \sigma \) to the graph of the standard normal distribution is given by

\((x, y) \to \left(\frac{x - \mu}{\sigma}, \sigma y\right)\)

This is a translation of \( \mu \) units in the negative direction of the x-axis, followed by a dilation of factor \( \frac{1}{\sigma} \) from the y-axis and a dilation of factor \( \sigma \) from the x-axis.

For example, if \( \mu = 100 \) and \( \sigma = 15 \), then this transformation is

\((x, y) \to \left(\frac{x - 100}{15}, 15y\right)\)

This transformation is area-preserving. In the following diagram, the rectangle ABCD is mapped to A' B' C' D'. Both rectangles have an area of 180 square units.

This property enables the probabilities of any normal distribution to be determined from the probabilities of the standard normal distribution.

The shaded regions are of equal area.

This leads to the general rule for the family of normal probability distributions.

The rule for the general normal distribution:

If \( X \) is a normally distributed random variable with mean \( \mu \) and standard deviation \( \sigma \), then the probability density function of \( X \) is given by:

and

where \( Z \) is the random variable of the standard normal distribution.

The general form of the normal density function involves two parameters, \( \mu \) and \( \sigma \), which are the mean (\( \mu \)) and the standard deviation (\( \sigma \)) of that particular distribution.

When a random variable has a distribution described by a normal density function, the random variable is said to have a normal distribution.

As with all probability density functions, the normal density function has the fundamental properties that:

• Probability corresponds to an area under the curve.
• The total area under the curve is 1.

However, it has some additional special properties:

• The graph of a normal density function is symmetric and bell-shaped:
• Its centre is determined by the mean of the distribution.
• Its width is determined by the standard deviation of the distribution.

Standardised Values

Clearly, the standard deviation is a natural measuring stick for normally distributed data. For example, a person who obtained a score of 112 on an IQ test with a mean of \( \mu = 100 \) and a standard deviation of \( \sigma = 15 \) is less than one standard deviation from the mean. Their score is typical of the group as a whole, as it lies well within the middle 68% of scores. In contrast, a person who scored 133 has done exceptionally well; their score is more than two standard deviations from the mean and this puts them in the top 2.5%.

Because of the additional insight provided, it is usual to convert normally distributed data to a new set of units which shows the number of standard deviations each data value lies from the mean of the distribution. These new values are called standardised values or \( z \)-values. To standardise a data value \( x \), we first subtract the mean \( \mu \) of the normal random variable from the value and then divide the result by the standard deviation \( \sigma \). That is,

Standardised value =
\[ \text{(data value − mean of the normal curve)} / \text{(standard deviation of the normal curve)} \]
or symbolically,
\[ z = \frac{(x − \mu)}{\sigma} \]

Standardised values can be positive or negative:

• A positive \( z \)-value indicates that the data value it represents lies above the mean.
• A negative \( z \)-value indicates that the data value lies below the mean.