Using Normal Distributions

If you missed the simulating introduction to Continuous Variables and Normal Distributions it might be worth taking a few minutes to read.

Z Value

The functional form for a normal distribution is a bit complicated. It can also be difficult to compare two variables if their mean and or standard deviations are different, for example heights in centimeters and weights in kilograms, even if both variable can be described by a normal distribution. To get around both of these conflicts we can define a new variable:

(1)

\begin{align} z=\frac{x-\mu}{\sigma} \end{align}

This variable gives a measure of how far the variable is from the mean $(x-\mu)$ then "normalizes" it by dividing by the standard deviation $(\sigma)$. This new variable gives us a way of comparing different variables. The z-value tells us how many standard deviations or "how many sigmas" the variable is from its respective mean.

When the distribution function is expressed in terms of the z-value it is sometimes called the "standard normal distribution." You've got to love the creativity!

(2)

\begin{align} f(x)=\frac{1}{ \sigma \sqrt{2 \pi}} e^{-\frac{1}{2} z ^2} \end{align}

Areas Under the Curve

To calculate the probability that a variable is within a range we have to find the area under the curve… And well, it turns out that there is no indefinite integral of the function! However, smart folks have figured out how to do definite integrals, but they are a bit complex so people who work with normal distributions rely on table, which you have in your formula booklet, or calculators.

An example of the table is shown in the collapsible box below. Its as close as I could find to the one the IB gives your…

+ Show Standard Normal Curve Areas

These tables can be a bit scary, but you simply need to know how to read them.

• The left most column tells you how many sigmas the above the the mean to 1 decimal place.
• The top row gives the second decimal place.
• The intersection of a row and column gives the probability.

For example, if we want to know the probability that a variable is no more than 0.51 sigmas above the mean we find select the 6th row down (corresponding to 0.5) and the 2nd column (corresponding to 0.01). The intersection of the 6th row and 2nd column is 0.6950. Which tells us that there is a 69.50% percent chance that a variable is less than 0.51 sigmas above the mean…

Notice that for 0.00 sigmas the probability is 0.5000. Thus showing that there is equal probability of being above or below the mean!

Example 1

Find $P(Z \leq 1.5)$

+ Show Example Solution

This problem essentially asks what is the probability that a variable is less than 1.5 sigmas above the mean. On the table of values find the row that corresponds to 1.5 and the row that column that corresponds to 0.00. Which gives a probability of 0.933.

Graphically this problem can be represented as such:

Example 2

Find $P(Z \geq 1.5)$

+ Show Example Solution

This problem essentially asks what is the probability that a variable is MORE than 1.17 sigmas above the mean. On the table of values find the row that corresponds to 1.1 and the row that column that corresponds to 0.07. Which gives a probability of 0.8790. However, this is the probability that the value is less than 1.17 sigmas above the mean. Since all the probabilities must sum to 1:

(3)

\begin{equation} P(Z>1.17)=1-P(Z<1.17) = 0.121 \end{equation}

Graphically this problem can be represented as such:

Example 3

Find $P( -1.16 \geq Z \leq 1.32)$

+ Show Example Solution

This example is a bit tougher… Graphically this problem can be represented as such:

This problem can be rewritten in the form below.

(4)

\begin{align} P( -1.16 \geq Z \leq 1.32) = P(Z\leq 1.32) - P(Z\leq -1.16) \end{align}

The difficulty comes in our table of values does not allow us to directly calculate $P(Z\leq -1.16)$. However we can use the symmetry of the distribution.

(5)

\begin{align} P(Z\leq -1.16) = 1 - P(Z\leq 1.16) = 0.1230 \end{align}

So we can say:

(6)

\begin{align} P( -1.16 \leq Z \leq 1.32) = 0.9066 - 0.1230 = 0.7836 \end{align}

Quantiles or k-Values

Coming soon. Maybe tomorrow. Its late and I'm tired.

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