Statistics and Probability

Furthermore

Official Guidance, clarification and syllabus links:

Awareness of the natural occurrence of the normal distribution.

Students should be aware that approximately

\( \quad 68\)% of the data lies between \( \mu \pm \sigma\)

\( \quad 95\)% lies between \( \mu \pm 2\sigma\)

\( \quad 99.7\)% lies between \( \mu \pm 3\sigma\).

Probabilities and values of the variable must be found using technology.

For inverse normal calculations mean and standard deviation will be given.

This does not involve transformation to the standardized normal variable z.

The normal distribution is a continuous probability distribution that is symmetrical around its mean, representing a bell-shaped curve where the bulk of the values lie close to the mean. The standard normal distribution has a mean of zero and a standard deviation of one. It is crucial in statistics because of its natural occurrence in various natural phenomena and its use in inferential statistics.

The key formula for the normal distribution is its probability density function (PDF):

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

Where:

• \( \mu \) is the mean of the distribution,
• \( \sigma \) is the standard deviation,
• \( \sigma^2 \) is the variance,
• \( x \) is the variable,
• \( e \) is the base of the natural logarithm.

An example of the normal distribution in use is finding the probability that a randomly selected value is less than a certain number. For instance, if the mean score of an exam is 60 with a standard deviation of 10, what is the probability of a student scoring less than 75?

$$ P(X < 75) = \int_{-\infty}^{75} \frac{1}{\sqrt{2\pi \cdot 10^2}} e^{- \frac{(x - 60)^2}{2 \cdot 10^2}} \, dx $$

This formula would never be required in an exam. Students need to be proficient in using the GDC for calculations involving the normal distribution that go beyond simple situations.

If you would like to interact with the graph of the standardised normal distribution you will find it waiting for you on the Graph Plotter.

The equation is:

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

Students should be aware that approximately 68% of the data lies between within one standard deviation away from the mean, 95% lies between 2 standard deviations away from the mean and 99.7% of the data lies between 3 standard deviations away from the mean. This is one of of those facts that needs memorising.

If you would like to interact with the graph of the normal distribution you will find it waiting for you on the Graph Plotter page (you will need to rescale the axes).

If you use the TI-Nspire calculator you can find instructions for using the Normal CDF function on the GDC Essentials page.

This Normal Distribution video is from Revision Village.

This Bicen Maths video clip shows everything you need to memorise on this topic for A Level Statistics.