The applications of the normal distribution vary widely and find themselves in many but are useful in estimating the true value of a specified parameter, often the mean, but also proportions. The central limit theorem allows us to make inferences about the overall population with the normal distribution using relatively small samples as long as they fulfill the requirement that they are independent and identically distributed. This means that we do not have to find out the heights of every US citizen over eighteen in order to accurately find their average height, which would be a lengthy never ending process. We can take a sample and infer the average height using the sample alone as long as we have the correct conditions.

This distribution is not only useful in finding true average height but all measurements of physical, chemical, biological, social, and industrial phenomena. Examples are the best way to demonstrate the use of the normal distribution.

Example: A production machine has normally distributed daily output in units. The average daily output is 4000 squeaky toys and daily output standard deviation is 500. What is the probability that the production of one random day will be below 3580? We construct the test statistic, called the “Z-score”, and then seeing how it compares on the curve. The Z-score in our example is constructed using the value of interest (3580), average (mean) production of 4000 squeaky toys with standard deviation of 500. Z= (3580-4000)/500=-0.84 Now we compare this value to the Normal distribution and see that 0.2 is below 3580 meaning that on any given day the probability that the squeaky toy producing machine will make 3580 toys or less is 20%.

Ever wondered what "grading on the curve"? Well it was the Gaussian or normal curve. Basically, your grade is determined by how you did compared to your classmates, not on your proportion of correct answers. The idea is that the class is being "treated" to the lecturers style of presentation and instruction and a measurement of that treatment (test scores) will follow a normal distribution.

Let's say you get a score of 42 out of 50, an 84%. But, when graded on a curve with mean 46.4 standard deviation 1 and 10 other students, you're not looking as good (plug it in above and see). Fortunately, most curves are done in classes with a lot more students than 10.