Question

Describe in your own words the standard normal distribution. Additionally, give a real-life example of where we may use the standard normal distribution.

Answer 1

Normal distributions do not necessarily have the same means and standard deviations. A normal distribution with a mean of 0 and a standard deviation of 1 is called a standard normal distribution.

Areas of the normal distribution are often represented by tables of the standard normal distribution.

The standard normal distribution, commonly referred to the Z-distribution, is a special case of a normal distribution with the following properties:

• It has a mean of zero.
• It has a standard deviation of one.
• Areas under the density curve can be found using a standard normal table

A score on the standard normal distribution is called a Z-Score. It should be interpreted as the number of standard deviations a data point is above or below the mean. A positive Z-Score indicates that a point is above the average, and a negative Z-Score indicates a score below the average.

Example:

Suppose, your midterm grades are in. You scored an 84 in Chemistry, a 93 in Spanish, and a 79 in Statistics. You run downstairs to share your grades. The time comes for you to blurt out your results, and suddenly a barrage of questions begin running through your head - which score will I share first? Which am I most proud of?

Surely the 93 is best… Right? 93 > 84 > 79, simple math. But, the Spanish test was easier. Plus, your friend just sent you a text message saying she scored a 61 in Chemistry. Statistics has challenged you all year, and your last test score was in 50, so the 79 was a major improvement. How will you decide which score was your biggest triumph?

What you need is a way to put these scores on the same scale. The solution to all of your problems lays within the Standard Normal Distribution, a distribution that assigns scores based on performance relative to how others performed in the same population.

In order to make the proper comparisons in the example above, let's assume (perhaps a bit unreasonably) that all midterms scores were normally distributed. However, there are infinitely many normal distributions in the world, each one centered around its own average, with its own standard deviation. For instance, Chemistry midterms had an average of 73 with a standard deviation of 8, Spanish midterms had an average of 87 with a standard deviation of 6, and Statistics midterms had an average of 70, with a standard deviation of 15. Recall that standard deviation is a measure of spread; it is an average distance of data points away from the mean.

o make an informed decision, we will covert all scores to the same scale. We will standardize the scores, converting them to the standard normal distribution, with a single mean and a single standard deviation.