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Standard Normal Distribution

Standard Normal Distribution

Author: Dan Laub

In this lesson, students will learn about z-scores in normal distribution.

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Source: All images created by Dan Laub

Video Transcription

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[MUSIC PLAYING] Hi. Dan Laub here. And in this lesson, we're going to discuss standard normal distributions.

But before we do so, let's cover the objective for this lesson. By the end of this lesson, we want to be able to estimate the approximate location of a z-score when given a standard normal distribution. So let's get started.

Recall from previous lessons that normal distributions have a bell-shaped curve in which the mean is the center point-- or top of the curve-- and the standard deviation indicates how spread out the distribution is. One of the goals of using normal distributions is to find the probability under the normal distribution curve that an event will occur. Since we are often concerned with finding a probability under the normal distribution curve and because normal distribution curves differ for the data we are considering, there is a need to create a standard normal distribution. This is because doing so will allow us to quickly determine the probabilities related to any normal distribution.

So as you look at the normal distribution here, you can see how we change things from a normal distribution to a standard normal distribution by simply changing the points of the standard deviations. And we do what's called "standardizing them." So in other words, we can change a normal distribution into a standard distribution by applying a formula to each value in the normal distribution. By converting a normal distribution into a standard normal distribution, each value is changed from the normal distribution into what are referred to as "z-scores" by using the following formula where a z-score is equal to the difference between an actual value and the mean and that difference is divided by the standard deviation.

As you see here, we use z-distributions to define the position of a value in terms of the mean and the standard deviation where the mean has a z-score equal to 0 due to it's being a location along the horizontal axis. And this axis is the one that provides us with the location of any z-score.

What is interesting about a z-distribution is that it's shape contains what are called "turning points." these turning points fall on the curve at the point where a z-score is equal to either negative 1 or 1. The left turning point has a z-score of negative 1, which indicates its location on the horizontal axis whereas the right turning point has a z-score of 1 since that is where it is located on the horizontal axis. Additionally, there are typically values of negative 3, negative 2, 2, and 3 on the z-distribution as well, which are highlighted on the graph you see here. What is especially useful about the z-distribution graph is that the areas between different z-scores are known-- making it relatively easy to determine the probability that a specific event will occur.

As you can see illustrated here, roughly 68% of the area under the z-distribution curve falls between the z-score values of negative 1 and 1. In addition, approximately 95% of the area under the z-distribution curve falls between the z-scores of negative 2 and 2 while approximately 99.7% of the area under the z-distribution curve falls between the z-score values of negative 3 and 3. A z-score less than negative 3 and greater than 3 account for the remaining 0.3% of the area under the z-distribution curve. Each one of these percentage values is related to a probability value that explains the chance of an event occurring if it happens to fall between two specific z-scores.

As you can see here, the mean is located on the point where z is equal to 0 on the horizontal axis of a z-distribution. There are also points on this distribution to indicate where the z-scores of negative 3, negative 2, negative 1, 1, 2, and 3 are located. When considering the area under the curve that falls between a z-score of negative 1 and 1, one can see here that the proportion of this area is approximately 68% of the total area under the curve.

In this graph, you can see the area between z equals negative 2 and z equals 2 highlighted, which represents approximately 95% of the area under the curve of the whole z-distribution. In the event that there are z-scores that fall within this range, such as z is equal to 0.75 and z is equal to 1.72, they represent values that are likely to occur as they fall within the middle 95% of the distribution. However, z-scores such as negative 3.14 and 4.87 fall somewhat far from the mean score of z is equal to 0. And therefore, these values are quite unlikely to occur.

So let's go back to our objective just to make sure we covered everything we said we would. By the end of this lesson, we wanted to be able to estimate the approximate location of a z-score when provided with a standard normal distribution, and we did that. It's relative likelihood. So again, my name is Dan Laub. But hopefully, you got some value from this lesson.

Formulas to Know

z equals fraction numerator A c t u a l space V a l u e space minus space M e a n over denominator S tan d a r d space D e v i a t i o n end fraction