Normal Distribution & Shifts in the Mean

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  • 0:05 Normal Distribution
  • 1:35 Characteristics
  • 3:04 Shifts to the Left & Right
  • 3:50 Randomness
  • 4:36 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, you will see what a normal distribution looks like and you will learn about the mean of a normal distribution. You will also see what happens when the mean is shifted.

Normal Distribution

When businesses use graphs to show people their data, they have words to describe how the graph looks and how the data is spread out over the graph. In this video lesson, we talk about the kinds of graphs that have a normal distribution where the data tends towards a middle value with equal amounts to the left and right.

A graph with a normal distribution looks like the outline of a bell, hence it is also referred to as a bell curve. You can see that most of the data is in the middle of the graph where the curve is the highest. There is less data to the left and right of the graph. There is also an equal amount of data to the left and right of the graph.

A bell curve graph
Example of normal distribution graph

In the real world, there are a surprising number of events that follow a normal distribution. For example, the height of people follows a normal distribution. This is why we can say that, in general, people will grow to a certain height. Because of the normal distribution, though, some people will be shorter and some will be taller, but most people will be the average height.

In business, a graph that shows the number of customers that come during the day might also show a normal distribution. For example, a restaurant that serves lunch will find a normal distribution that shows that the majority of its customers come at lunch time while a few come before and after the lunch rush.


Because there is an equal amount of data to the left and right, the mean or average of this graph is the middle of the graph, the point where the curve is the highest. If you folded this graph in half, the left and right sides would mirror each other with the graph trailing off to a zero to the left and right. A normally distributed graph has half of the data to the left and the other half to the right. There is an equal number of data to either side. The data does not prefer one side over the other.

We also have another term we use when describing graphs. It is called the standard deviation. It is a measure of how spread out the data is. For a graph with a normal distribution, 34 percent of the data is within 1 standard deviation from the middle. If we take 1 standard deviation to the left and right, we will have 68 percent of our data. If we take 2 standard deviations to the left and right, we will have 95 percent of our data. Within 3 standard deviations, we will have 99.7 percent of our data. These standard deviations divide each side of our graph into fourths. The first quarter is the one closest to the middle while the fourth quarter covers the trailing off to zero part.

Shifts to the Left and Right

When you see a graph where the tip of the curve is shifted to either the left or the right, we no longer have a normal distribution. We now have a graph that shows a preference for one side or the other. Now our average from the data will not be the exact middle of the graph. Instead, our average now is shifted to the left or right depending on which direction our tip of the curve is shifted. If our curve tip is shifted to the left, then so is our average. If our curve tip is shifted to the right, then our average is shifted to the right, as well.

With these kinds of graphs, we still see the data tending to a particular value. The particular value just won't be the middle value.

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