In this video we'll be learning about the normal distribution and the 68-95-99.7 rule. When we talk about normal distributions, we refer to data we get from a population or sample. So before we actually talk about the normal distribution, we need to first distinguish the difference between a parameter and a statistic.
A parameter is a number that describes the data from a population. whereas a statistic is a number that describes the data from a sample. Examples of parameters and statistics are the mean and standard deviation, but because of the definitions we just talked about, we have to be very careful with what symbols we use to represent these numbers. When we are dealing with a sample, we use the symbol x-bar to represent the sample mean, and we use the letter s to represent the sample standard deviation.
These are statistics. When we are dealing with a population, we use the Greek letter mu to represent the population mean, and we use the Greek letter sigma to represent the population standard deviation. These are parameters.
The population parameters mu and sigma are very important when we talk about normally distributed populations. So what is a normal distribution anyways? A normal distribution is a special type of density curve that is bell shaped. For this reason, the normal distribution is sometimes called the bell curve or the normal curve.
The normal distribution describes the tendency for data to cluster around a central value. In fact, this central value is the population mean mu, which is always located in the middle of the curve. So for any normal distribution, we can say that some data points will fall below the mean, other data points will fall above the mean, but most of the data values are located near the mean.
The normal distribution and its shape actually arises from many different variables found in nature such as weight, height, volume, blood pressure, and many more. This is why the normal distribution is commonly studied. For example, exam scores are known to follow a normal distribution. Some people do great on exams, some people do poorly on exams, but a large majority of people score near the average or the mean.
In this example, the average exam score is 50 because it is located in the middle of the curve. Now that you know what a normal distribution looks like, we need to talk about the population mean mu and the population standard deviation sigma. Both of these tell us important information about how the normal distribution looks. We'll talk about the population mean mu first. The population mean mu characterizes the position of the normal distribution.
If you increase the mean, the curve will follow and move towards the right, and if you decrease the mean, the curve will still follow and move towards the left. This happens because the data will always cluster around the mean in normally distributed populations. As a result, the value of the mean determines the position of the normal distribution.
On the other hand, the population standard deviation sigma characterizes the spread of the normal distribution. The larger the standard deviation, the more spread out the distribution will be, and the smaller the standard deviation, the less spread out it will be. Notice that when the spread increases the curve gets much flatter, and when the spread decreases the curve gets taller. The reason for this is because the normal distribution is a density curve, and the total area of any density curve must remain equal to 1 or 100%.
So changes in the width of the curve must be compensated for by changes in the height of the curve, and vice versa. Overall here are some points about the normal distribution. The normal distribution is unimodal. This means that the distribution has a single peak.
The normal curve is symmetric about its mean, so you can clearly see that the distribution can be cut into two equal halves. The parameters mu and sigma completely characterize the normal distribution. The population mean mu determines the location of the distribution and where the data tends to cluster around.
The population center deviation sigma determines how spread out the distribution will be. The notation given to a population that follows a normal distribution can be written like this. Although it looks scary, it means what it says. For the variable x, it follows a normal distribution and has the mean mu with a standard deviation of sigma.
Now that you've been introduced to the normal distribution, we can talk about the 68-95-99.7 rule. If we were measuring the heights of all students at a local university, and found that it was normally distributed with a mean height of 5.5 feet and a standard deviation of half a foot or 0.5, we can construct a normal distribution as follows. From here, we can create intervals that increase by the standard deviation, so we'll have 6, 6.5, and 7, and on the other side we'll have 5, 4.5, and 4. So what the 68-95-99.7 rule says is that, within one standard deviation away from the mean, it contains a total area of 0.68 or 68%. Because of this, we can say that 68% of the population are between 5 and 6 feet tall.
And if you go two standard deviations away from the mean, it contains an area of 95%. This means that 95% of the people in the population have a height between 4.5 and 6.5 feet. And finally, within 3 standard deviations away from the mean, it contains a total area of 99.7%. This means that, for the population we are studying, 99.7% of the people are between 4 and 7 feet tall. Now you might be wondering, what happens if we go 4 standard deviations away from the mean, or 5 or 6 standard deviations away from the mean?
And to answer that, you actually can. A normal distribution actually never touches the x-axis. it continues on to infinity. So you can go as many standard deviations away from the mean as you want, but the area contained within these regions will be very very small.
The 68-95-99.7 rule is a great way for approximating the areas of a normal distribution, and this works for any normal distribution no matter what shape and size. So let's do some practice questions. Feel free to pause the video at any point so you can try these questions for yourself. Question number one.
The normal distribution below has a standard deviation of 10. Approximately what area is contained between 70 and 90? In this question, we know that the population mean is equal to 70 because it's in the center of the distribution. We also know from the question that one standard deviation is equal to 10, and we can see this because each interval goes up by 10. According to the 68-95-99.7 rule, we know that there is an area of 95% contained within two standard deviations of the mean. 2 standard deviations to the right gets us to 90, and 2 standard deviations to the left gets us to 50. According to the 68-95-99.7 rule, this means that there is an area of 95% contained within this interval. However, we are only interested in the area from 70 to 90, so dividing this area by 2 gives us our area of interest.
95% divided by 2 gives us an area of 47.5%, and that is our answer. Question number 2. For the normal distribution below, approximately what area is contained between negative 2 and 1? In this example, we know that we have a mu of 0 because 0 is in the center of the distribution, and we know that we have a sigma of 1 because each interval goes up by 1. To approximate the area between negative 2 and 1, we will use the 68-95-99.7 rule.
We can strategically divide this area into two parts so that we can easily incorporate this rule. We'll start with the right half, which goes from 0 to 1. We know that one standard deviation away from the mean gives us 68%, and half of this is 34%, giving us our area from 0 to 1. The next half goes from 0 to negative 2, but we know that within two standard deviations from the mean we have an area of 95%. Dividing this by 2 gives us the area from 0 to negative 2, which is equal to 47.5%.
And finally, To get the total area contained between negative 2 and 1, all we have to do is add these two areas together, and when we do, we get a total area of 81.5%. If you found this video helpful, consider supporting us on Patreon to help us make more videos. You can also visit our website at simplelearningpro.com to get access to many study guides and practice questions.