The Gaussian Distribution - Online Tutor, Practice Problems & Exam Prep

The Gaussian distribution, also known as the normal distribution, is characterized by its smooth curve, where the mean (μ) is at the center and the standard deviation (σ) determines the curve's width. Increasing the number of measurements shifts the mean, while altering the standard deviation affects the curve's broadness. A higher σ results in a wider curve, indicating greater variability, whereas a lower σ leads to a narrower curve, reflecting less variability. Understanding these concepts is crucial for analyzing data in statistics and various scientific fields.

Gaussian Distribution

Hey everyone. So in this series of videos we're going to talk about the Gaussian distribution. First realize that performing an experiment numerous times with no systematic error results in a smooth curve called the Gaussian distribution. Here we have an image of a typical curve. With this curve we have our f(x), which serves as our function. x here represents our population as a whole. With this we have our formula, but what's most important in terms of this formula is what do these variables represent.

If we take a look at this curve in terms of the Gaussian distribution curve, increasing the number of measurements in the experiments changes the mean. Remember mean is just our x with a line on top. It's going to transition to μ, so μ looks like this. This represents the population mean or average. Here is our μ, and one thing about it is that it's always in the exact center of our curve.

Next, we're going to say it changes our standard deviation, which is typically s, to Sigma. So Sigma is this. If we take a look here at our curve, here goes Sigma, and Sigma looks like it is the distance from the exact center of our curve, which is μ, to the edge of one of the other parts of the curve. Here on this side, it also represents Sigma, and this is going to represent our population standard deviation.

Now the shape of the Gaussian distribution curve can occur here if we change our μ, it'll shift the population distribution curve to the left or to the right. Remember we said that μ represents the exact center of our curve, our population mean. So if I made μ here, that would mean that the curve shifts now the exact center more or less is right here. If I made here, then it would be like this. If we change our standard deviation, our population standard deviation, which is Sigma, it's going to increase or decrease the broadness of the distribution curve. So if we had a very high population standard deviation, that means we'd have a very broad curve.

Right? So here's μ, and then here goes our standard deviations on either side. If I had a very small population standard deviation, a very, very low one, that means our curve would be very thin, very narrow. So here's μ again, and since our Sigma is so small, the distance from μ to one side of the curve will be very, very narrow. Okay. So just remember these are the key variables associated with any typical type of Gaussian distribution curve.

Gaussian Distribution

Now, normally, each distributed variable has its own mean and standard deviation. We're going to say the standard normal distribution simplifies this by setting the mean at 0 and standard deviations in units of 1. So here, let's take a look at this Gaussian distribution curve. If we look, we have our standard deviations, so 0 here is the center of it, which also represents our population mean or \( \mu \), and each standard deviation to the left or to the right is just in units of 1. So to the left of it, we have -1, -2, -3. To the right of it, we have +1, +2, and +3. Right underneath that that matches up with it is our z score. And then here we have our cumulative percentages. So that way, help us understand the percentages involved with any exception of the Gaussian distribution curve. So right in the middle our mean, our average is 50%.

Now, if we take a look here, we have our standard normal distribution formula and we have our z score formula. Now for the z score formula, it's value minus mean divided by standard deviation. So, we'd have some value x minus our population mean, which is \( \mu \), divided by our population standard deviation, which is \( \sigma \). Now, how would we figure this out? Well, this is what we're going to say. So let's say we're looking at the average IQ of the human population, and let's say the average IQ is equal to 100. And let's say that our population standard deviation is 15. So here what would that mean in terms of this Gaussian distribution curve?

Alright, so our average IQ is 100, so that's our population \( \mu \). Alright, so we put 100 there, and the standard deviation is 15, which means we add 15 on the right, so 115, 130, 145. And on the other side, we subtract, so this would be minus 15 from 100 is 85, minus 15 from 85 is 70, minus 15 from 70 is 55. We're going to say, if we're looking at this curve, we'll say that the percentage from 0 to +1 here, that percentage is equal to 34.13%.

Okay, so these are just agreed upon percentages in terms of a Gaussian distribution curve. And remember, if 0 to +1 is the same amount of space as 0 to -1, that would mean that this side is also 34.13% as well. So this would mean that about 68 percent probability of the data falls between minus 1 to plus 1 area. So together they add up to about 68%. Now, let's say we're looking next at +1 to +2, and also -1 to -2. What would they be? Well, here, this would come out to about 13.59% or so on both sides here. So if we took into consideration the +1 to -1, and then added these -1 to -2, and then +1 to +2. All four of these shaded regions would add up to about 95% of the data falling between -2 to +2. That's the probability involved.

And then finally, let's say we included this last chunk here for both sides. So right now, we're covering almost the entire curve. When we take into consideration from -3 to +3, this accounts for about 99.7%. So 99.7%, possibility probability of a measurement falling within this range. So if we take a look here at our intelligence, so we'd say -1 or +1 area, that means 68% of the population would have an IQ between 85 and 115. Right? Because from -1 to +1 means that we're matching up 85 to 115. 95%. So from -2 to +2, would mean we'll take into account 70 all the way to 130. And then finally if we're doing -3 to +3 we're taking into consideration a lot, so that's 99.7% of the population has an IQ between 55 to 145. So that's how we can deal with the probability of someone's intelligence where you fall within each one of these groupings.

Now, you may say 99.7% is not 100%. What about the missing 0.3 percent? Well, remember, with anything there's going to be some uncertainty involved. You can't cover every single possibility, so there's going to be a little bit of measurements or points that lie outside the range of -3 to +3. So, a good example, Einstein. Einstein is seen as one of the smartest individuals known individuals that was on earth. Right? He had a reported intelligence of about 160. So off scale, way off scale, 145 is the peak that we have within this Gaussian distribution curve, but Einstein tested beyond that. So you're going to have individuals that test beyond the 145 IQ, and unfortunately, you're going to have individuals that are below the 55 IQ. Just realize that this is just a way of giving us the probability of a particular point or measurement falling within one of these regions of a typical Gaussian distribution curve. So keep that in mind when we start looking more in-depth on figuring out the percentages or probability of a person, a measurement, what have you, within a particular portion of the Gaussian distribution curve.

The Gaussian Distribution & Z-Table

Hey everyone. So with the discussion of the Gaussian distribution curve, it's important to realize that the use of z tables is essential in the determination of probabilities. Now, probabilities we're going to represent by p, and probability just represents the probability of a population fitting within a certain parameter of the Gaussian distribution curve. If we take a look here, I want to know what's the probability of a certain percentage of the population that fits within this shaded region of my Gaussian distribution curve. So to do that, we need to first figure out what our z score will be. Here, let's just say I give us a z score of negative 1.65. Well, remember that z equals your measurement or value, which is x, minus your population mean, which is μ, divided by a population standard deviation, which is σ. Here we don't have to use this formula because I'm already giving us a value of negative 1.65. We're going to use this negative 1.65 to figure out our p value, which is what all these numbers here represent. Alright, so we just have to figure out where z equals negative 1.65 gives us. We're gonna say here's z. We're gonna say in this vertical column from negative 3.4 all the way to 0. This deals with the first two digits of any z value given to you. So our first two digits here are negative 1.6. Alright, so negative 1.6 means we're here, and then the last digit we get that from this portion here. So the last digit here is 5, right? And it's in the 2nd decimal place. So we're looking for 0.05, which is right here. So all we gotta do is trace and see where they match up. So right here, and then right here, this is where they match up. So it looks like it matches up with this value of 0.0495. So that represents our probability, which is p, so 0.0495. But remember we don't want the decimal form, we want the percentage form. So how do we get the percentage form? We multiply by a 100, so that comes out to 4.95%. So this tells me that 4.95 percent of the population resides within the shaded region of this Gaussian distribution curve. Here I just chose a number at random for Z. It's not drawn to scale because 4.95 seems like a very small percentage, so we shouldn't expect it to be so much shaded region within the Gaussian distribution curve. What was important here is understanding how to use the z table, your z score, to find your probability p. From there multiply it by a 100 and you'll have your percentage of the population.

The Gaussian Distribution & Z-Table

Hey everyone. So here we have to determine what is the percentage or probability of the population fitting within this shaded region of the Gaussian distribution curve. So let's just say that this shaded region had associated with it a z-score of 1.82. So again, I'm just pulling a number at random. Here is 1.82. We look at the chart here or table. Remember the first two numbers of our z-score come from the column. So we want 1.82, which means we're right here. The last digit comes from up here. So we're looking for 0.02 because it's in the second decimal place. So right here. Then we look to see where they line up. So if we look, they're going to meet here at 0.9656. So that'll give us our PR probability. If we want it as a percentage, we multiply by 100. So that's 96.56%. So here, that would mean that this shaded region of our Gaussian distribution curve represents 96.56%. Again, not drawn to scale. Here, we're learning how to use the information given to us to find our p value. Now, let's say they wanted so our z-score here we'll say is 1.82. Now, let's say that they wanted us to find instead of z equals 1.82, let's say they wanted us to find the percentage from z equals 0 to z equals 1.82. So we're only looking at this portion of our Gaussian distribution curve. We don't want the rest of it. How would we figure that out? Well, remember that that 0 there represents our population mean or our mu. It's always dead center of our Gaussian distribution curve and because of that it represents 50%, which would mean that this entire region here represents 50%. So we do 96.56% minus 50%, which would mean that our region that's in green, the part that's 0 to 1.82 equals 46.56%. So that there would represent our additional answer in terms of this question. Okay. So remember, that's the approach you need to take if you're looking for the entire shaded region or you're looking for a portion of the shaded region.

Gaussian Distribution & Z-Table

Here in this example question it says, suppose there are 100 students in your analytical lecture and at the end of the semester the class average is 80 with a standard deviation of 5.3. Determine the distribution and probability of grades based on your understanding of the Gaussian distribution curve.

Alright. So first of all, they're telling us that the class average is 80, which represents our population average. So, remember μ here is dead center at 0, so we have 80%. We have a standard deviation of 5.3, so we're going to add 5.3 here, so this becomes 85.3%. This one comes to 90.6%, and then this one is 95.9%. On the other side, we subtract, so we're going to subtract 5.3 each time, so this one comes here to be 74.7%. This becomes 69.4%, and finally this one here is 64.1%. So we have our percentages involved.

Now, they want us to determine the distribution and probability of grades. Alright. Let's look. Let's say we're looking at the minus 1 to plus 1 area of our Gaussian distribution curve. Remember, we talked about this within our concept videos. This represents about 68% or, in this case, 68 people. Here, this represents our distribution. Let's keep going. Negative 2 to plus 2 area, remember we said that this represents 95%, or in this case 95 people, so this is another distribution. And then negative 3 to plus 3 area, remember we said that this represents 99.7%, but you can't have 99.7% of people or 99.7 people. So we'll just say that this represents 100% of the class, 100 people. So there goes our distributions, 68 people, 95 people, all the people.

Now we need to figure out the probability. So remember our probability was represented by p. Okay. So our probability p here, so from negative one to plus one, that means we're going from 74.7% to 85.3%. That's the probability of a person residing within that region, those percentages.

Next, from negative 2 to positive 2, that would be a percentage of 69.4% to 90.6%. And then finally, from minus 3 to plus 3, the probability is 64.1% to 95.9%. So those would be the probabilities that our population fits within these parameters of our Gaussian distribution curve.

Alright, keep in mind some of these terminologies as well as the way we look at a typical Gaussian distribution curve in terms of the population mean and standard deviation.

Gaussian Distribution & Z-Table

Determine the percentage of final grades that would lie below 71. Remember, we have to figure out the probability of grades that lie below 71. To figure out your probability \( p \), you first need to know what your \( z \) score will be. So, remember, \( z \) equals your value, which is \( x \), minus your population mean, divided by your population standard deviation, which is \( \sigma \). Here our value of 71 is going to be \( x \) minus our population mean, which remember from the previous question is 80, divided by our standard deviation, which is 5.3. This equation would yield -1.6981 on your calculator, but remember, with our \( z \) tables, we're only allowed to deal with 3 digits, so this gets rounded to -1.70. Go back and look at our \( z \) tables. If we use the \( z \) table and we line up the numbers correctly, that will give us a \( p \)-value equal to 0.046. Multiplying that by a 100 gives us our percentage of 4.46%. So, 4.46% of the classroom population would have a score below 71.

From EXAMPLE 1, determine the percentage of final grades that would lie between 88 to 92. 

• A. 5.36 %
• B. 5.33 %
• C. 5.39 %
• D. 5.49 %

Here’s what students ask on this topic:

What is the Gaussian distribution and why is it important in statistics?

The Gaussian distribution, also known as the normal distribution, is a probability distribution characterized by its bell-shaped curve. It is defined by two parameters: the mean (μ), which is the center of the distribution, and the standard deviation (σ), which measures the spread or width of the curve. The Gaussian distribution is important in statistics because it describes how data points are distributed around the mean in many natural phenomena. It is used in various fields such as physics, biology, and social sciences to model real-world data, perform hypothesis testing, and make predictions.

How do the mean (μ) and standard deviation (σ) affect the shape of the Gaussian distribution curve?

The mean (μ) and standard deviation (σ) are crucial in determining the shape of the Gaussian distribution curve. The mean (μ) is the central point of the curve, and changing it shifts the entire curve left or right along the x-axis. The standard deviation (σ) affects the curve's width: a larger σ results in a wider, flatter curve, indicating greater variability in the data, while a smaller σ leads to a narrower, taller curve, reflecting less variability. These parameters help in understanding the distribution and spread of data in a given dataset.

What is the significance of the area under the Gaussian distribution curve?

The area under the Gaussian distribution curve represents the total probability and is always equal to 1. This area is significant because it allows us to determine the probability of a random variable falling within a specific range. For example, the area within one standard deviation (σ) of the mean (μ) covers approximately 68% of the data, within two standard deviations covers about 95%, and within three standard deviations covers about 99.7%. This property is essential for statistical inference and hypothesis testing.

How is the Gaussian distribution used in hypothesis testing?

The Gaussian distribution is fundamental in hypothesis testing, particularly in tests involving large sample sizes. It is used to determine the likelihood of observing a sample mean given a null hypothesis. By comparing the sample mean to the population mean (μ) and considering the standard deviation (σ), we can calculate a z-score, which tells us how many standard deviations the sample mean is from the population mean. This z-score is then used to find the p-value, which helps in deciding whether to reject the null hypothesis. This process is crucial for making data-driven decisions in research.

What are some real-world applications of the Gaussian distribution?

The Gaussian distribution has numerous real-world applications across various fields. In finance, it is used to model stock prices and returns. In quality control, it helps in monitoring manufacturing processes and detecting defects. In psychology, it is used to analyze test scores and behavioral data. In natural sciences, it models phenomena such as measurement errors and biological traits. The Gaussian distribution's ability to describe data variability and central tendency makes it a versatile tool for analyzing and interpreting data in many disciplines.