4 & 5. Statistics, Quality Assurance and Calibration Methods
The Gaussian Distribution
Video duration:
6mRelated Videos
Related Practice
Skip to main content
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 and 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 -one, -two, -three, to the right of it we have plus 1, plus 2, and plus 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 within each section 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 a 100, so that's our population mu. Alright, so we put a 100 there, and 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're going to say that the percentage from 0 to plus 1 here, that percentage is equal to 34.13%. Okay. So these are just, agreed upon percentages in terms of a Gaussian distribution curve. Alright, and remember if 0 to plus 1 is the same amount of space as 0 to negative one, that would mean that this side is also 34.13% as well. So this would mean that 6 about 68% probability of the data falls between minus 1 to plus 1 area. Okay. So together they add up to about 68%. Now, let's say we're looking next at plus 1 to plus 2, and also minus 1 to minus 2. What would they be? Well here, this would come out to about 13.5 9% or so on both sides here. So if we took into consideration the plus 1 to minus 1, and then added these negative one to negative 2, and then plus 1 to plus 2. All 4 of these shaded regions would add up to about 98% or 95%. Sorry. 95% of the data falling between negative 2 to positive 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 negative 3 to positive 3, this is about 99.7%. So 99.7 percent, possibility, probability of measurement falling within this range. So if we took a look here at our intelligence, so we'd say minus 1 or plus 1 area, that means 68% of the population would have an IQ between 85 and 115. Right? Because from negative 1 to plus 1 means that we're matching up 85 to 115. 95%. So from negative 2 to plus 2, so negative 2 to plus 2 would mean we're taking into account 70 all the way to 130. And then finally if we're doing negative 3 to plus 3 we're taking into consideration a lot. So that's 99.7% of 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 a 100%. What about the missing 0.3%. 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 of the range of negative 3 to plus 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, the peak that we have within this Gaussian distribution curve, but Einstein tested beyond that. So you're gonna have individuals that test beyond the 145 IQ, and unfortunately, you're gonna 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.
03:01
