Mathematical Measurements - Video Tutorials & Practice Problems

Mathematical Measurements

Hi in this video, we're gonna be talking about mathematical measurements. So a lot of these measurements, you've gone over in some kind of basic math class. But they're also important in biology and specifically genetics because genetics uses these measurements to analyze phenotype. And so generally this is a sample of a phenotype or phenotype, an entire population or a phenotype over all organisms in the world. But in order to be able to analyze those large numbers of phenotype, so we have to be able to Kwan them and say, you know, this phenotype differs by this one by this much. And so I'm going to introduce some measurements which you are likely familiar with from some math class, but I just want to kind of present them in the way of genetics. So the first is the mean and so I mean is an average. And so there's two types of means that we can take in genetics. One is the population mean. So what is the population? A population is going to be all individuals within the group that you're measuring? So all humans in florida or all, you know, sloths in costa rica. So that could be an entire population. But generally a population is a very large number. And it's not necessarily possible if we were looking at what's the main height of all humans, then we can't really go around to every human on earth and measure their height. And so typically how the mean is studied is in the form of a sample and this is a representative subset. So instead of taking all humans on Earth, we take, you know, 30 humans from each continent. And that could potentially represent a sample of the population on earth. Now you have to be careful with using a sample because you could, even though usually samples are random. So you just sort of randomly pick people. It could end up being um not super accurate, right? Because if we were selecting 30 individuals from every continent on earth to measure the average height of humans in the world, that could be very misleading if we happen just to accidentally choose um 30 really tall people in America and then 30 really short people in um africa. And that would necessarily be representative of the entire population. Now generally samples they're way statistically that I'm not going to show you to make sure you're doing an accurate sample. Um So usually scientists do that. But for now we're just going to talk about the mean. So how you calculate the mean is this stands for mean. You may also see it written like this and it's the exact same thing. But I would know both notations of what the mean represents. And so this is some. So this means you're adding right and Xterra the point. So if we're thinking about this in terms of genetics, what are some phenotype that we could measure using the mean? Well, one I've already talked about his height, but it could also be wait, it could be number of legs. If you're looking at insects or something, it could be a number of spots, It could be foot length, it could be a number of ridges. It could be tail length, could be head size. Um It could be organ size. All of these different traits are very their numbers right? And so we can very easily measure them and get the average using the main. So you take the sum of all the individuals in your sample size. Um so if you have 100 lizards and you're looking at tail length and then you take the measurement of tail length of all 100 and then you divide it by the number of individuals. So in the case of lizards, there would be a 100. So that's how you calculate the main. Now a second statistic which may not be so intuitive is actually variance. And so variance is a measurement that says how far something is from the mean. So let's say we're looking at spots on a Dalmatian and we say that the mean is um 100 spots. And then we look at an individual and we say that, Okay, well that individual only has 73 spots and the variance calculates how far away this value is from the mean. And it usually gives it a In terms of a portion because we're looking at a set of values. So it's not just one individual, but just generally, how is the variance of every single individual? That was used to calculate this 100 mean? So if we have 50 individual, 50 Dalmatians that were used, the variance is going to calculate that for every single one of those 50 and then give us a proportion of how varied is the population that we use to calculate the main. So um when it comes to traits, variance is going to measure one trait, but we can actually do this with multiple traits and that's co variance is measuring the how much variation is common to two traits. So how you do this is sometimes it's written as s sometimes it's written like this um And either way it's the same and I would know both in case it comes up in a test or quiz. So notice here that in both of these cases it's squared. So variances always is squared. So what you get is how you calculate this is you do the sum, so again we're adding but this time you're taking the individual score, so 73 in our example and you're minus sing it by the mean, so that would be this and you square it Then you do in -1, which in our sample was 50. Right? So in is 50 -1. And so if you were to calculate this and I haven't calculated before and I'm not gonna do it off the top of my head. But if you were to calculate this you would get the variance for this one individual here, and generally because it's the sum of this number has to be repeated. This thing has to be repeated 50 times, right? Because you have to and then you add them all together. So um so where the 73 is, you would input 49 other values that you use to get this mean, and you would add them all together, and that would equal the variance. So in addition to the variance, a common statistic that uses standard deviation. And this measures the amount of variation that exists within a set of data. And then this differs from standard error, and standard error is telling you how accurate the sample mean is. So remember we calculated a so we said dalmatians, right? And let's say we were interested in dalmatians in Connecticut And we wanted to know how many spots Dalmatians in Connecticut had. Well we only took a sample of 50, right? And when we took that sample, that could have been, we could have just accidentally chosen a sample of really spotted um dalmatians. And there could have been this whole other section of dalmatians in Connecticut that had less spots. So the standard error is measuring how accurate our sample mean, was by taking into consideration that there's this larger population that we took a sample of. So standard deviation is here, and standard deviation notice is s or you can write it like this. Now notice this differs from variants, right? Because variance is squared, it's either s square or um the square, depending on how you write it. So if you have the variance, the way to get standard deviation is just take the square root. So that's what you're seeing here. You're seeing the sum of the individual point minus the mean. But notice it's written different than above, squared over in minus one. And this is the exact same formula is here, the only difference is this square root right there. So when you have the variance, you can easily get the standard deviation by square rooting it. Or if you have the standard deviation, then you can easily get the variance by squaring it, which makes math on test easier. Now, finally, a lot of these values are written in the form of a normal distribution. So this is the bell curve that we all hear about and hate when we go take our test. But essentially this is just a graphical depiction of variation in a phenotype and specifically focusing on the range of variation. Now, sometimes you You see this is a frequency hissed a gram because on the Y-axis you see that this is these are frequencies right? So we were to turn these into percent, it would be 20%, and pretty much this is saying 20% of the population has a value here, 40%, has a value here, 80% has a value here. And so for this we're looking at four different traits. It doesn't say what traits they are, they're just by different colors. And so um here's our sort of our peak point here which is zero for the blue, red and gold and negative two for the green. And then over here, what is this? This is variance, right, Because we know it has that weird symbol and its square. So we're saying that this is the peak point here, 00 negative two. And then a variance is either 20.215 or 0.5. Now if we take a look at the blue at the variance of 0.2, then we can see that this bell curve is very short meaning that the majority of the population is falls like very close to the mean, to the very or to the center here, if we take a look at the red, one who has a much bigger variant, then the majority of the population falls here, right? And that's um that's a much farther distance from the mean. And if we keep going out to this gold color here, which I'm going to write in blue just so you can easily see it five. That means that the majority of the population comes here but it is much less condensed the mean And then this one has a very different, it doesn't always have to be zero, right? And so um again with this, this is .5, so we say here and here, and so that is just different ways. So we're saying that this is sort of the normal range of variants in the population depending on what it's measuring. And I know that can get a little confusing because we're not talking about one particular trait, but you'll probably see these examples in one particular trait, and it's important just to recognize like a bell curve and how to look at it and read it. So with that let's not move on.

The table shows a distribution of bristle numbers in a Drosophila population. What is the variance? 

• A. 1.0
• B. 1.2
• C. 5.5
• D. 3.0

Using the variance calculated in problem #2, what is the standard deviation?

• A. 1.0
• B. 1.1
• C. 2.3
• D. 1.7