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In this video, we are going to be talking about the normal curve and inferential statistics set of properties. The normal curve is a theoretical curve that never actually hits to one. It is bell-shaped, is new. In a nominal is, move, is symmetrical and skewed. Pell ends to infinity. And the mode, median and mean are the same. And so it's theoretical because these are assumptions that we can rarely actually meet. That we try our best to compare them to a sample size in order to standardize and make inferences based on the normal curve. So the way that we do this is to use z-scores. Z-scores are values that pick different points along the horizontal axis that the unit normal distribution and encompasses certain portions of it. Different proportions of normal curves area are associated with different z-scores. These proportions of curves are standardized and do not change among distributions examined. Raw scores may be converted into z-scores, and z scores may be converted into raw scores. Mean and standard deviation are learned already. And key to the standardization process. Normal distributions of scores have much cuteness. Z essentially equals the raw score minus the mean divided by the standard deviation. A standard normal table, also called the unit normal table or Z-score table. It's a mathematical table through the US by which are the values, the cumulative distribution function at the normal distribution. So here we have a sample of what a normal curve would look like. And we look at the mean would be 0 and a standard deviation would be one or negative one. And so the distance between negative 11 are always the same. And so the same between negative two and negative one would always be the same amount of distance. And so as we go on out to three as well. And so this is what allows for standardization process. So I like this chart a little bit better because it shows more of the breakdown between each of them. So they will always have a 34 percent between the mean and one standard deviation. And 35 percent align, the other half are between the mean and one standard deviation. And so that is why we're able to make inferences that it is perfectly symmetrical. So also note that it also changes based off of the standard deviation. So one standard deviation away from the mean is not proportional to two standard deviations away from the mean. So one is only 34 percent and then from one to two is 13.5%, and then from two to three is only 2.35. And so that's very important to remember that while it is standardized, their deviations are not as fun as far as between the first deviation then the second deviation. So why is the normal distribution curve important? Is important because it allows for generalizability, is important because it allows for us to make inferences. And the normal distribution is a statistical test. The assumption that generalizability function, the scores for my random sample are assumed to be normally distributed. This allows for a sample scores to estimate what the score would be for an entire population. That statistical inference function. Knowing the normal distribution helps us to understand statistical inferences. The normal distribution as a statistical tests assumption. If normally, normality is not achieved and the cisco test is applied anyway, then the statistical test results will be unreliable. Nature of the test assumption violations would always be taken into account when interpreting numerical results and hypothesis testing. So z-scores are scores that have been standardized to the theoretical normal curve. Z scores are present how different a raw score is from the mean and standard deviation units. To find an area first, compute the z-score. The z-score formula changes a raw score to a standardized score. And so we've kind of already went over this term right here, where it is the raw score minus the mean divided by standard deviation. So standard scores and the normal distribution, we have two parameters. And so right here we have mu equals 0 and sigma equal to one. So two z scores are certain numbers. The segments are either to the left or to the right on the horizontal axis of the normal distribution. And so essentially that is important because we don't have non-academics, even though we may have a negative z score, we can never have a negative proportion or a negative percentage. And so positives and negatives are just directions. And so that this negative and right as positive. So I like this particular chart because it shows the cutoff, so certain proportions of the normal distribution. And so here, if we wanted to, again look at any normal distribution, we would see that we would have 0 to one and mu two sigma. And so this is any normal distribution and this is the standard normal distribution. Okay, so here we have the z chart and I have provided the Z chart for you guys and the required reading section. So the way that we look at the z score chart and try and figure out where the proportion lies, would be, that we would go down the score first. And so if I had a 1.96, so 1.96 as I would need to go down to 1.9 and then I would need to go over 2.6. And so here I have it. Dopamine night. Here we go at 0.4750. And so if I was to continue going down through this, I could do a number of them. So your book talks about going to the array. So we can essentially go to infinity. That beyond 3.869, we would have to say that it is 0.499999 continuing because it is a theoretical curve. And so we can never actually hit one. And that would be something that we would have to take into consideration when looking at the Z-scores as well. So you guys also see that this is only one sided and sense it is perfectly symmetrical. We only need one side of the chart. And so if I had a negative 1.96, then I would be able to do the same. And it would be in the same proportion of 0.47, 500. Because again, remember that negative is a direction and not actually, I'm missing something. Or determining proportions of curved areas above and below Z scores. If we know a particular c squared, we also know the proportion of scores found above and below the z score. And so we're going to kind of go over here, I'm just a minute how to get that number? That essentially, if we have a z-score of negative 2.92.95 and negative 1.68. We know that from the center to negative 1.68 is 0.4686. And we also know that from the center to negative 2.95, it is 0.9.4986. And so if I want to find out what this particular area is, I would have to take the 0.4984 and subtract it from the 0.4968. And so if you guys remember back into like your high school statistics class, your teacher had you to do a whole lot of shading. Well, this is why your teacher had to deal, had you do all the shading is to help you understand. Be adding and subtraction between the normal curve. So the tells of a normal curve are important because these are the ones that we use for particular cutoffs. Said a man under the curve area remaining either to the left, I'm going over to the right of a positive amount remaining in the tail of the distribution tells us the distribution have significance for us statistical decision-making. Determining the amount of curve area either in the electorate, right? Tell, involves take an enum value and solving for an unknown value. So let's say that we have here a couple of Z scores and said, we want to consider what is the proportion of the curve below a particular z-score. And so here, if I have a negative, 1 is 0. So if we go back to your chart here, and I want to look at my 1, I would have 0.3413. And so I'm wanting to find everything that is looking for again. So the portion of the curve below that. And so I would need to be tagged by from negative 4.13 and that would give me 1.587. And so if I wanted to add, then I would need to and then I would need to add five to this. And so 3.3413 plus 0.5 is 0.8413. And so that will give me the proportion of the curve above and the standard deviation of a negative one. And so we could do it for the two as well. So if we went down to two, we would see that our proportion at the curve with a negative two is associated with a 0.4772. So if I was to subtract that, then I will get 0.0228. That if I was to add 0.5 to the 0.4772, then I will get 9 way this eye, 0.9772 and so on and so forth. And so remember that once we hit the five, then it's always going to be 0.9999 or 10 000 001, because we can never reach the end of the line and it goes on until infinity. All right, so if we're wanting to look at the proportions curve area between two scores. If we want to look at between a positive and a negative Z score, then we need to add the two proportions together. So let's say that f has a negative 1.83 and a positive 2.66. Then I would need to add the two together to get 0.9 65 as my proportion between the curves. And so it would be the same as if I was to have negative one and a positive one. And so I would pick my point 3, 4, 1, 3, and then I would add that as well. Then that would give me 0.6826. And so again, this is where the shading comes in. And so if you shade them together, then that helps you to remember that you need to add them. All right, so if you have a proportion and a negative, then we need to, or are positive, then you need to subtract the two proportions. And so let's say that we have a positive 0.96. And so, and we also have a positive 2.05. And so I would need to go on my chart here and find out where 800 white 96 sub B. 109 and then go over to my settings here. That would give me a proportion and I like the array of five by four and then subtract it from. Okay, so let's say that I have a negative one, y 93, and a negative 1.5. So since now I have two negative, that means that I need to subtract the two proportions. And so I would take the point 4, 7, 3, 2. And I would want to subtract it from the point 4, 6, 7, 8. And then that will give me a proportion of 0.0054. And that's because if we have, in that case we had two negatives. And so if it was two positives, then we would go through and do the same. Alright, so converting raw scores to z-scores. Here we have a Z I best represents the standard score ever particular raw score, that she represents the mean of the distribution of scores. X hat equals a particular raw score from the distribution of scores. And S is the standard deviation from the deviation of scores. And so let's assume that we would do the top one here, Assessment, Measure. And we're going to look at the Smith inventory. And with this met the inventory at a 120 minus 50 divided by 30, which would give us negative 30 divided by 30, or a Z score of negative one. And so if we did our deviation scores, we would get a negative to a five and a negative three, which would allow for us to confirm that we do have 0, where they all add up with our regular deviation scores. So if we wanted to convert a z-score and to a raw score, we would have to use the DI, which is the standard score for a particular raw score. The XI, the mean of the distribution of scores. That she, again, as a particular raw score distribution scores and S is the standard deviation from the distribution scores. And so let's say that we wanted to do the same one, that we wanted to find out where they raw score would be for that particular score. And so we would take the 150, which is then being, and we would add to the 350, 300 times the negative 1-zero. But she would give us a 150 plus a negative 30. And so every time we place a negative, that just means that we subtract instead, 150 minus 30 is 120. So 120 would be our raw score and that particular case. So transformers for us to stay under crimes had other than the normal distribution. So essentially we need to eliminate the negative values. So new raw scores is in you mean plus a new standard deviation times the original Z-score. And this gets read the normal distribution. And so essentially in our library touched about before, we cannot have a negative proportion. So the new score is precisely the same point on the new distribution as the old z-score is on the normal distribution. So we just have to converted over into the new mean plus the new standard deviation times the original z score, which will give us the same exact spot on the curve. Areas under the curve can also be expressed as probabilities. So probabilities are proportions and ranges from 0 to one. The higher the value, the greater the probability by the more likely that the event will occur. And that pretty much covers everything on the normal curve and inferential statistics. Thank you.