title

Introduction to the normal distribution | Probability and Statistics | Khan Academy

description

Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/more-on-normal-distributions/v/introduction-to-the-normal-distribution
Exploring the normal distribution
Watch the next lesson: https://www.khanacademy.org/math/probability/statistics-inferential/normal_distribution/v/normal-distribution-excel-exercise?utm_source=YT&utm_medium=Desc&utm_campaign=ProbabilityandStatistics
Missed the previous lesson?
https://www.khanacademy.org/math/probability/regression/regression-correlation/v/covariance-and-the-regression-line?utm_source=YT&utm_medium=Desc&utm_campaign=ProbabilityandStatistics
Probability and statistics on Khan Academy: We dare you to go through a day in which you never consider or use probability. Did you check the weather forecast? Busted! Did you decide to go through the drive through lane vs walk in? Busted again! We are constantly creating hypotheses, making predictions, testing, and analyzing. Our lives are full of probabilities! Statistics is related to probability because much of the data we use when determining probable outcomes comes from our understanding of statistics. In these tutorials, we will cover a range of topics, some which include: independent events, dependent probability, combinatorics, hypothesis testing, descriptive statistics, random variables, probability distributions, regression, and inferential statistics. So buckle up and hop on for a wild ride. We bet you're going to be challenged AND love it!
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to KhanAcademy’s Probability and Statistics channel:
https://www.youtube.com/channel/UCRXuOXLW3LcQLWvxbZiIZ0w?sub_confirmation=1
Subscribe to KhanAcademy: https://www.youtube.com/subscription_center?add_user=khanacademy

detail

{'title': 'Introduction to the normal distribution | Probability and Statistics | Khan Academy', 'heatmap': [{'end': 555.434, 'start': 536.064, 'weight': 1}, {'end': 762.76, 'start': 710.286, 'weight': 0.736}, {'end': 1063.883, 'start': 980.103, 'weight': 0.709}], 'summary': 'Provides a comprehensive understanding of the normal distribution in inferential statistics, covering its application in probability density functions and calculus, numerical evaluation, significance in the central limit theorem and binomial distribution, z-score formula, cumulative distribution function, and practical applications in financial modeling.', 'chapters': [{'end': 180.152, 'segs': [{'end': 35.331, 'src': 'embed', 'start': 1.866, 'weight': 0, 'content': [{'end': 6.229, 'text': 'The normal distribution is arguably the most important concept in statistics.', 'start': 1.866, 'duration': 4.363}, {'end': 13.614, 'text': 'Everything we do, or almost everything we do in inferential statistics, which is essentially making inferences based on data points,', 'start': 6.409, 'duration': 7.205}, {'end': 16.356, 'text': 'is to some degree based on the normal distribution.', 'start': 13.614, 'duration': 2.742}, {'end': 27.265, 'text': 'And so what I want to do in this video and in this spreadsheet is to essentially give you as deep an understanding of the normal distribution as possible.', 'start': 16.396, 'duration': 10.869}, {'end': 32.448, 'text': "And just the rest of your life you're always, if someone says, oh, we're assuming a normal distribution, you're like, oh, I know what that is.", 'start': 27.325, 'duration': 5.123}, {'end': 35.331, 'text': 'This is the formula, and I understand how to use it, et cetera, et cetera.', 'start': 32.488, 'duration': 2.843}], 'summary': 'Understanding the normal distribution is key in statistics, influencing inferential statistics and data assumptions.', 'duration': 33.465, 'max_score': 1.866, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY1866.jpg'}, {'end': 133.378, 'src': 'embed', 'start': 105.876, 'weight': 2, 'content': [{'end': 110.059, 'text': 'probability distribution or a continuous probability density function.', 'start': 105.876, 'duration': 4.183}, {'end': 113.062, 'text': "you can't just say what is the probability of me getting a 5?", 'start': 110.059, 'duration': 3.003}, {'end': 119.967, 'text': "You have to say what is the probability of me getting between, let's say, a 4.5 and a 5.5?", 'start': 113.062, 'duration': 6.905}, {'end': 122.069, 'text': 'You have to give it some range.', 'start': 119.967, 'duration': 2.102}, {'end': 125.872, 'text': "And then your probability isn't given by just reading this graph.", 'start': 122.289, 'duration': 3.583}, {'end': 129.475, 'text': 'The probability is given by the area under that curve.', 'start': 126.432, 'duration': 3.043}, {'end': 133.378, 'text': 'It would be given by this area.', 'start': 130.555, 'duration': 2.823}], 'summary': 'Probability is determined by the area under the curve, not just reading the graph.', 'duration': 27.502, 'max_score': 105.876, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY105876.jpg'}], 'start': 1.866, 'title': 'Understanding the normal distribution', 'summary': 'Provides a comprehensive understanding of the normal distribution in inferential statistics, as well as an introduction to its application in probability density functions and calculus.', 'chapters': [{'end': 180.152, 'start': 1.866, 'title': 'Understanding the normal distribution', 'summary': 'Provides a comprehensive understanding of the normal distribution and its significance in inferential statistics, as well as an introduction to its application in probability density functions and calculus.', 'duration': 178.286, 'highlights': ['The normal distribution is fundamental in inferential statistics, underpinning much of the analysis based on data points.', 'The chapter aims to equip the audience with a deep understanding of the normal distribution, including its formula and practical applications.', 'Probability density functions in the context of the normal distribution involve calculating probabilities within specific ranges, requiring the integration of the probability density function over the designated interval.', 'The area under the probability density function curve, determined through calculus, represents the probability within a given range and is essential to understanding continuous probability distributions.']}], 'duration': 178.286, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY1866.jpg', 'highlights': ['The normal distribution is fundamental in inferential statistics, underpinning much of the analysis based on data points.', 'The chapter aims to equip the audience with a deep understanding of the normal distribution, including its formula and practical applications.', 'The area under the probability density function curve, determined through calculus, represents the probability within a given range and is essential to understanding continuous probability distributions.', 'Probability density functions in the context of the normal distribution involve calculating probabilities within specific ranges, requiring the integration of the probability density function over the designated interval.']}, {'end': 457.64, 'segs': [{'end': 266.965, 'src': 'embed', 'start': 233.308, 'weight': 2, 'content': [{'end': 234.549, 'text': 'So it might be a pretty good approximation.', 'start': 233.308, 'duration': 1.241}, {'end': 236.51, 'text': "And that's actually what I do in the other video,", 'start': 234.569, 'duration': 1.941}, {'end': 244.776, 'text': 'just to approximate the area under the curve and give you a good sense that the normal distribution is what the binomial distribution becomes,', 'start': 236.51, 'duration': 8.266}, {'end': 247.998, 'text': 'essentially if you have many, many, many, many trials.', 'start': 244.776, 'duration': 3.222}, {'end': 254.422, 'text': "And what's interesting about the normal distribution, just so you know, I don't know if I mentioned this already, this right here, this is the graph.", 'start': 248.418, 'duration': 6.004}, {'end': 258.647, 'text': 'And this is just another word.', 'start': 256.221, 'duration': 2.426}, {'end': 265.221, 'text': 'People might talk about the central limit theorem, but this is really kind of one of the most important or interesting things about our universe.', 'start': 258.666, 'duration': 6.555}, {'end': 266.965, 'text': 'Central limit theorem.', 'start': 265.321, 'duration': 1.644}], 'summary': 'Normal distribution approximates binomial distribution with many trials.', 'duration': 33.657, 'max_score': 233.308, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY233308.jpg'}, {'end': 309.385, 'src': 'embed', 'start': 282.275, 'weight': 1, 'content': [{'end': 285.316, 'text': 'if you got ahead every time and if you were to take the sum of them?', 'start': 282.275, 'duration': 3.041}, {'end': 289.138, 'text': 'as you approach an infinite number of flips, you approach the normal distribution.', 'start': 285.316, 'duration': 3.822}, {'end': 295.742, 'text': "And what's interesting about that is each of those trials, in the case of flipping a coin, each trial is a flip of the coin.", 'start': 290.139, 'duration': 5.603}, {'end': 299.239, 'text': "each of those trials don't have to have a normal distribution.", 'start': 296.457, 'duration': 2.782}, {'end': 309.385, 'text': 'So we could be talking about molecular interactions and every time compound X interacts with compound Y,', 'start': 299.739, 'duration': 9.646}], 'summary': 'As the number of coin flips approaches infinity, it approaches a normal distribution.', 'duration': 27.11, 'max_score': 282.275, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY282275.jpg'}, {'end': 373.904, 'src': 'embed', 'start': 324.495, 'weight': 0, 'content': [{'end': 333.581, 'text': 'And if people are trying to kind of, if you do take data points from something that is very, very complex and that it is the sum of arguably many,', 'start': 324.495, 'duration': 9.086}, {'end': 340.045, 'text': "many, almost infinite individual independent trials, it's a pretty good assumption to assume the normal distribution.", 'start': 333.581, 'duration': 6.464}, {'end': 344.248, 'text': "We'll do other videos where we talk about when it is a good assumption and when it isn't a good assumption.", 'start': 340.105, 'duration': 4.143}, {'end': 348.918, 'text': 'But anyway, just to digest this a little bit, and let me actually rewrite it.', 'start': 344.935, 'duration': 3.983}, {'end': 354.983, 'text': "This is what you'll see on Wikipedia, but this could be rewritten as 1 over sigma times.", 'start': 348.938, 'duration': 6.045}, {'end': 360.267, 'text': 'the square root of 2 pi times exp is just e to that power.', 'start': 354.983, 'duration': 5.284}, {'end': 363.989, 'text': "so it's just e to this whole thing over here.", 'start': 360.267, 'duration': 3.722}, {'end': 372.662, 'text': 'minus x, minus the mean squared over 2 sigma squared.', 'start': 363.989, 'duration': 8.673}, {'end': 373.904, 'text': 'This is a standard deviation.', 'start': 372.742, 'duration': 1.162}], 'summary': 'Normal distribution is a good assumption for complex data with many independent trials.', 'duration': 49.409, 'max_score': 324.495, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY324495.jpg'}, {'end': 470.508, 'src': 'embed', 'start': 440.972, 'weight': 5, 'content': [{'end': 445.314, 'text': 'You would actually say how many people are between 5.1 inches and 4.9 inches taller than the average.', 'start': 440.972, 'duration': 4.342}, {'end': 446.454, 'text': 'You have to give it a little bit of range.', 'start': 445.334, 'duration': 1.12}, {'end': 452.457, 'text': "Because no one is exactly, or it's almost infinitely impossible to the atom to be exactly 5 foot 9.", 'start': 446.474, 'duration': 5.983}, {'end': 456.039, 'text': "Even the definition of an inch isn't defined that particularly.", 'start': 452.457, 'duration': 3.582}, {'end': 457.64, 'text': "So that's how you use this function.", 'start': 456.339, 'duration': 1.301}, {'end': 464.883, 'text': 'You know, this is so heavily used in one it shows up in nature, but in all of inferential statistics,', 'start': 458.757, 'duration': 6.126}, {'end': 470.508, 'text': 'I think it behooves you to become as familiar with this formula as possible and, I guess, to make that happen.', 'start': 464.883, 'duration': 5.625}], 'summary': "Use a range when describing people taller than average, as exact measurements are practically impossible to obtain. it's important to become familiar with this formula in inferential statistics.", 'duration': 29.536, 'max_score': 440.972, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY440972.jpg'}], 'start': 180.532, 'title': 'Normal distribution', 'summary': 'Covers the numerical evaluation of normal distribution, its significance in the central limit theorem and binomial distribution, its appearance in independent trials like flipping coins, its function for determining probabilities within specified ranges, and its prevalence in nature and complex data analysis.', 'chapters': [{'end': 266.965, 'start': 180.532, 'title': 'Normal distribution approximation', 'summary': 'Discusses the numerical evaluation of the normal distribution and provides methods for approximation using trapezoids and rectangles, emphasizing its importance in the context of the central limit theorem and its relation to the binomial distribution.', 'duration': 86.433, 'highlights': ['The normal distribution is numerically evaluated due to the complexity of the analytical evaluation.', 'Approximation methods for the area under the curve involve using trapezoids and rectangles, providing a practical approach for understanding the normal distribution.', 'The normal distribution is emphasized as the result of many trials in the binomial distribution and is linked to the central limit theorem, highlighting its significance in the broader context of statistics.']}, {'end': 344.248, 'start': 267.907, 'title': 'Normal distribution in nature', 'summary': 'Explains the concept of normal distribution and how it appears as the sum of independent trials, such as in flipping coins, and its significance in complex data analysis, with relevance to molecular interactions and its prevalence in nature.', 'duration': 76.341, 'highlights': ['The normal distribution appears as the sum of independent trials, demonstrated through the example of flipping coins, wherein the sum of flips approaches the normal distribution as the number of flips approaches infinity.', 'In the context of molecular interactions, individual trials may not be normally distributed, but the sum of a large number of interactions results in a normal distribution, highlighting the significance of normal distribution in complex data analysis.', 'The normal distribution is prevalent in nature, making it a significant distribution to consider when dealing with complex and diverse data sets.']}, {'end': 457.64, 'start': 344.935, 'title': 'Normal distribution function', 'summary': 'Explains the normal distribution function, 1 over sigma times the square root of 2 pi times exp to the power of e, and its application in determining the probability of finding individuals within a specified range of height from the mean using the standard deviation and variance.', 'duration': 112.705, 'highlights': ['The normal distribution function can be rewritten as 1 over sigma times the square root of 2 pi times exp to the power of e, minus x minus the mean squared over 2 sigma squared.', 'The function is utilized to determine the probability of finding individuals within a specified range of height from the mean, using the standard deviation and variance.', 'The need to provide a range when calculating probabilities, as exact measurements are almost infinitely impossible due to the nature of measurement units and human characteristics.']}], 'duration': 277.108, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY180532.jpg', 'highlights': ['The normal distribution is prevalent in nature, making it a significant distribution to consider when dealing with complex and diverse data sets.', 'The normal distribution appears as the sum of independent trials, demonstrated through the example of flipping coins, wherein the sum of flips approaches the normal distribution as the number of flips approaches infinity.', 'The normal distribution is emphasized as the result of many trials in the binomial distribution and is linked to the central limit theorem, highlighting its significance in the broader context of statistics.', 'Approximation methods for the area under the curve involve using trapezoids and rectangles, providing a practical approach for understanding the normal distribution.', 'The normal distribution function can be rewritten as 1 over sigma times the square root of 2 pi times exp to the power of e, minus x minus the mean squared over 2 sigma squared.', 'The need to provide a range when calculating probabilities, as exact measurements are almost infinitely impossible due to the nature of measurement units and human characteristics.']}, {'end': 780.564, 'segs': [{'end': 512.182, 'src': 'embed', 'start': 477.502, 'weight': 3, 'content': [{'end': 486.289, 'text': 'So if I were to take this and I like to just maybe help you memorize it this could be rewritten as if we take the sigma into the square root sign.', 'start': 477.502, 'duration': 8.787}, {'end': 493.612, 'text': 'if we take the standard deviation in there, it becomes 1 over the square root of 2 pi Sigma squared.', 'start': 486.289, 'duration': 7.323}, {'end': 496.033, 'text': "I've never seen it written this way, but it gives me a little intuition.", 'start': 493.892, 'duration': 2.141}, {'end': 500.395, 'text': "Sigma squared, it's always written as sigma squared, but it's really just the variance.", 'start': 496.753, 'duration': 3.642}, {'end': 503.557, 'text': 'And the variance is what you calculate before you calculate the standard deviation.', 'start': 500.435, 'duration': 3.122}, {'end': 504.918, 'text': "So that's interesting.", 'start': 504.017, 'duration': 0.901}, {'end': 506.619, 'text': 'And then this top right here.', 'start': 505.418, 'duration': 1.201}, {'end': 512.182, 'text': 'this could be written as e to the minus 1 half times.', 'start': 506.619, 'duration': 5.563}], 'summary': 'Reformulating the standard deviation formula, emphasizing variance as a precursor, and introducing an alternative representation for better understanding.', 'duration': 34.68, 'max_score': 477.502, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY477502.jpg'}, {'end': 569.377, 'src': 'heatmap', 'start': 536.064, 'weight': 0, 'content': [{'end': 541.305, 'text': 'x minus mu is the mean.', 'start': 536.064, 'duration': 5.241}, {'end': 541.965, 'text': "So that's here.", 'start': 541.385, 'duration': 0.58}, {'end': 543.006, 'text': "So that's this distance.", 'start': 541.985, 'duration': 1.021}, {'end': 546.306, 'text': 'And this is the standard deviation, which is this distance.', 'start': 543.986, 'duration': 2.32}, {'end': 552.794, 'text': 'in here tells me how many standard deviations I am away from the mean.', 'start': 549.093, 'duration': 3.701}, {'end': 555.434, 'text': "And that's actually called the standard z-score I talked about in the other video.", 'start': 552.834, 'duration': 2.6}, {'end': 557.275, 'text': 'And then we square that.', 'start': 555.914, 'duration': 1.361}, {'end': 560.355, 'text': 'And then we take this to the minus 1 half.', 'start': 557.775, 'duration': 2.58}, {'end': 561.435, 'text': 'Well, let me rewrite that.', 'start': 560.395, 'duration': 1.04}, {'end': 569.377, 'text': "If I were to write e to the minus 1 half times a, that's the same thing as e to the a to the minus 1 half.", 'start': 561.475, 'duration': 7.902}], 'summary': 'Explanation of standard z-score and its calculation.', 'duration': 33.313, 'max_score': 536.064, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY536064.jpg'}, {'end': 692.891, 'src': 'embed', 'start': 665.561, 'weight': 1, 'content': [{'end': 670.623, 'text': 'we just say 2 pi times our variance times e to the number of standard deviations.', 'start': 665.561, 'duration': 5.062}, {'end': 671.684, 'text': 'we are away from the mean.', 'start': 670.623, 'duration': 1.061}, {'end': 672.464, 'text': 'You square that.', 'start': 671.784, 'duration': 0.68}, {'end': 674.245, 'text': 'You take the square root of that thing.', 'start': 672.884, 'duration': 1.361}, {'end': 677.86, 'text': "And inverted, and that's the normal distribution.", 'start': 674.938, 'duration': 2.922}, {'end': 681.323, 'text': "So anyway, I wanted to do that just because I thought it was neat and it's interesting to play around with it.", 'start': 677.88, 'duration': 3.443}, {'end': 685.206, 'text': "And that way, if you see it in any of these other forms in the rest of your life, you won't say what's that?", 'start': 681.363, 'duration': 3.843}, {'end': 687.828, 'text': 'I thought the normal distribution was this, or was this?', 'start': 685.426, 'duration': 2.402}, {'end': 688.508, 'text': 'And now you know.', 'start': 687.888, 'duration': 0.62}, {'end': 692.891, 'text': "With that said, let's play around a little bit with this normal distribution.", 'start': 689.329, 'duration': 3.562}], 'summary': 'Formula for normal distribution: 2πσe^(±z), to understand and apply it in practical scenarios.', 'duration': 27.33, 'max_score': 665.561, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY665561.jpg'}, {'end': 766.559, 'src': 'heatmap', 'start': 700.739, 'weight': 2, 'content': [{'end': 705.322, 'text': "So right now it's plotting it with a mean of 0 and a standard deviation of 4.", 'start': 700.739, 'duration': 4.583}, {'end': 707.944, 'text': 'And I just write the variance here just for your information.', 'start': 705.322, 'duration': 2.622}, {'end': 709.685, 'text': 'The variance is just the standard deviation squared.', 'start': 707.984, 'duration': 1.701}, {'end': 720.993, 'text': "And so what happens when you change the mean? So if the mean goes from 0 to let's say it goes to 5, notice this graph just shifted to the right by 5.", 'start': 710.286, 'duration': 10.707}, {'end': 721.813, 'text': 'It was centered here.', 'start': 720.993, 'duration': 0.82}, {'end': 722.874, 'text': "Now it's centered over here.", 'start': 721.953, 'duration': 0.921}, {'end': 730.849, 'text': 'Minus 5, what happens? The whole bell curve just shifts 5 to the left from the center.', 'start': 725.123, 'duration': 5.726}, {'end': 733.352, 'text': 'Now what happens when you change the standard deviation?', 'start': 731.33, 'duration': 2.022}, {'end': 740.517, 'text': 'The standard deviation is a measure of The variance is the average squared distance from the mean.', 'start': 733.372, 'duration': 7.145}, {'end': 741.997, 'text': 'The standard deviation is the square root of that.', 'start': 740.537, 'duration': 1.46}, {'end': 746.198, 'text': "So it's kind of, not exactly, but kind of the average distance from the mean.", 'start': 742.037, 'duration': 4.161}, {'end': 751.419, 'text': 'So the smaller the standard deviation, the closer a lot of the points are going to be to the mean.', 'start': 746.258, 'duration': 5.161}, {'end': 753.539, 'text': 'So we should get kind of a narrower graph.', 'start': 751.439, 'duration': 2.1}, {'end': 754.699, 'text': "And let's see if that happens.", 'start': 753.559, 'duration': 1.14}, {'end': 758, 'text': 'So when the standard deviation is 2, we see that.', 'start': 754.719, 'duration': 3.281}, {'end': 762.76, 'text': "The graph, you're more likely to be really close to the mean than further away.", 'start': 758.7, 'duration': 4.06}, {'end': 766.559, 'text': "And if you make the standard deviation, I don't know, if you make it 10.", 'start': 762.78, 'duration': 3.779}], 'summary': 'Changing mean and standard deviation shifts and narrows the bell curve graph.', 'duration': 65.82, 'max_score': 700.739, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY700739.jpg'}, {'end': 801.233, 'src': 'embed', 'start': 766.559, 'weight': 4, 'content': [{'end': 768.28, 'text': 'All of a sudden, you get a really flat graph.', 'start': 766.559, 'duration': 1.721}, {'end': 769.92, 'text': 'And this thing keeps going on forever.', 'start': 768.3, 'duration': 1.62}, {'end': 771.821, 'text': "And that's a key difference.", 'start': 770.52, 'duration': 1.301}, {'end': 774.002, 'text': 'The binomial distribution is always finite.', 'start': 771.881, 'duration': 2.121}, {'end': 775.842, 'text': 'You can only have a finite number of values.', 'start': 774.022, 'duration': 1.82}, {'end': 780.564, 'text': 'While the normal distribution is defined over the entire real number line.', 'start': 776.262, 'duration': 4.302}, {'end': 791.848, 'text': 'So the probability, if you have a mean of minus 5 and a standard deviation of 10, the probability of getting 1, 000 here is very, very low.', 'start': 780.984, 'duration': 10.864}, {'end': 793.849, 'text': 'But there is some probability.', 'start': 791.888, 'duration': 1.961}, {'end': 801.233, 'text': "there's some probability that all of the atoms in my body just arrange perfectly, that I fall through the seat I'm sitting on.", 'start': 795.25, 'duration': 5.983}], 'summary': 'Binomial distribution is finite, normal distribution spans real numbers, low probability of extreme values.', 'duration': 34.674, 'max_score': 766.559, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY766559.jpg'}], 'start': 458.757, 'title': 'Z-score formula and normal distribution', 'summary': 'Discusses the z-score formula, emphasizing its components and their role in calculating standard deviation and variance. it also explores the properties and applications of the normal distribution, showcasing its formula variations and demonstrating the impact of mean and standard deviation on its graph.', 'chapters': [{'end': 561.435, 'start': 458.757, 'title': 'Understanding z-score formula', 'summary': 'Discusses the z-score formula, highlighting its components and their role in calculating the standard deviation and variance, with emphasis on the relationship between mean, data point, and standard deviation.', 'duration': 102.678, 'highlights': ['The z-score formula is explained, breaking down its components and their significance in calculating standard deviation and variance, clarifying the relationship between mean, data point, and standard deviation.', 'The formula is rearranged to show the standard deviation within the square root sign, providing a different perspective on its structure and aiding in better understanding.', 'The variance is described as the value calculated prior to the standard deviation, offering insight into its role in the z-score formula.']}, {'end': 780.564, 'start': 561.475, 'title': 'Understanding normal distribution', 'summary': 'Explores the properties and applications of the normal distribution, demonstrating its formula variations and the impact of mean and standard deviation on its graph, showcasing its infinite range in contrast to the finite nature of the binomial distribution.', 'duration': 219.089, 'highlights': ['The formula for the normal distribution is explored, showcasing its various forms and the significance of pi and e within it.', 'The impact of mean and standard deviation on the graph of the normal distribution is demonstrated, with clear visual shifts observed as these parameters are altered.', 'Comparison of the normal distribution to the binomial distribution, highlighting the infinite range of the former in contrast to the finite nature of the latter.']}], 'duration': 321.807, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY458757.jpg', 'highlights': ['The z-score formula is explained, breaking down its components and their significance in calculating standard deviation and variance, clarifying the relationship between mean, data point, and standard deviation.', 'The formula for the normal distribution is explored, showcasing its various forms and the significance of pi and e within it.', 'The impact of mean and standard deviation on the graph of the normal distribution is demonstrated, with clear visual shifts observed as these parameters are altered.', 'The formula is rearranged to show the standard deviation within the square root sign, providing a different perspective on its structure and aiding in better understanding.', 'Comparison of the normal distribution to the binomial distribution, highlighting the infinite range of the former in contrast to the finite nature of the latter.', 'The variance is described as the value calculated prior to the standard deviation, offering insight into its role in the z-score formula.']}, {'end': 1015.001, 'segs': [{'end': 828.704, 'src': 'embed', 'start': 780.984, 'weight': 0, 'content': [{'end': 791.848, 'text': 'So the probability, if you have a mean of minus 5 and a standard deviation of 10, the probability of getting 1, 000 here is very, very low.', 'start': 780.984, 'duration': 10.864}, {'end': 793.849, 'text': 'But there is some probability.', 'start': 791.888, 'duration': 1.961}, {'end': 801.233, 'text': "there's some probability that all of the atoms in my body just arrange perfectly, that I fall through the seat I'm sitting on.", 'start': 795.25, 'duration': 5.983}, {'end': 805.955, 'text': "It's very unlikely, and it probably won't happen in the life of the universe, but it can happen.", 'start': 801.273, 'duration': 4.682}, {'end': 812.978, 'text': 'And that could be described by a normal distribution, because it says anything can happen, although it could be very, very, very unprobable.', 'start': 806.375, 'duration': 6.603}, {'end': 820.422, 'text': 'So the thing I talked about at the beginning of the video is when you figure out a normal distribution.', 'start': 813.919, 'duration': 6.503}, {'end': 823.083, 'text': "you can't just look at this point on the graph.", 'start': 820.422, 'duration': 2.661}, {'end': 824.564, 'text': 'Let me get the pen tool back.', 'start': 823.123, 'duration': 1.441}, {'end': 828.704, 'text': 'You have to figure out the area under the curve between two points.', 'start': 825.202, 'duration': 3.502}], 'summary': 'Mean of -5, std dev of 10, probability of 1,000 very low, described by normal distribution.', 'duration': 47.72, 'max_score': 780.984, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY780984.jpg'}, {'end': 885.066, 'src': 'embed', 'start': 851.323, 'weight': 1, 'content': [{'end': 864.393, 'text': "So you have to say the probability between, let's say, minus, and actually I can type it in here I can say the probability between, let's say,", 'start': 851.323, 'duration': 13.07}, {'end': 870.775, 'text': 'minus 0.005 and plus 0.05..', 'start': 864.393, 'duration': 6.382}, {'end': 872.156, 'text': 'is, well, it rounded.', 'start': 870.775, 'duration': 1.381}, {'end': 873.557, 'text': "So it says they're close to 0.", 'start': 872.196, 'duration': 1.361}, {'end': 873.918, 'text': 'Let me do it.', 'start': 873.557, 'duration': 0.361}, {'end': 878.161, 'text': 'Between minus 1 and between 1.', 'start': 873.958, 'duration': 4.203}, {'end': 879.822, 'text': 'It calculated at 7%.', 'start': 878.161, 'duration': 1.661}, {'end': 881.764, 'text': "And I'll show you how I calculated this in a second.", 'start': 879.822, 'duration': 1.942}, {'end': 885.066, 'text': 'So let me get the screen drawn to a log.', 'start': 883.485, 'duration': 1.581}], 'summary': 'Probability between -0.005 and +0.05 is around 7%.', 'duration': 33.743, 'max_score': 851.323, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY851323.jpg'}, {'end': 991.227, 'src': 'embed', 'start': 942.183, 'weight': 2, 'content': [{'end': 950.167, 'text': "Our mean is minus 5, so it's x plus 5 over the standard deviation squared, which is the variance.", 'start': 942.183, 'duration': 7.984}, {'end': 953.877, 'text': "So that's 100.", 'start': 950.207, 'duration': 3.67}, {'end': 955.241, 'text': 'squared dx.', 'start': 953.877, 'duration': 1.364}, {'end': 961.739, 'text': 'This is what this number is right here, this 7%, or actually 0.07 is the area right under there.', 'start': 955.823, 'duration': 5.916}, {'end': 968.498, 'text': "Now, unfortunately for us in the world, this isn't an easy integral to evaluate analytically, even for those of us who know our calculus.", 'start': 962.256, 'duration': 6.242}, {'end': 971.019, 'text': 'So this tends to be done numerically.', 'start': 968.959, 'duration': 2.06}, {'end': 973.92, 'text': 'And kind of an easy way to do this.', 'start': 971.419, 'duration': 2.501}, {'end': 980.103, 'text': 'well, not an easy way, but a function has been defined called the cumulative distribution function.', 'start': 973.92, 'duration': 6.183}, {'end': 982.544, 'text': 'that is a useful tool for figuring out this area.', 'start': 980.103, 'duration': 2.441}, {'end': 988.846, 'text': 'So what the cumulative distribution function is, is essentially, let me call it the cumulative distribution function.', 'start': 982.564, 'duration': 6.282}, {'end': 991.227, 'text': "It's a function of x.", 'start': 989.666, 'duration': 1.561}], 'summary': 'Mean is -5, variance is 100, using cumulative distribution function for analysis.', 'duration': 49.044, 'max_score': 942.183, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY942183.jpg'}], 'start': 780.984, 'title': 'Normal distribution and cumulative distribution function', 'summary': 'Covers the concept of normal distribution, including how to calculate probability between two points on the curve, using the example of a mean of -5 and a standard deviation of 10, demonstrating a 7% probability between -1 and 1. it also explores the calculation of a normal distribution function and explains the concept of cumulative distribution function, commonly used in numerical analysis.', 'chapters': [{'end': 914.461, 'start': 780.984, 'title': 'Understanding the normal distribution', 'summary': 'Explains the concept of a normal distribution and how to calculate the probability between two points on the curve, using the example of a mean of -5 and a standard deviation of 10, and demonstrates the calculation of the probability between -1 and 1 as 7%.', 'duration': 133.477, 'highlights': ['The probability of getting 1,000 with a mean of -5 and a standard deviation of 10 is very low, but not impossible.', 'Describing the concept of a normal distribution and how it encompasses all probabilities, no matter how unlikely.', 'Demonstrating the calculation of the probability between -1 and 1 on the normal distribution curve, yielding a result of 7%.']}, {'end': 968.498, 'start': 914.461, 'title': 'Normal distribution calculation', 'summary': 'Explores the calculation of a normal distribution function with a mean of -5 and a standard deviation of 10, resulting in an area of 0.07 under the curve.', 'duration': 54.037, 'highlights': ['The standard deviation is 10 times the square root of 2 pi times e to the power of -0.5 times (x plus 5) over the variance, which is 100, squared dx.', 'The area under the curve for the given normal distribution is 0.07.']}, {'end': 1015.001, 'start': 968.959, 'title': 'Cumulative distribution function', 'summary': 'Explains the concept of cumulative distribution function which is a useful tool for finding the area under the curve and determining the probability of landing at a value less than a given x, aiding in understanding probability density function. it is commonly used in numerical analysis.', 'duration': 46.042, 'highlights': ['Cumulative distribution function is a useful tool for finding the area under the curve and determining the probability of landing at a value less than a given x.', 'Understanding probability density function is aided by the concept of cumulative distribution function.', 'It is commonly used in numerical analysis.']}], 'duration': 234.017, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY780984.jpg', 'highlights': ['The probability of getting 1,000 with a mean of -5 and a standard deviation of 10 is very low, but not impossible.', 'Demonstrating the calculation of the probability between -1 and 1 on the normal distribution curve, yielding a result of 7%.', 'The area under the curve for the given normal distribution is 0.07.', 'Cumulative distribution function is a useful tool for finding the area under the curve and determining the probability of landing at a value less than a given x.', 'Describing the concept of a normal distribution and how it encompasses all probabilities, no matter how unlikely.']}, {'end': 1582.547, 'segs': [{'end': 1046.878, 'src': 'embed', 'start': 1017.042, 'weight': 0, 'content': [{'end': 1020.188, 'text': "dx And there's actually an Excel.", 'start': 1017.042, 'duration': 3.146}, {'end': 1029.892, 'text': 'When you actually use the Excel normal distribution function, you say norm distribution, you have to give it your x value.', 'start': 1020.468, 'duration': 9.424}, {'end': 1032.113, 'text': 'You give it the mean.', 'start': 1031.152, 'duration': 0.961}, {'end': 1034.054, 'text': 'You give it the standard deviation.', 'start': 1032.673, 'duration': 1.381}, {'end': 1040.876, 'text': 'And then you say whether you want the cumulative distribution, in which case you say true, or you want just this normal distribution,', 'start': 1034.374, 'duration': 6.502}, {'end': 1041.637, 'text': 'which you say false.', 'start': 1040.876, 'duration': 0.761}, {'end': 1046.878, 'text': 'So if you wanted to graph this right here, you would say false in caps.', 'start': 1041.656, 'duration': 5.222}], 'summary': "Excel's norm distribution function uses x value, mean, and standard deviation to calculate cumulative or normal distribution.", 'duration': 29.836, 'max_score': 1017.042, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY1017042.jpg'}, {'end': 1096.116, 'src': 'embed', 'start': 1066.726, 'weight': 2, 'content': [{'end': 1072.593, 'text': 'So this is a cumulative distribution function for the same For this, this is a normal distribution.', 'start': 1066.726, 'duration': 5.867}, {'end': 1073.833, 'text': "Here's a cumulative distribution.", 'start': 1072.633, 'duration': 1.2}, {'end': 1074.373, 'text': 'And just so you get.', 'start': 1073.853, 'duration': 0.52}, {'end': 1079.735, 'text': 'the intuition is, if you want to know what is the probability that I get a value less than 20, right?', 'start': 1074.373, 'duration': 5.362}, {'end': 1082.496, 'text': 'So I could get any value less than 20, given this distribution.', 'start': 1079.755, 'duration': 2.741}, {'end': 1086.258, 'text': 'The cumulative distribution right here.', 'start': 1084.357, 'duration': 1.901}, {'end': 1088.799, 'text': 'let me make it so you can see the.', 'start': 1086.258, 'duration': 2.541}, {'end': 1096.116, 'text': "if you go to 20, You just go right to that point there and you say wow, the probability of getting 20 or less, it's pretty high.", 'start': 1088.799, 'duration': 7.317}], 'summary': 'Cumulative distribution function showing probability of getting a value less than 20 in a normal distribution.', 'duration': 29.39, 'max_score': 1066.726, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY1066726.jpg'}, {'end': 1244.058, 'src': 'embed', 'start': 1213.101, 'weight': 4, 'content': [{'end': 1217.282, 'text': "Now, one thing that shows up a lot is what's the probability that you land within the standard deviation?", 'start': 1213.101, 'duration': 4.181}, {'end': 1220.843, 'text': 'And just so you know this graph, the central line right here.', 'start': 1218.202, 'duration': 2.641}, {'end': 1221.563, 'text': 'this is the mean.', 'start': 1220.843, 'duration': 0.72}, {'end': 1228.665, 'text': 'And then these two lines I drew right here, these are one standard deviation below and one standard deviation above the mean.', 'start': 1222.443, 'duration': 6.222}, {'end': 1234.552, 'text': "Some people think, what's the probability that I land within one standard deviation of the mean? Well, that's easy to do.", 'start': 1230.569, 'duration': 3.983}, {'end': 1236.333, 'text': "What I can do is I'll just click on this.", 'start': 1234.572, 'duration': 1.761}, {'end': 1244.058, 'text': "And I could call this, what's the probability that I land between, let's see, one standard deviation, the mean is minus 5.", 'start': 1237.093, 'duration': 6.965}], 'summary': 'Explaining probability within one standard deviation of the mean.', 'duration': 30.957, 'max_score': 1213.101, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY1213101.jpg'}, {'end': 1430.035, 'src': 'embed', 'start': 1404.328, 'weight': 1, 'content': [{'end': 1410.471, 'text': 'and I really want you to play with this and play with the formula and get an intuitive feeling for this, the cumulative distribution function,', 'start': 1404.328, 'duration': 6.143}, {'end': 1412.772, 'text': 'and think a lot about how it relates to the binomial distribution.', 'start': 1410.471, 'duration': 2.301}, {'end': 1414.133, 'text': 'And I covered that in the last video.', 'start': 1412.792, 'duration': 1.341}, {'end': 1417.895, 'text': 'To plot this, I just took each of these points.', 'start': 1415.213, 'duration': 2.682}, {'end': 1423.853, 'text': 'I went to plot the points between minus 20 and 20, and I just incremented by 1.', 'start': 1417.935, 'duration': 5.918}, {'end': 1425.413, 'text': 'I just decided to increment by 1.', 'start': 1423.853, 'duration': 1.56}, {'end': 1426.934, 'text': "So this isn't a continuous curve.", 'start': 1425.413, 'duration': 1.521}, {'end': 1430.035, 'text': "It's actually just plotting a point at each point and connecting it with a line.", 'start': 1426.974, 'duration': 3.061}], 'summary': 'Understanding cumulative distribution function and its relation to binomial distribution.', 'duration': 25.707, 'max_score': 1404.328, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY1404328.jpg'}], 'start': 1017.042, 'title': 'Using excel for normal distribution and understanding cumulative distribution function', 'summary': "Discusses using excel's normal distribution function and specifying parameters such as x value, mean, and standard deviation, along with the choice between cumulative and normal distribution. it also explains the concept of cumulative distribution function for a normal distribution, highlighting probability calculations for specific values and the relationship between standard deviation and probability, with emphasis on practical application in financial modeling.", 'chapters': [{'end': 1066.706, 'start': 1017.042, 'title': 'Using excel for normal distribution', 'summary': 'Discusses using the excel normal distribution function, specifying parameters such as x value, mean, and standard deviation, as well as choosing between cumulative and normal distribution.', 'duration': 49.664, 'highlights': ['The Excel normal distribution function requires specifying the x value, mean, and standard deviation, and choosing between cumulative and normal distribution.', "When using the Excel normal distribution function, 'false' is used to graph the normal distribution, while 'true' is used for the cumulative distribution function."]}, {'end': 1582.547, 'start': 1066.726, 'title': 'Understanding cumulative distribution function', 'summary': 'Explains the concept of cumulative distribution function for a normal distribution, highlighting the probability calculations for specific values and the relationship between standard deviation and probability, with emphasis on the practical application of the concept in financial modeling.', 'duration': 515.821, 'highlights': ['The chapter explains the concept of cumulative distribution function for a normal distribution, highlighting the probability calculations for specific values.', 'The relationship between standard deviation and probability is emphasized, with practical application in financial modeling discussed.', 'The process of calculating the cumulative distribution function and its practical application in financial modeling are explained.']}], 'duration': 565.505, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hgtMWR3TFnY/pics/hgtMWR3TFnY1017042.jpg', 'highlights': ['The Excel normal distribution function requires specifying the x value, mean, and standard deviation, and choosing between cumulative and normal distribution.', 'The process of calculating the cumulative distribution function and its practical application in financial modeling are explained.', 'The chapter explains the concept of cumulative distribution function for a normal distribution, highlighting the probability calculations for specific values.', "When using the Excel normal distribution function, 'false' is used to graph the normal distribution, while 'true' is used for the cumulative distribution function.", 'The relationship between standard deviation and probability is emphasized, with practical application in financial modeling discussed.']}], 'highlights': ['The normal distribution is fundamental in inferential statistics, underpinning much of the analysis based on data points.', 'The normal distribution is prevalent in nature, making it a significant distribution to consider when dealing with complex and diverse data sets.', 'The z-score formula is explained, breaking down its components and their significance in calculating standard deviation and variance, clarifying the relationship between mean, data point, and standard deviation.', 'The Excel normal distribution function requires specifying the x value, mean, and standard deviation, and choosing between cumulative and normal distribution.', 'The area under the probability density function curve, determined through calculus, represents the probability within a given range and is essential to understanding continuous probability distributions.']}