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How to Calculate Standard Deviation
 
03:55
Follow these five steps to calculate standard deviation. Also includes the standard deviation formula. Here's the video transcript: "How to Calculate Standard Deviation How many vegetables do you have in your fridge? Is that a common amount or are you an outlier? We can use standard deviation to know whether someone’s behavior is normal or extraordinary. Standard deviation, often calculated along with the mean of a data set, tells us how spread out the data is. It is used for data that is normally distributed and can be easily calculated using a graphing calculator or spreadsheet software, but it can also be calculated with a few math operations. We’re going to use an example involving the number of vegetables five of our friends have in their fridges. They have 2, 3, 4, 7, and 9 vegetables. To calculate the standard deviation, the first step is to calculate the mean of the data set, denoted by x with a line over it, also called x-bar. In this case, the mean would be (2 + 3 + 4 + 7 + 9) / 5 = 5. Our average friend has 5 vegetables in their fridge. The second step is to subtract the mean from each data point to find the differences. It’s helpful to use a table like this. 2 - 5 = -3, 3 - 5 = -2, 4 - 5 = -1, 7 - 5 = 2, and 9 - 5 = 4. The third step is to square each difference. (This makes all the differences positive so they don’t cancel each other out and it also magnifies larger differences and minimizes smaller differences.) -32 = 9, -22 = 4, -12 = 1, 22 = 4, and 42 = 16. The fourth step is to calculate the mean of the squared differences. (9 + 4 + 1 + 4 + 16) / 5 = 6.8. The final step is to take the square root. (This counteracts the squaring we did earlier and allows the standard deviation to be expressed in the original units.) The square root of 6.8 is about 2.6 and that’s the standard deviation. We're done! The mean number of vegetables is 5 with a standard deviation of 2.6 veggies. Knowing that about ⅔ of the data fall within one standard deviation of the mean (assuming the data is normally distributed), we can say that about ⅔ of our friends have between 2.4 and 7.6 vegetables in their fridges. To recap, these are the five steps for calculating standard deviation: 1. Calculate the mean. 2. Subtract the mean from each data point. 3. Square each difference. 4. Calculate the mean of the squared differences. 5. Take the square root. Using symbols, the equation for calculating standard deviation looks like this [see video]... Lower case sigma stands for standard deviation of a population. Upper case sigma tells us to calculate the sum for each instance. X is each data point. X bar is the mean of the data points. And n is the number of data points. Keep in mind that there is a similar formula that divides by n-1. That formula is used when you only have data for a sample of the population. Hope this was helpful. See you next time!"
Views: 239853 Jeremy Jones
Law of Large Numbers - Explained and Visualized
 
04:18
Video transcript: How does an insurance company decide how much to charge for car insurance? How do casinos set up payout structures to make sure that they make a profit? How does a poker player decide whether or not to fold? How do we efficiently measure the appeal of a political candidate? The answers to these questions are informed by the law of large numbers. The law of large numbers states that as the number of trials or observations increases, the actual or observed probability approaches the theoretical or expected probability. This is important to understand because it allows us to predict and have confidence in how events will play out in the long run. Let’s take a common example—flipping a coin. Assuming the coin is fair, we know that the theoretical probability of flipping heads is .5 or 50%. However, that doesn’t guarantee that if we flip a coin 10 times we’ll get 5 heads. But we can be confident that as we continue flipping a coin indefinitely the cumulative proportion of heads flipped should get closer and closer to 50%. Looking at this graphically helps to illustrate the concept. I just flipped a coin 20 times and these are the results. After each flip, we’ll calculate the cumulative proportion of heads so far. So the first flip is tails, so our current proportion is 0 heads out of 1 flip—0%. The second flip is also tails, so now it’s 0 heads out of 2 flips— still 0% heads. Next flip is tails again— 0%. Then heads, now we have 1 heads out of 4 flips— 25% heads. Flip again and it’s heads—40%. Heads again and we finally hit our theoretical probability of 50% for the first time. Let’s keep going…(video plays out the rest of the 20 flips). This graph shows the observed probability approaching the theoretical probability. One common misconception, referred to as the gambler’s fallacy, is that if the first four flips were tails, you’re more likely to get heads on the next flip because the proportion is supposed to even out to 50% heads. This is not the case because each flip of a coin is an independent event, its outcome is unaffected by all previous events. So if you start out with four tails in a row, it’s not that you are more likely to get heads, it’s just that in the grand scheme of things, four tails flips will get averaged with a huge number of flips that are expected to yield an even number of heads and tails, causing the proportion to approach 50% as the number of trials increases. Another version of the law of large numbers explains that the more people from a population that you sample, so the larger your sample size, assuming your sample is free from bias, the closer your sample average will be to the population average. Let’s say you have a group of 100 people. Each has some number of dollars in their wallet. If we ask one person how much money she has in her wallet, we’ll get our first observation ($49), which might be pretty far from the average of the group. After asking the second person ($29) and averaging that value with the first ($30), we are likely to have a better estimate of the group average. As we continue this process of adding observations and thereby increasing our sample size, we’ll generally get better and better estimates of the group’s average. SUMMARY So the law of large numbers gives us a compass with which to navigate the randomness around us. Even though we can never predict the outcome of a single coin flip, we can know that over time about half of the flips will be heads. This knowledge underpins insurance, gambling, and investing. And in general, the principle supports the idea that a well-founded strategy that is followed consistently should win out over time, even though it might result in a few negative events along the way.
Views: 66400 Jeremy Jones
Standard Deviation - Explained and Visualized
 
03:43
This video's more focused on the concept. This one explains how it's calculated: https://www.youtube.com/watch?v=WVx3MYd-Q9w Video transcript: "Have we discovered a new particle in physics? Is a manufacturing process out of control? What percentage of men are taller than Lebron James? How about taller than Yao Ming? All of these questions can be answered using the concept of standard deviation. For any set of data, the mean and standard deviation can be calculated. For example, five people may have the following amounts of money in their wallets: 21, 50, 62, 85, and 90. The mean is $61.60 and the standard deviation is $28.01. How much does the data vary from the average? Standard deviation is a measure of spread, that is, how spread out a set of data is. A low standard deviation tells us that the data is closely clustered around the mean (or average), while a high standard deviation indicates that the data is dispersed over a wider range of values. It is used when the distribution of data is approximately normal, resembling a bell curve. Standard deviation is commonly used to understand whether a specific data point is “standard” and expected or unusual and unexpected. Standard deviation is represented by the lowercase greek letter sigma. A data point’s distance from the mean can be measured by the number of standard deviations that it is above or below the mean. A data point that is beyond a certain number of standard deviations from the mean represents an outcome that is significantly above or below the average. This can be used to determine whether a result is statistically significant or part of expected variation, such as whether a bottle with an extra ounce of soda is to be expected or warrants further investigation into the production line. The 68-95-99.7 rule tells us that about 68% of the data fall within one standard deviation of the mean. About 95% of data fall within two standard deviations of the mean. And about 99.7% of data fall within 3 standard deviations of the mean. The average height of an American adult male is 5’10, with a standard deviation of 3 inches. Using the 68-95-99.7 rule, this means that 68% of American men are 5’10 plus or minus 3 inches, 95% of American men are 5’10 plus or minus 6 inches, and 99.7% of American men are 5’10 plus or minus 9 inches. So, this means only about .3% of American men deviate more than 9 inches from the average, with .15% taller than 6’7 and .15% shorter than 5’1. This reasoning suggests that Lebron James is 1 in 2500 and Yao Ming is 1 in 450 million. In particle physics, scientists have what are called 5-sigma results, results that are five standard deviations above or below the mean. A result that varies this much can signify a discovery as it has only a 1 in 3.5 million chance that it is due to random fluctuation. In summary, standard deviation is a measure of spread. Along with the mean, the standard deviation allows us to determine whether a value is statistically significant or part of expected variation."
Views: 947371 Jeremy Jones
How to Calculate Permutations and Combinations - Probability
 
09:22
Animated explanation of how to calculate permutations with repetition, permutations without repetition, and combinations without repetition. Useful when trying to calculate probabilities. How much more secure is a 6-digit passcode than a 4-digit passcode? How many different two-card hands could you get in Texas Hold ‘em? How many different outcomes are there for the 100m final? All of these questions can be answered using permutations and combinations, which are part of the field of mathematics awesomely called combinatorics and less awesome called counting. When working with permutations and combinations, we need to ask two questions to begin solving the problem: “Does order matter?” and “Is repetition possible?”
Views: 7104 Jeremy Jones
Monty Hall Problem Explained with Four Solutions
 
08:46
Explanation of the Monty Hall Problem with four solutions. Play the game at http://www.stayorswitch.com
Views: 25813 Jeremy Jones
Beginner Blender Tutorial: Wood Table (Boolean Modifier and Wood Texture)
 
13:13
This tutorial shows how to use the boolean modifier and add a wood grain texture.
Views: 19994 Jeremy Jones
Open Rocket Tutorial
 
19:28
Views: 13790 Jeremy Jones
Airfoil Blender Tutorial for 3D Printing
 
08:56
How to model an airfoil fin to be 3D printed for a model rocket.
Views: 1840 Jeremy Jones
Low Poly Rose Timelapse (Blender v2.76)
 
05:22
Tutorial video: https://www.youtube.com/watch?v=V_iNNrvonxo Music: Don't Stop by Audionautix is licensed under a Creative Commons Attribution license (https://creativecommons.org/licenses/by/4.0/) Artist: http://audionautix.com/
Views: 1543 Jeremy Jones
Blender Beginniner Tutorial: Low Poly Rose
 
42:50
This is a tutorial showing how to model a low poly rose.
Views: 4450 Jeremy Jones
Blender Beginner Tutorial - St. Louis Gateway Arch (Blender 2.76b)
 
14:19
This Blender 3D modeling tutorial goes over how to create the St. Louis Gateway Arch.
Views: 1382 Jeremy Jones
Beginner Blender Tutorial - Introduction to Camera Tracking - aka How to Showcase Your 3D Model
 
17:03
This tutorial demonstrates how to show off a completed 3D model using camera tracking.
Views: 1327 Jeremy Jones
The Multiplication Rule of Probability - Explained
 
03:12
Any time you want to know the chance of two events happening together, you can use the multiplication rule of probability. Independent events: P(A and B) = P(A) x P(B) Dependent events: P(A and B) = P(A) x P(B | A) ...where P(B | A) is the probability of event B given that event A happened. Have other topics you'd like to see a video on? Let me know in the comments! Video transcript: What’s the chance of rolling snake eyes? What’s the chance of flipping heads three times in a row? When calculating the probability of two or more events happening together, we can use the multiplication rule of probability. We’ll start with independent events where one event’s outcome has no effect on the other event’s outcome. For example, what’s the probability of rolling snake eyes? Each roll is an independent event because the value on one die has no influence on the value of the second die. The multiplication rule of probability says that the probability of two events A and B happening together is the probability of event A multiplied by the probability of event B - in this case, the probability of rolling a 1 on the first die, multiplied by the probability of rolling a 1 on the second die. This is the case if you’re rolling the dice together or one at a time. The probability of rolling a 1 on a die is one out of six, so the probability of rolling a 1 on both dice is ⅙ times ⅙. Across all 36 possible rolls of two dice, one of them is snake eyes. Let’s look at a second example - what’s the probability of flipping heads three times in a row? Well, it’s the probability of the first flip landing heads, multiplied by the probability of the second flip landing heads, multiplied by the probability of the third flip landing heads. ½ x ½ x ½ = ⅛. What if the events are dependent? What if the second event’s probability is based on the outcome of the first event? In this case, the probability of the events happening together is a little more complicated - the probability of event A happening multiplied by the probability of event B happening given that event A happened. For example, what’s the probability that you’ll draw an ace, hold onto it, and then draw a king? In this case, we’ll start with the probability of drawing an ace: four out of 52 cards are aces. Then, we need to calculate the probability of drawing a king, given that we’ve already drawn an ace, which is different than the probability of drawing a king from a full deck. There are four kings left in the deck, and there are 51 cards remaining since we’re holding onto an ace. We multiply these two probabilities together and we get a probability of 16/2652, about a 1 in 167 chance. In summary, if you want to know the likelihood of event A and event B happening, you can use the multiplication rule of probability. Make sure to identify whether the events are independent or dependent and adjust your calculation accordingly.
Views: 77 Jeremy Jones
Math Puzzle: The Miller and the Butcher
 
00:46
A miller uses a 40 lb rock to measure grain with a two-sided scale. He lends the rock to his friend the butcher. The butcher comes to return the rock and says, “I’m sorry but I dropped the rock and it broke into 4 pieces.” Upon seeing the pieces, the miller says, “Actually, this is great. With these 4 pieces I can measure any amount of grain from 1 to 40 lbs!” What are the weights of the 4 pieces?
Views: 566 Jeremy Jones

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