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Boundless Lecture Slides Available on the Boundless Teaching Platform Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Using Boundless Presentations The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. Get started now at: http://boundless.com/teaching-platform Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. If you have any questions or problems please email: [email protected] Free to share, print, make copies and changes. Get yours at www.boundless.com About Boundless Boundless is an innovative technology company making education more affordable and accessible for students everywhere. The company creates the world’s best open educational content in 20+ subjects that align to more than 1,000 popular college textbooks. Boundless integrates learning technology into all its premium books to help students study more efficiently at a fraction of the cost of traditional textbooks. The company also empowers educators to engage their students more effectively through customizable books and intuitive teaching tools as part of the Boundless Teaching Platform. More than 2 million learners access Boundless free and premium content each month across the company’s wide distribution platforms, including its website, iOS apps, Kindle books, and iBooks. To get started learning or teaching with Boundless, visit boundless.com. Free to share, print, make copies and changes. Get yours at www.boundless.com Continuous Random Variables The Normal Curve Normal Approximation Measurement Error Expected Value and Standard Error Normal Approximation for Probability Histograms Continuous Random Variables Free to share, print, make copies and changes. Get yours at www.boundless.com Continuous Random Variables > The Normal Curve The Normal Curve • Continuous Probability Distributions • The Uniform Distribution • The Exponential Distribution • The Normal Distribution • Graphing the Normal Distribution • The Standard Normal Curve • Finding the Area Under the Normal Curve Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/the-normal-curve-39/ Continuous Random Variables > Normal Approximation Normal Approximation • The Normal Approximation to the Binomial Distribution • The Scope of the Normal Approximation • Calculating a Normal Approximation • Change of Scale Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/normal-approximation-40/ Continuous Random Variables > Measurement Error Measurement Error • Bias • Chance Error • Outliers Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/measurement-error-41/ Continuous Random Variables > Expected Value and Standard Error Expected Value and Standard Error • Expected Value • Standard Error Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/expected-value-and-standard-error-42/ Continuous Random Variables > Normal Approximation for Probability Histograms Normal Approximation for Probability Histograms • Probability Histograms • Probability Histograms and the Normal Curve • Conclusion Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/normal-approximation-for-probabilityhistograms-43/ Appendix Free to share, print, make copies and changes. Get yours at www.boundless.com Continuous Random Variables Key terms • Accuracy the degree of closeness of measurements of a quantity to that quantity's actual (true) value • bell curve In mathematics, the bell-shaped curve that is typical of the normal distribution. • best fit line A line on a graph showing the general direction that a group of points seem to be heading. • binomial distribution the discrete probability distribution of the number of successes in a sequence of independent yes/no experiments, each of which yields success with probability • Box–Muller transformation A pseudo-random number sampling method for generating pairs of independent, standard, normally distributed (zero expectation, unit variance) random numbers, given a source of uniformly distributed random numbers. • central limit theorem The theorem that states: If the sum of independent identically distributed random variables has a finite variance, then it will be (approximately) normally distributed. • central limit theorem The theorem that states: If the sum of independent identically distributed random variables has a finite variance, then it will be (approximately) normally distributed. • central limit theorem The theorem that states: If the sum of independent identically distributed random variables has a finite variance, then it will be (approximately) normally distributed. • correlation One of the several measures of the linear statistical relationship between two random variables, indicating both the strength and direction of the relationship. • cumulant Any of a set of parameters of a one-dimensional probability distribution of a certain form. • cumulative distribution function The probability that a real-valued random variable with a given probability distribution will be found at a value less than or equal to . • datum A measurement of something on a scale understood by both the recorder (a person or device) and the reader (another person or device). 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Get yours at www.boundless.com Continuous Random Variables • discrete random variable obtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, …. • empirical rule That a normal distribution has 68% of its observations within one standard deviation of the mean, 95% within two, and 99.7% within three. • empirical rule That a normal distribution has 68% of its observations within one standard deviation of the mean, 95% within two, and 99.7% within three. • entropy A measure which quantifies the expected value of the information contained in a message. • Erlang distribution The distribution of the sum of several independent exponentially distributed variables. • independent not dependent; not contingent or depending on something else; free • integral the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed • interquartile range The difference between the first and third quartiles; a robust measure of sample dispersion. • law of large numbers The statistical tendency toward a fixed ratio in the results when an experiment is repeated a large number of times. • Lebesgue measure The unique complete translation-invariant measure for the -algebra which contains all -cells—in and which assigns a measure to each -cell equal to that -cell's volume (as defined in Euclidean geometry: i.e., the volume of the -cell equals the product of the lengths of its sides). • normal approximation The process of using the normal curve to estimate the shape of the distribution of a data set. • normal approximation The process of using the normal curve to estimate the shape of the distribution of a data set. 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Get yours at www.boundless.com Continuous Random Variables • normal probability plot a graphical technique used to assess whether or not a data set is approximately normally distributed • normal probability plot a graphical technique used to assess whether or not a data set is approximately normally distributed • normalization The process of removing statistical error in repeated measured data. • outlier a value in a statistical sample which does not fit a pattern that describes most other data points; specifically, a value that lies 1.5 IQR beyond the upper or lower quartile • p-value The probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. • Poisson process A stochastic process in which events occur continuously and independently of one another. • Precision the ability of a measurement to be reproduced consistently • random error an error which is a combination of results both higher and lower than the desired measurement; precision error • random error an error which is a combination of results both higher and lower than the desired measurement; precision error • random variable a quantity whose value is random and to which a probability distribution is assigned, such as the possible outcome of a roll of a die • real number An element of the set of real numbers; the set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers. • regression An analytic method to measure the association of one or more independent variables with a dependent variable. 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Get yours at www.boundless.com Continuous Random Variables • regression line A smooth curve fitted to the set of paired data in regression analysis; for linear regression the curve is a straight line. • standard normal distribution The normal distribution with a mean of zero and a standard deviation of one. • standard score The number of standard deviations an observation or datum is above the mean. • systematic error an error which consistently yields results either higher or lower than the correct measurement; accuracy error • systematic error an error which consistently yields results either higher or lower than the correct measurement; accuracy error • weighted average an arithmetic mean of values biased according to agreed weightings • z-score The standardized value of observation from a distribution that has mean and standard deviation . • z-score The standardized value of observation from a distribution that has mean and standard deviation . • z-score The standardized value of observation from a distribution that has mean and standard deviation . Free to share, print, make copies and changes. Get yours at www.boundless.com Continuous Random Variables Normal Probability Density The normal distribution is described by this probability density function. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Normal distribution." CC BY-SA https://en.wikipedia.org/wiki/Normal_distribution View on Boundless.com Continuous Random Variables Low Accuracy, High Precision This target shows an example of low accuracy (points are not close to center target) but high precision (points are close together). In this case, there is more systematic error than random error. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "High precision Low accuracy." CC BY-SA http://en.wikipedia.org/wiki/File:High_precision_Low_accuracy.svg View on Boundless.com Continuous Random Variables Probability Histogram This probability histogram shows the probabilities that 0, 1, 2, 3, or 4 heads will show up on four tosses of a fair coin. Free to share, print, make copies and changes. Get yours at www.boundless.com Hofstra University. "Histogram Generator." CC BY http://people.hofstra.edu/stefan_waner/stats/histogram.html View on Boundless.com Continuous Random Variables Normal Probability Plot The data points do not deviate far from the straight line, so we can assume the distribution is approximately normal. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Normprob." CC BY-SA http://en.wikipedia.org/wiki/File:Normprob.png View on Boundless.com Continuous Random Variables Memoryless Exponential Distributions If a random variable T is exponentially distributed, its conditional probability obeys this formula. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Exponential distribution." GNU FDL http://en.wikipedia.org/wiki/Exponential_distribution View on Boundless.com Continuous Random Variables The Normal Distribution This image shows the equation for the normal distribution. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Normal distribution." CC BY-SA https://en.wikipedia.org/wiki/Normal_distribution View on Boundless.com Continuous Random Variables High Accuracy, Low Precision This target shows an example of high accuracy (points are all close to center target) but low precision (points are not close together). In this case, there is more random error than systematic error. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "High accuracy Low precision." CC BY-SA http://en.wikipedia.org/wiki/File:High_accuracy_Low_precision.svg View on Boundless.com Continuous Random Variables Low Accuracy, High Precision This target shows an example of low accuracy (points are not close to center target) but high precision (points are close together). In this case, there is more systematic error than random error. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "High precision Low accuracy." CC BY-SA http://en.wikipedia.org/wiki/File:High_precision_Low_accuracy.svg View on Boundless.com Continuous Random Variables Approximately Normal - Probability Plot This is a sample of size 50 from a normal distribution, plotted as a normal probability plot. The plot looks fairly straight, indicating normality. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Normprob." CC BY-SA http://en.wikipedia.org/wiki/File:Normprob.png View on Boundless.com Continuous Random Variables Catching a Bus The Uniform Distribution can be used to calculate probability problems such as the probability of waiting for a bus for a certain amount of time. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Arriva T6 nearside." Public domain https://en.wikipedia.org/wiki/File:Arriva_T6_nearside.JPG View on Boundless.com Continuous Random Variables High Accuracy, Low Precision This target shows an example of high accuracy (points are all close to center target) but low precision (points are not close together). In this case, there is more random error than systematic error. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "High accuracy Low precision." CC BY-SA http://en.wikipedia.org/wiki/File:High_accuracy_Low_precision.svg View on Boundless.com Continuous Random Variables -Score Table The -score table is used to calculate probabilities for the standard normal distribution. Free to share, print, make copies and changes. Get yours at www.boundless.com Mandy's Notes. "Z-Score Chart." CC BY http://www.mandysnotes.com/Statistics/Z-Scores/Z-Score-Chart View on Boundless.com Continuous Random Variables Normal Area 1 This graph shows the area below 8.5. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX. "David Lane, Normal Approximation to the Binomial. June 12, 2013." CC BY 3.0 http://cnx.org/content/m11162/latest/ View on Boundless.com Continuous Random Variables The Bell Curve The graph of a normal distribution is known as a bell curve. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikispaces. "killianhma0809 - Normal Distribution." CC BY-SA http://killianhma0809.wikispaces.com/Normal+Distribution View on Boundless.com Continuous Random Variables Height of a Bell Curve The height of the graph at any x value can be found through this equation. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikispaces. "killianhma0809 - Normal Distribution." CC BY-SA http://killianhma0809.wikispaces.com/Normal+Distribution View on Boundless.com Continuous Random Variables Normal Area 2 This graph shows the area below 7.5. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX. "David Lane, Normal Approximation to the Binomial. June 12, 2013." CC BY 3.0 http://cnx.org/content/m11162/latest/ View on Boundless.com Continuous Random Variables Normal Approximation Approximation for the probability of 8 heads with the normal distribution. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX. "David Lane, Normal Approximation to the Binomial. June 12, 2013." CC BY 3.0 http://cnx.org/content/m11162/latest/ View on Boundless.com Continuous Random Variables Boxplot Versus Probability Density Function Boxplot and probability density function of a normal distribution . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia. "Boxplot vs PDF." CC BY-SA http://commons.wikimedia.org/wiki/File:Boxplot_vs_PDF.svg View on Boundless.com Continuous Random Variables Mean of Exponentially Distributed Random Variable Random variable X and rate parameter of . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Exponential distribution." GNU FDL http://en.wikipedia.org/wiki/Exponential_distribution View on Boundless.com Continuous Random Variables Variance of an Exponentially Distributed Random Variable Random variable X and rate parameter of . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Exponential distribution." GNU FDL http://en.wikipedia.org/wiki/Exponential_distribution View on Boundless.com Continuous Random Variables Graph 1 Bell curve visualizing a normal distribution with a relatively small standard deviation. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikispaces. "killianhma0809 - Normal Distribution." CC BY-SA http://killianhma0809.wikispaces.com/Normal+Distribution View on Boundless.com Continuous Random Variables Areas Under the Normal Curve This table gives the cumulative probability up to the standardized normal value . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikidot. "Using Normal Distributions - IB Math Stuff." CC BY-SA http://ibmathstuff.wikidot.com/usingnormaldistributions View on Boundless.com Continuous Random Variables -table The -score table is used to calculate probabilities for the standard normal distribution. Free to share, print, make copies and changes. Get yours at www.boundless.com http://statistics.wdfiles.com/local--files/ch7/normDistTable.pdf. CC BY-SA http://statistics.wdfiles.com/local--files/ch7/normDistTable.pdf View on Boundless.com Continuous Random Variables Normal Approximation The normal approximation to the binomial distribution for 12 coin flips. The smooth curve is the normal distribution. Note how well it approximates the binomial probabilities represented by the heights of the blue lines. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX. "David Lane, History of Normal Distribution. June 12, 2013." CC BY 3.0 http://cnx.org/content/m11164/latest/ View on Boundless.com Continuous Random Variables Central Limit Theorem A distribution being "smoothed out" by summation, showing original density of distribution and three subsequent summations Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Central limit thm." CC BY-SA http://en.wikipedia.org/wiki/File:Central_limit_thm.png View on Boundless.com Continuous Random Variables Normal Distribution and Scales Compares the various grading methods in a normal distribution. Includes: standard deviations, cumulative percentages, percentile equivalents, -scores, scores, and standard nine. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Normal distribution and scales." Public domain http://en.wikipedia.org/wiki/File:Normal_distribution_and_scales.gif View on Boundless.com Continuous Random Variables SDM This is the formula for the true standard deviation of the sample mean. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Standard error." GNU FDL http://en.wikipedia.org/wiki/Standard_error View on Boundless.com Continuous Random Variables Graph 2 Bell curve visualizing a normal distribution with a relatively large standard deviation. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikispaces. "killianhma0809 - Normal Distribution." CC BY-SA http://killianhma0809.wikispaces.com/Normal+Distribution View on Boundless.com Continuous Random Variables Law of Large Numbers An illustration of the law of large numbers using a particular run of rolls of a single die. As the number of rolls in this run increases, the average of the values of all the results approaches 3.5. While different runs would show a different shape over a small number of throws (at the left), over a large number of rolls (to the right) they would be extremely similar. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Largenumbers." Public domain http://en.wikipedia.org/wiki/File:Largenumbers.svg View on Boundless.com Continuous Random Variables Expected Value The computation of the expected value in our example. Free to share, print, make copies and changes. Get yours at www.boundless.com P.Mean Website. "Stats: Expected value and moments (July 29, 2005)." CC BY http://www.pmean.com/05/Moments.asp View on Boundless.com Continuous Random Variables Probability of Number of Girls The probabilities of the number of girls in a family of three children. Free to share, print, make copies and changes. Get yours at www.boundless.com P.Mean Website. "Stats: Expected value and moments (July 29, 2005)." CC BY http://www.pmean.com/05/Moments.asp View on Boundless.com Continuous Random Variables Expected Value of Girl Bonus The computation of the expected value of the girl bonus in our example. Free to share, print, make copies and changes. Get yours at www.boundless.com P.Mean Website. "Stats: Expected value and moments (July 29, 2005)." CC BY http://www.pmean.com/05/Moments.asp View on Boundless.com Continuous Random Variables Finite Population Correction The error should be multiplied by the FPC when the sampling fraction is large. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Standard error." GNU FDL http://en.wikipedia.org/wiki/Standard_error View on Boundless.com Continuous Random Variables Correlation Correction This factor results in an unbiased estimate of the true standard error when correlation exists. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Standard error." GNU FDL http://en.wikipedia.org/wiki/Standard_error View on Boundless.com Continuous Random Variables SEM SEM is usually estimated by the sample estimate of the population standard deviation divided by the square root of the sample size. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Standard error." GNU FDL http://en.wikipedia.org/wiki/Standard_error View on Boundless.com Continuous Random Variables Approximately Normal - Histogram This is a sample of size 50 from a normal distribution, plotted out as a histogram. The histogram looks somewhat bell-shaped, indicating normality. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Normhist." CC BY-SA http://en.wikipedia.org/wiki/File:Normhist.png View on Boundless.com Continuous Random Variables Non-Normality - Probability Plot This is a sample of size 50 from a right-skewed distribution, plotted as a normal probability plot. Notice that the points deviate on the, indicating the distribution is not normal. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Normexpprob." CC BY-SA http://en.wikipedia.org/wiki/File:Normexpprob.png View on Boundless.com Continuous Random Variables Non-Normality - Histogram This is a sample of size 50 from a right-skewed distribution, plotted as a histogram. Notice that the histogram is not bell-shaped, indicating that the distribution is not normal. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Normexphist." CC BY-SA http://en.wikipedia.org/wiki/File:Normexphist.png View on Boundless.com Continuous Random Variables Outliers This graph shows a best fit line to fit the data points, as well as two extra lines that are two standard deviations above and below the best fit line. Anything outside those lines can be considered an outlier. Free to share, print, make copies and changes. Get yours at www.boundless.com OER Commons. CC BY http://www.oercommons.org/courses/linear-regression-and-correlation-outliers/view View on Boundless.com Continuous Random Variables Statistical outliers This graph shows a best-fit line (solid blue) to fit the data points, as well as two extra lines (dotted blue) that are two standard deviations above and below the best fit line. Highlighted in orange are all the points, sometimes called "inliers", that lie within this range; anything outside those lines—the dark-blue points—can be considered an outlier. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia. "RANSAC_Inliers_and_Outliers.png." CC BY-SA 4.0 https://commons.wikimedia.org/wiki/File:RANSAC_Inliers_and_Outliers.png View on Boundless.com Continuous Random Variables Attribution • Wiktionary. "datum." CC BY-SA 3.0 http://en.wiktionary.org/wiki/datum • Wikipedia. "standard score." CC BY-SA 3.0 http://en.wikipedia.org/wiki/standard%20score • Wiktionary. "normalization." CC BY-SA 3.0 http://en.wiktionary.org/wiki/normalization • Wikipedia. "Standard score." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Standard_score • Wikipedia. "Normalization (statistics)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Normalization_(statistics) • Wiktionary. "Lebesgue measure." CC BY-SA 3.0 http://en.wiktionary.org/wiki/Lebesgue_measure • Wikipedia. "Probability distribution." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Probability_distribution • Wikipedia. "Probability density function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Probability_density_function • Wikipedia. "Box." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Box?Muller+transformation • Wikipedia. "cumulative distribution function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/cumulative%20distribution%20function • Wikipedia. "p-value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/p-value • Wikipedia. "Uniform distribution (continuous)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Uniform_distribution_(continuous) • Wikipedia. "P-value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/P-value • OpenStax CNX. "Susan Dean and Barbara Illowsky, Continuous Random Variables: The Uniform Distribution. September 17, 2013." CC BY 3.0 http://cnx.org/content/m16819/latest/ • OpenStax CNX. "Susan Dean and Barbara Illowsky, Continuous Random Variables: The Uniform Distribution. September 17, 2013." CC BY 3.0 http://cnx.org/content/m16819/latest/ • Wiktionary. "Poisson process." CC BY-SA 3.0 http://en.wiktionary.org/wiki/Poisson_process • Wikipedia. "Erlang distribution." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Erlang%20distribution Free to share, print, make copies and changes. Get yours at www.boundless.com Continuous Random Variables • Wikipedia. "Exponential distribution." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Exponential_distribution • OpenStax CNX. "Susan Dean and Barbara Illowsky, Continuous Random Variables: The Exponential Distribution. September 17, 2013." CC BY 3.0 http://cnx.org/content/m16816/latest/ • Wikipedia. "Exponential distribution." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Exponential_distribution • Wiktionary. "empirical rule." CC BY-SA 3.0 http://en.wiktionary.org/wiki/empirical_rule • Wikipedia. "entropy." CC BY-SA 3.0 http://en.wikipedia.org/wiki/entropy • Wiktionary. "cumulant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/cumulant • Wikispaces. "CEP932 - Final Paper of Normal Distribution." CC BY-SA http://cep932.wikispaces.com/Final+Paper+of+Normal+Distribution • Wikipedia. "Normal distribution." CC BY-SA 3.0 https://en.wikipedia.org/wiki/Normal_distribution • Wiktionary. "empirical rule." CC BY-SA 3.0 http://en.wiktionary.org/wiki/empirical_rule • Wiktionary. "bell curve." CC BY-SA 3.0 http://en.wiktionary.org/wiki/bell_curve • Wiktionary. "real number." CC BY-SA 3.0 http://en.wiktionary.org/wiki/real_number • Wikipedia. "Percentile." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Percentile • Wikispaces. "killianhma0809 - Normal Distribution." CC BY-SA http://killianhma0809.wikispaces.com/Normal+Distribution • Wiktionary. "standard normal distribution." CC BY-SA 3.0 http://en.wiktionary.org/wiki/standard_normal_distribution • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/z-score • Wikidot. "Chapter 7: Normal distribution - Statistics." CC BY-SA http://statistics.wikidot.com/ch7 • Wikispaces. "mrschasesstatspage - Chapter 2-The Normal Distributions." CC BY-SA http://mrschasesstatspage.wikispaces.com/Chapter+2-The+Normal+Distributions • Wikipedia. "Standard score." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Standard_score • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/z-score Free to share, print, make copies and changes. Get yours at www.boundless.com Continuous Random Variables • Wikidot. "Using Normal Distributions - IB Math Stuff." CC BY-SA http://ibmathstuff.wikidot.com/usingnormaldistributions • Wiktionary. "central limit theorem." CC BY-SA 3.0 http://en.wiktionary.org/wiki/central_limit_theorem • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/normal-approximation • OpenStax CNX. "David Lane, History of Normal Distribution. September 17, 2013." CC BY 3.0 http://cnx.org/content/m11164/latest/ • Wiktionary. "law of large numbers." CC BY-SA 3.0 http://en.wiktionary.org/wiki/law_of_large_numbers • Wiktionary. "central limit theorem." CC BY-SA 3.0 http://en.wiktionary.org/wiki/central_limit_theorem • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/normal-approximation • Wikipedia. "Law of large numbers." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Law_of_large_numbers • Wikipedia. "Central limit theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Central_limit_theorem • Wiktionary. "binomial distribution." CC BY-SA 3.0 http://en.wiktionary.org/wiki/binomial_distribution • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/z-score • OpenStax CNX. "David Lane, Normal Approximation to the Binomial. September 17, 2013." CC BY 3.0 http://cnx.org/content/m11162/latest/ • Wikipedia. "Precision." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Precision • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/systematic-error • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/random-error • Wikispaces. "IBChemistrySL - Errors and Uncertainty in Experimental Data." CC BY-SA http://ibchemistrysl.wikispaces.com/Errors+and+Uncertainty+in+Experimental+Data • Wikipedia. "Random and systematic errors." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Random_and_systematic_errors • Wikipedia. "Random error." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Random_error • Wikipedia. "Accuracy." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Accuracy Free to share, print, make copies and changes. Get yours at www.boundless.com Continuous Random Variables • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/systematic-error • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/random-error • Wikispaces. 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