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Continuous
Random Variables
The Normal Curve
Normal Approximation
Measurement Error
Expected Value and Standard Error
Normal Approximation for Probability Histograms
Continuous Random Variables
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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
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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
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Continuous Random Variables > Measurement Error
Measurement Error
• Bias
• Chance Error
• Outliers
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Continuous Random Variables > Expected Value and Standard Error
Expected Value and Standard Error
• Expected Value
• Standard Error
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Continuous Random Variables > Normal Approximation for Probability Histograms
Normal Approximation for Probability Histograms
• Probability Histograms
• Probability Histograms and the Normal Curve
• Conclusion
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Appendix
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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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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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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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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 .
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Continuous Random Variables
Normal Probability Density
The normal distribution is described by this probability density function.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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 .
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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 .
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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 .
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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.
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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 .
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Wikimedia. "RANSAC_Inliers_and_Outliers.png." CC BY-SA 4.0 https://commons.wikimedia.org/wiki/File:RANSAC_Inliers_and_Outliers.png View on
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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
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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
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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
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Free to share, print, make copies and changes. Get yours at www.boundless.com
Continuous Random Variables
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Continuous Random Variables
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• sds. CC BY http://sds
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