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Chapter 10

Continuous Random Variables

Book Version 1
By Boundless
Boundless Statistics
Statistics
by Boundless
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Section 1
The Normal Curve
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Continuous Probability Distributions

A continuous probability distribution is a representation of a variable that can take a continuous range of values.

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The Uniform Distribution

The continuous uniform distribution is a family of symmetric probability distributions in which all intervals of the same length are equally probable.

The Exponential Distribution

The exponential distribution is a family of continuous probability distributions that describe the time between events in a Poisson process.

The Normal Distribution

The normal distribution is symmetric with scores more concentrated in the middle than in the tails.

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Graphing the Normal Distribution

The graph of a normal distribution is a bell curve.

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The Standard Normal Curve

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

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Finding the Area Under the Normal Curve

To calculate the probability that a variable is within a range in the normal distribution, we have to find the area under the normal curve.

Section 2
Normal Approximation
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The Normal Approximation to the Binomial Distribution

The process of using the normal curve to estimate the shape of the binomial distribution is known as normal approximation.

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The Scope of the Normal Approximation

The scope of the normal approximation is dependent upon our sample size, becoming more accurate as the sample size grows.

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Calculating a Normal Approximation

In this atom, we provide an example on how to compute a normal approximation for a binomial distribution.

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Change of Scale

In order to consider a normal distribution or normal approximation, a standard scale or standard units is necessary.

Section 3
Measurement Error
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Bias

Systematic, or biased, errors are errors which consistently yield results either higher or lower than the correct measurement.

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Chance Error

Random, or chance, errors are errors that are a combination of results both higher and lower than the desired measurement.

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Outliers

In statistics, an outlier is an observation that is numerically distant from the rest of the data.

Section 4
Expected Value and Standard Error
Expected Value

The expected value is a weighted average of all possible values in a data set.

Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic.

Section 5
Normal Approximation for Probability Histograms
Probability Histograms

A probability histogram is a graph that shows the probability of each outcome on the yyy-axis.

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Probability Histograms and the Normal Curve

Many different types of distributions can be approximated by the normal curve.

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Conclusion

Many distributions in real life can be approximated using normal distribution.

You are in this book
Boundless Statistics by Boundless
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Chapter 9
Probability and Variability
  • Discrete Random Variables
  • The Binomial Random Variable
  • Other Random Variables
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Chapter 10
Continuous Random Variables
  • The Normal Curve
  • Normal Approximation
  • Measurement Error
  • Expected Value and Standard Error
  • Normal Approximation for Probability Histograms
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Chapter 11
Correlation and Regression
  • Correlation
  • More About Correlation
  • Regression
  • The Regression Line
  • R.M.S. Error for Regression
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