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Probability Density Function | The general formula for the probability density function of the normal distribution is where is the location parameter and is the scale parameter. The case where = 0 and = 1 is called the standard normal distribution. The equation for the standard normal distribution is Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the standard normal probability density function. | ||||||||||||||||
Cumulative Distribution Function | The formula for the cumulative distribution function of the normal distribution does not exist in a simple closed formula. It is computed numerically.The following is the plot of the normal cumulative distribution function. | ||||||||||||||||
Percent Point Function | The formula for the percent point function of the normal distribution does not exist in a simple closed formula. It is computed numerically.The following is the plot of the normal percent point function. | ||||||||||||||||
Hazard Function | The formula for the hazard function of the normal distribution is where is the cumulative distribution function of the standardnormal distribution and is the probability density function of the standard normal distribution. The following is the plot of the normal hazard function. | ||||||||||||||||
Cumulative Hazard Function | The normal cumulative hazard function can be computed from the normal cumulative distribution function.The following is the plot of the normal cumulative hazard function. | ||||||||||||||||
Survival Function | The normal survival function can be computed from the normal cumulative distribution function.The following is the plot of the normal survival function. | ||||||||||||||||
Inverse Survival Function | The normal inverse survival function can be computed from the normal percent point function.The following is the plot of the normal inverse survival function. | ||||||||||||||||
Common Statistics |
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Parameter Estimation | The location and scale parameters of the normal distribution can be estimated with the sample mean and sample standard deviation, respectively. | ||||||||||||||||
Comments | For both theoretical and practical reasons, the normal distribution is probably the most important distribution in statistics. For example,
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Theroretical Justification - Central Limit Theorem | The normal distribution is widely used. Part of the appeal is that it is well behaved and mathematically tractable. However, the central limit theorem provides a theoretical basis for why it has wide applicability.The central limit theorem basically states that as the sample size (N) becomes large, the following occur:
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Software | Most general purpose statistical software programs support at least some of the probability functions for the normal distribution. |
Learning Objectives Create and interpret frequency polygons Create and interpret cumulative frequency polygons Create and interpret overlaid frequency polygons Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful for comparing sets of data. Frequency polygons are also a good choice for displaying cumulative frequency distributions . To create a frequency polygon, start just as for histograms , by choosing a class interval. Then draw an X-axis representing the values of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle value represented by the class. Draw the Y-axis to indicate the frequency of each class. Place a point in the
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