Continuous Probability Distributions

 

1).  Normal Distribution Curve (Lind 2005)

Here is a link to  University of Leicester and a good explanation of Normally Distributed Populations.    http://www.le.ac.uk/biology/gat/virtualfc/Stats/normal.htm 

  1. The normal curve is bell-shaped and has a single peak at the center of the distribution. 

  2. The arithmetic mean, median, and mode of the distribution are equal and located at the peak. Thus half the area under the curve is above the mean and half is below it. 

  3. The normal probability distribution is symmetrical about its mean.

  4. The normal probability distribution is asymptotic. That is the curve gets closer and closer to the X-axis but never actually touches it.

  5. Theoretically, curve extends to infinity.

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

  7. It is also called the z distribution.

A z-value is the distance between a selected value, designated X, and the population mean , divided by the population standard deviation, .

 

Z- Value

 

by

Forest Edward Thompson III (UoPhx 2008)

 

 

A normal distribution can be transformed into a standard normal distribution by discovering the z-value. The z-value is determined by subtracting the (X) value from the mean, and divided by the standard deviation.

 

The z-value is also known as:

§  z scores   

§  standard normal values

§  normal deviate

§  standard normal deviate

§  z statistics

 

 

The Standard Normal Value formula is:

·       X is the value of any observation, measurements, or numbers

·       Mu is the mean of the distribution

·       S is the standard deviation of the distribution

 

 

The Standard Normal Value formula is used to find the z-value. The z-value is then used to determine the probability for the standard normal probability distribution. This is discovered by using the statistical graph for the “Areas under the Normal Curve” or the “Standard Normal Probabilities”.  

 

This is an example of the “Areas under the Normal Curve” or the “Standard Normal Probabilities” chart:

 

Lind, D. A. & Marchal, W. G. & Wathen, S. A. (2004). Statistical techniques in business and economics, 12e: Appendix D, pg. 720: Areas Under the Normal Curve. New York: The McGraw-Hill Companies  

 

 

To understand how to use the chart:

1.     Use the Standard Normal Value formula to discover the z-value

2.     Once the z-value is discovered refer to your chart to understand its probability

3.     Locate the z-value number on the left side of the chart going vertically (using this chart as an example the z-value numbers are from point 0.0 to 3.0 in the vertical column)

4.     Once the z-value number has been discovered in the vertical column, then use the horizontal row to discover the second part of the z-value (using this chart as an example the z-value numbers are from point 0.00 to 0.09 in the horizontal row)

Example: the z-value is 2.62

Using the chart, find 2.6 in the vertical column and 0.02 in the horizontal row. If z = 2.62, then P (0 to z) = 0.4956


 

   These are internet links that will help with more detailed information on normal probability distribution, bell-shape curve, z-value, charts, and statistical calculators:  

 

http://www.cas.buffalo.edu/classes/psy/segal/2072001/z-dist&corr/zdist.htm

http://www.math.com/tables/stat/distributions/z-dist.htm

http://stattrek.com/Lesson2/Normal.aspx

http://davidmlane.com/hyperstat/z_table.html

http://www.math.csusb.edu/faculty/stanton/m262/normal_distribution/normal_distribution.html

http://faculty.uncfsu.edu/dwallace/sz-score.html

 

Reference

 

Lind, D. A. & Marchal, W. G. & Wathen, S. A. (2004). Statistical techniques in business and economics, 12e: Chapter 7: Continuous Probability Distributions. New York: The McGraw-Hill Companies

 

2).  The Empirical Rule for A Normally Distributed Population

  1. 68% of measurements are within 1 Standard Deviation from the mean 

  2. 95% of measurement are within 2 Standard Deviations from the mean 

  3. 99.7% of the measurements are within 3 Standard Deviations from the mean 

  4. Nearly Every measurement is within +/-3 Standard Deviation form mean Except Outliers.  And that is only 3 sigma...think about 6 sigma.

3).  The Normal Approximation to the Binomial

  1. The normal distribution (a continuous distribution) yields a good approximation of the binomial distribution (a discrete distribution) for large values of n.

  2. The normal probability distribution is generally a good approximation to the binomial probability distribution when n(pie) and n(1- (pie) ) are both greater than 5.

With the binomial experiment:

  1. There are only two mutually exclusive outcomes (success or failure) on each trial.

  2. A binomial distribution results from counting the number of successes.

  3. Each trial is independent.

  4. The probability is fixed from trial to trial, and the number of trials n is also fixed.

The value .5 subtracted or added, depending on the problem, to a selected value when a binomial probability distribution (a discrete probability distribution) is being approximated by a continuous probability distribution (the normal distribution).