The Normal Distribution

• Mathematicians have noticed that some histograms represent basic ways nature behaves.
• There are some general classes
  • The uniform or rectangular distribution will be generated by rolling a die a bunch of times.
    • Each outcome is equally likely.
    • So a histogram of of the results should be uniformly distributed.
  • But if we changed the experiment to roll three dice and find the sum
    • We could create the normal distribution.
  • Or four dice find the sum of the three highest
    • We would get a distribution skewed to the left.
  • Summing the three lowest would lead to a distribution skewed to the right
  • Tests frequently produce a bimodal distribution.
    • Test1 for this class :
• The normal or Gaussian distribution is important because many datasets follow this distribution.
• This is named after Carl Fredrich Gauss, one of my academic ancestors and sometimes called the "Greatest mathematician since antiquity".
• In the normal curve, the mean, median and mode are all the same.
• The curve is known as the "bell shaped curve" is symmetric about the mean.
• 50% of the data is below the mean and 50% of the data is above the mean.
• The empirical rule states
  • 68% of the data is within 1 standard deviation of the mean.
  • 95% of the data is within 2 standard deviations of the mean.
  • 99.7% of the data is within 3 standard deviations of the main.
• If SAT scores are normally distributed with a mean of 500 and a standard distribution of 100
  • What percentage of the scores are greater than 500?
  • What percentage of the scores are between 400 and 600
  • What percentage of the scores are less than 400 or greater than 600?
  • What percentage of the scores are between 500 and 600
  • What percentage of the scores are greater than 600?
  • What range contains 95% of the scores?
  • What can you say about 99.7% of the data?
• The empirical rule is nice, but we need more precision.
• There is a table that tells us how a normal distribution with a mean of 0 and a standard deviation of 1 behaves.
  • This is on page 822-823
  • And will be available to you on the test.
  • Be aware that different books present this table in different forms.
  • This is called the standard normal distribution
• A study of a weight loss pill includes a control group that has taken a placebo. It is expected that the weight change results for this group will be normal distributed with a mean of 0 and a standard deviation of one pound.
  • What percent of the group do you expect to gain weight?
  • What percent of the group do you expect to lose weight?
  • What percent will lose 1 or more pounds?
  • What percent will gain .5 pounds or more?
  • What percent will have a weight change between -.3 and .3 pounds?
  • What percent will have a weight change greater than 1.7 pounds?
• Since we don't want to generate a new table for every distribution we can reduce other distributions to the standard normal distribution
  • To do this, we compute the z-score or standard score
  • z_x = (x-μ)/σ
• If the z score is
  • Negative, the result is below the mean
  • Positive, the result is above the mean
• Do some problems page 829