The Normal Distribution

• In the normal distribution the mean, median and mode all have the same value.
• This is also called the Gaussian distribution.
• 68% of all data is within 1 standard deviation of the mean.
• 95% of all data is within 2 standard deviations of the mean
• 99.7% of all data is within 3 standard deviations of the mean
• Example:

  Problem 2, page 727
     mean 1700
     SD = 500
  
     1700+2*500 = 2700
  
  Problem 6
     17000 - 1 * 500 = 1200
  
  Problem 12
      Between 16,000 and  18,000
      This is 1700 +/- 1000  or within 2 standard deviations of the mean.
      This is 95% of the data.
  
  Problem 14, 
      Between 17000 and 18,000 
      This is 1/2 the data between 2 standard deviations.  Or 95/2 = 47.5
  
  Problem 16
      This is one standard deviation.  Or 1/2 of 68% or 34%
  
  Problem 20
      More than 1800.
      50% is less than 17000
      50% is more than 17000
      50+47.5% is less than 18000
      100-50+47.5 is more than 18000
      2.5% is more than 18000
  
     

• It is convenient to use a normalized normal distribution.
  • mean = 0
  • sd = 1
• To do this, we compute the z score of a data item
• z = (data-mean)/ sd
• Example:

  Problem 32.   mean = 60, sd = 8
      x = 76,   z = 76-60/8 = 16/8 = 2.
  Problem 38
      x = 72,   z = (72-60)/8 = 12/8 = 1.5
  Problem 44
      x = 44    z = (44-60)/8 = -16/8 = -2
         
  Problem 50,   mean = 12.4, sd = 20.4
      x = 48.1, z = (48.1-12.4)/20.4 = 1.75
  
  Problem 54
      x = -2.9, z = (-2.5-12.4)/20.4 = -.73
  
  Problem 60
      z = 3, mean = 400, sd = 50
      3 = (x-400)/50
      3*50 = x-400
      150+400 = x 
      550 = x
     

• Table 12.3, given a z score, we can find the data that is above and below that zscore.
• Look at a z score of 0.
• Examples

  Problem 68
     z = .8    Table value 78.81
  
  Problem   76
     z = .3        61.79
     z = 2.1  98.21
  
     Between is 98.21 - 61.79 =  36.42
     

Homework