LECTURE 4 FRIDAY SEPTEMBER 5 2008 CHAPTER 3 NORMAL DISTRIBUTIONS DENSITY CURVES A smooth curve showing the pattern of variation for a variable It usually refers to a very large data set or to an entire population A density curve is an idealized description of the pattern of variation for the variable The area under the density curve and above the horizontal scale should be thought of as the entire data set or the entire population ie 100 or 1 00 The horizontal scale x axis shows the values for the variable and the area under the curve between any two x axis points represents the proportion or percent of the entire population which lies between these two values Density curves are described by SYMMETRIC Mean and Standard Deviation SKEWED 5 Number Summary Terminology data sets vs populations The mean for a data set is called XBAR but the mean for a population is called The std dev for a data set is called s but the std dev for a population is called The MEAN represents the balance point of the distribution The MEDIAN represents the equal areas point or the 50 point or the 50th percentile NORMAL DISTRIBUTIONS The normal distributions are a family of density curves which are symmetric about center single peaked and bell shaped The mean and the standard deviation completely describe a normal distribution Changing the mean slides the distribution along the horizontal scale Changing the standard deviation changes the spread about the center Regardless of the specific values of and the normal distributions have the characteristic that the same proportion of the population always lies between any two standard deviation values 68 95 99 7 APPROXIMATION Approximately 68 of the entire population lies within 1 standard deviation either way from the mean ie between 1 and 1 The theoretical value is 68 26 Approximately 95 of the entire population lies within 2 standard deviations either way from the mean ie between 2 and 2 The theoretical value is 95 45 Approximately 99 7 of the entire population lies within 3 standard deviations either way from the mean ie between 3 and 3 The theoretical value is 99 73 The normal distribution can be used a good approximation of real variable populations providing we have an idea of the proper values for the mean and standard deviation of the variable THE STANDARD NORMAL DISTRIBUTION A normal distribution whose mean 0 and whose std dev 1 is a standard normal distribution and the area under the curve is tabulated in Table A on Page 684 in your textbook In Table A z represents the number of standard deviations away from the mean Z values are shown from 3 49 to 3 49 Theoretically z runs from infinity to infinity but 99 95 of the distribution is within 3 49 to 3 49 standard deviations from the mean So very little is left out by cutting the table off at these values For a given value of z table A shows the area below that value of z To obtain the proportion of the distribution which lies between two z values one would use Table A to determine the area below each z value and then subtract the smaller from the larger STANDARDIZING To apply the standard normal distribution to other normal distributions whose mean is not equal to 0 and whose standard deviation is not equal to 1 a process called standardizing must be used We convert a variable s X value to its equivalent Z value with the formula Z X Examples going from X to Z to AREA Examples going from AREA to Z to X
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