Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Normal distributions can have different mean and varianceNormal distributions can have different mean and varianceNormal distribution has specific relative frequenciesApproximate percentages in the normal distribution Normal DistributionNormal DistributionNormal DistributionSlide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Use of Normal tables (page 493)Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Health Behavior Statistical Methods HP 340L Lecture 6 The Normal Distribution Chapter 6 We covered material in Kiess & Green Chapter 5  Descriptive statistics: Variability  Measures of variability • Range • Interquartile range (IQR) • Variance  Population variance  Sample variance  Estimated population variance Last Lecture We will cover material in Kiess & Green Chapter 6  The normal distribution  The standard normal distribution  Using normal tables  Z-scores Today’s Lecture We use histograms to display the distribution of scores in a sample of data.  We can draw a smooth curve over the histogram to approximate the distribution of the population scores.  Draw 200 males from a normal distribution with mean height = 69 in, sd = 3  Plot histogram  Draw a smooth curve over the histogram Distributions Density curves  A theoretical model for a distribution  Properties of Density Curves  The total area under the curve is 1, corresponding to 100% of the distribution  The area under the curve between any two points is equal to the proportion of the distribution in that interval Distributions Normal density  Theoretical distribution that specifies the frequencies of a set of scores in a population  Many ‘real life’ continuous variables are well approximated by a normal distribution, e.g., height, weight, IQ  The sampling distributions of many test statistics follow a normal distribution (this will make sense later in the course) Distributions The normal curve  Characteristics of a normal distribution:  smooth  symmetrical  bell-shaped  unimodal  mean, median, & mode coincide  asymptotic (tails never touch base line)  continuous Distributions The horizontal axis (x-axis)  Gives the values of the variable of interest, e.g., height, IQ, blood pressure, etc. Distributions Vertical axis (y-axis)  The height of the normal curve represents the relative frequencies for the scores/values on the horizontal axis.  The tall center of the curve is the mode; indicates more people have those scores.  The relative frequency gets smaller and smaller until it is nearly zero as the tails taper off Distributions Normal distributions can have different mean and variance N(μ, σ2) Location Dispersion Denotes a Normal distribution with population mean μ and variance σ2 DistributionsHP-340: Fall 2016 11  Normal distributions can have different mean and variance Distributions Normal distribution has specific relative frequencies Distributions Approximate percentages in the normal distribution DistributionsHP-340: Fall 2016 14   3   2          2   3Normal Distribution The sum of all these probabilities = 1 Highly probable events occur closer to the center (where the mean is). Approximately: 68% of the area lies within 1 standard deviation of μ Least probable events occur in the tails of the curve (where the area is small). Mean=Median=ModeHP-340: Fall 2016 15   3   2          2   3Normal Distribution The sum of all these probabilities = 1 Highly probable events occur closer to the center (where the mean is). Approximately: 68% of the area lies within 1 standard deviation of μ 95% of the area lies within 2 standard deviations of μ Least probable events occur in the tails of the curve (where the area is small). Mean=Median=ModeHP-340: Fall 2016 16   3   2          2   3Normal Distribution The sum of all these probabilities = 1 Highly probable events occur closer to the center (where the mean is). Approximately: 68% of the area lies within 1 standard deviation of μ 95% of the area lies within 2 standard deviations of μ 99.7% of the area lies within 3 standard deviations Least probable events occur in the tails of the curve (where the area is small). Mean=Median=Mode All normal distributions have the same shape, differing only by the scale of the X axis (by the size of σ)  To simply calculations involving the normal distribution, we can use a standard scale of σ = 1 and shift the distribution to center it at µ = 0 Standard Normal Distribution To center the distribution, simply subtract off the mean  Example: Standard Normal Distribution x x – x 66 -1.5 69 1.5 68 0.5 68 0.5 63 -4.5 64 -3.5 68 0.5 74 6.5 x To standardize the scale, divide each score by the standard deviation (s)  Example: Standard Normal Distribution x x – x 66 -1.5 -0.44 69 1.5 0.44 68 0.5 0.15 68 0.5 0.15 63 -4.5 -1.33 64 -3.5 -1.04 68 0.5 0.15 74 6.5 1.92 x( ) = xxsx−′Standard Normal Distribution A normal distribution with:  Mean: μ = 0  Variance: σ2 = 1  Denoted: N(0,1)  A score from a normally distributed variable with any mean and standard deviation can be transformed into a score on the standard normal distribution  Called a z-score  Measures how many standard deviation units the corresponding raw score is above or below the mean Standard Normal Distribution ( ) = Xz−µσ Converting scores (x) into standard deviation units (z)  Example:  X = height (inches)  Distribution: X ~ N(μ = 55, σ2 = (2.5)2)  Convert to Z score:  For a child 60 inches tall:  The child’s height is two standard deviations above the mean Standard Normal Distribution ( )55 = 2.5xz−( )60 55 = = 22.5z−N(55, (2.5)2) N(0, 1) Standard Normal DistributionArea = 72% 4.3 6.8 N(5, 1) Standard Normal Distribution  How to