DISTRIBUTION GOAL frequency of scores plotted against score describe summarize distributions Introduction to Statistics in Psychology PSY 201 shape unimodal bimodal skew Professor Greg Francis 40 Lecture 07 Describing everyone s height Frequency normal distribution central tendency mode median mean 30 variation range variance standard deviation 20 10 summarizing forces you to lose information 0 25 30 35 40 45 50 55 60 65 70 Score frequency likelihood probability more later some theoretical distributions are special a few numbers completely specify the distribution 2 3 NORMAL DISTRIBUTION PARAMETERS PARAMETERS 2 2 1 Y e X 2 2 a family of distributions changing changes the spread of the curve Y height of the curve for any given value of X in the distribution of scores mathematical value of the ratio of the circumference of a circle to its diameter A constant 3 14159 e base of the system of natural logarithms A constant 2 7183 mean of the distribution of scores standard deviation of a distribution of scores member of the family is designated by the mean and standard deviation changing shifts the curve to the left or the right shape remains the same compare normal distributions for 1 and 2 both with 3 0 4 0 3 0 4 0 2 0 3 0 1 0 2 0 1 0 5 0 5 Score 0 sometimes written as 1 Y exp X 2 2 2 2 4 0 2 4 6 8 10 12 14 Score 5 6 10 15 PARAMETERS PROPERTIES STANDARD NORMAL changing and together produces predictable results all normal distributions have the following in common remember z scores 0 mean 1 standard deviation 1 Unimodal symmetrical bell shaped maximum height at the mean if the z scores are normally distributed 0 4 0 3 2 2 1 Y e X 2 2 becomes 1 z 0 2 2 12 Y e 1 2 or 1 z 2 2 Y e 2 2 A normal distribution is continuous X must be a continuous variable and there is a corresponding value of Y for each X value 0 2 0 1 0 5 0 5 10 15 Score 3 A normal distribution asymptotically approaches the X axis 7 8 9 STANDARD NORMAL SIGNIFICANCE SIGNIFICANCE looks like It turns out that lots of frequency distributions can be described as a normal distribution It turns out that lots of frequency distributions can be described as a normal distribution 0 4 0 3 for example 0 2 give me your height 0 1 intelligence scores weight reaction times judgment of distance 0 4 2 0 2 4 Score rating of personality reasons are subtle and mathematical higher level graduate statistics course 10 11 12 SIGNIFICANCE STANDARD NORMAL USE when the distribution is a normal distribution we can describe the distribution by just specifying assume you have a standard normal distribution don t worry about where it came from 2 1 Y e z 2 2 same as all other distributions Mean X Standard deviation s Noting it is a normal distribution percentiles percentile rank 0 4 proportion of scores within a range 0 3 0 2 make it easier to interpret data significance that s all we need That s part of our goal describe distributions identify key aspects of the data 0 1 0 4 2 0 2 4 Score if your distribution is normal you can create a standard normal by converting to z scores 13 14 15 STANDARD NORMAL STANDARD NORMAL CONCLUSIONS total area under the curve always equals 1 0 area under the curve one standard deviation away from the mean is approximately 0 3413 normal distribution area under the curve from the mean 0 to one tail equals 0 5 area under the curve two standard deviations away from the mean is approximately 0 4772 area under the curve three standard deviations away from the mean is approximately 0 4987 0 4 0 3 equations properties standard normal equations 0 4 0 2 0 3 0 1 0 2 0 4 2 0 2 4 0 1 Score 0 4 2 0 2 4 Score 16 17 18 NEXT TIME area under the curve proportions percentiles percentile ranks and Old and new 19