Introduction to Statistics inPsychologyPSY 201Professor Greg FrancisLecture 07normal distributionDescribing everyone’s height.DISTRIBUTIONfrequency of scores plotted againstscore25 30 35 40 45 50 55 60 65 70Score010203040Frequencyfrequency → likelihood, probability(more later)2GOALdescribe (summarize) distributions• shape: unimodal, bimodal, skew,...• central tendency: mode, median, mean• variation: range, variance, standarddeviationsummarizing forces you to loseinformationsome theoretical distributions arespecial!a few numbers completely specify thedistribution3NORMAL DISTRIBUTIONY =1σ√2πe−(X−µ)2/2σ2• Y height of the curve for any givenvalue of X in the distribution of scores• π mathematical value of the ratio ofthe circumference of a circle to itsdiameter. A constant (3.14159.....)• e base of the system of natural loga-rithms. A constant (2.7183...)• µ mean of the distribution of scores• σ standard deviation of a distribu-tion of scoressometimes written asY =1σ√2πexp−(X − µ)2/2σ24PARAMETERSa family of distributionsmember of the family is designated bythe mean µ and standard deviation σchanging µ shifts the curve to the leftor the rightshape remains the same0 2 4 6 8 10 12 14Score00.10.20.30.45PARAMETERSchanging σ changes the spread of thecurvecompare normal distributions forσ = 1 and σ = 2, both with µ =3-5 0 5 10 15Score00.10.20.30.46PARAMETERSchanging µ and σ together producespredictable results-5 0 5 10 15Score00.10.20.30.47PROPERTIESall normal distributions have thefollowing in common1. Unimodal, symmetrical, bell shaped,maximum height at the mean.2. A normal distribution is continuous.X must be a continuous variable,and there is a corresponding value ofY for each X value.3. A normal distribution asymptoticallyapproaches the X axis.8STANDARD NORMALremember z-scores: 0 mean, 1standard deviationif the z-scores are normally distributedY =1σ√2πe−(X−µ)2/2σ2becomesY =11√2πe−(z−0)2/2(12)orY =1√2πe−z2/29STANDARD NORMALlooks like-4 -2 0 2 4Score00.10.20.30.410SIGNIFICANCEIt turns out that lots of frequencydistributions can be described as anormal distributionfor examplegive me your height11SIGNIFICANCEIt turns out that lots of frequencydistributions can be described as anormal distribution• intelligence scores• weight• reaction times• judgment of distance• rating of personality• ...reasons are subtle and mathematicalhigher level (graduate) statistics course12SIGNIFICANCEwhen the distribution is a normaldistribution, we can describe thedistribution by just specifying• Mean: X• Standard deviation: s• Noting it is a normal distributionthat’s all we need!That’s part of our goal: describedistributions13STANDARD NORMALassume you have a standard normaldistribution (don’t worry about whereit came from)Y =1√2πe−z2/2-4 -2 0 2 4Score00.10.20.30.4if your distribution is normal, you can create astandard normal by converting to z-scores14USEsame as all other distributionsidentify key aspects of the datapercentilespercentile rankproportion of scores within a range...make it easier to interpret datasignificance!15STANDARD NORMALtotal area under the curve alwaysequals 1.0area under the curve from the mean(0) to one tail equals 0.5-4 -2 0 2 4Score00.10.20.30.416STANDARD NORMALarea under the curve one standard deviationaway from t he mean is approximately 0.3413area under the curve two standard deviationsaway from t he mean is approximately 0.4772area under the curve three standard deviationsaway from t he mean is approximately 0.4987-4 -2 0 2 4Score00.10.20.30.417CONCLUSIONSnormal distributionequationspropertiesstandard normal equations18NEXT TIMEarea under the curveproportionspercentilespercentile ranksandOld and