Normal CurveNormal CurveAreas Under the Normal CurveExamplesStandardized Scores and Forward LookupStandard UnitsThe Normal approximation for dataPercentile Ranks and Backward LookupPercentile RankRecap, Bits and PiecesChange of scaleFinal PointsChapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsStat220, Part IIDescriptive StatisticsLecture 10Chapter 5: T he Normal Approximation for DataChapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsNormal curveThe famous normal curve can often be used as an ’ideal’distribution, to which histograms from real data can becompared. Its equation isy =1√2πe−x22× 100%We will work with it through diagrams and tables, without everusing the equation itself.Chapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsNormal CurveIt was discovered in 1720 by Abraham de Moivre, but is alsocalled Gaussian curve or bell curve.Figure: Carl Friedrich GaussChapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsNormal curveThe graph is symmetric about 0, stretches to infinity on bothends, and the total area equals 100%.The curve is always above the horizontal axis.The area under the normal curve between -1 and 1 is about68%.Chapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsNormal curveThe graph is symmetric about 0, stretches to infinity on bothends, and the total area equals 100%.The curve is always above the horizontal axis.The area under the normal curve between -2 and 2 is about95%.Chapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsNormal curveThe graph is symmetric about 0, stretches to infinity on bothends, and the total area equals 100%.The curve is always above the horizontal axis.The area under the normal curve between -3 and 3 is about99.7%.Chapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsUsing Normal TablesOther areas can be found using tables (page A104 in textbook).The textbook table gives the areas for regions symmetricaround zero.Chapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsUsing Normal TablesYou will not encounter the textbook table (almost) anywhereelse. The more standard tables gives areas to the left of somespecified point.You can use either; there is no “statistical controversy” here.Looking up normal tables is just a technicality!Chapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsExercises1 Find the area between 0 and 1 under the normal curve.2 Find the area to the left of 1 under the normal curve.3 Find the area to the right of 0.45 under the normal curve.Use either table. Always start a normal-table exercise bydrawing a picture of the problem!Chapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsStandard unitsMany histograms are similar in shape to the normal curve,provided they are drawn to the same scale. Making thehorizontal scales match up between histogram and normal curveinvolves standard units.(Standardized Score) =(Original Value) - AverageSDA value is converted to standard units by checking how manySDs it is above or below the average. We will call this convertedvalue the Standardized Score (it is also known as the z-score).Chapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsExample 1: The HANES study•HANES: Health And Nutrition Examination Survey(1976-1980)•A representative cross-section of 20,322 Americans age 1to 74 was examined.•Data were obtained on•demographic variables: age, education, income•physiological variables: height, weight, blood pressure,cholesterol levels•dietary habits•levels of lea d and pesticides in the blood•prevalence of diseasesChapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap, Bitsand PiecesChange ofscaleFinal PointsExample 1: Find StandardizedScoresThe height of women age 18-74 in the HANES study hasaverage 63.5 inches and SD 2.5 inches1 What is the standardized score for 68.5 inches?2 What is the standardized score for 61 inches?Chapter 5:Normal Ap-proximationNormalCurveNormal CurveAreas Underthe NormalCurveExamplesStandardizedScores andForwardLookupStandardUnitsThe Normalapproximationfor dataPercentileRanks andBackwardLookupPercentileRankRecap,