Chapter VI: The Normal Distribution

Chapter Outline

6.1 The Importance of Normal Distribution

6.2 Properties of the Normal Distribution

6.3 Areas Under the Normal Curve

The Importance of Normal DistributionNormal limits – for biological markers that are normally distributed, the observations fall within the 2.5% of the lower and upper tails of a large set of observations obtained from a group of presumably healthy persons

Calculated by adding and subtracting two standard deviations from the mean

Normal distribution is also called Gaussian distributionWhy the normal distribution is popular

Possess certain mathematical properties that make it attractive and easy to manipulateThe assumption of much statistical theory and methodologyBasic for the use of inferential statistics

Normal limits Normal limits –– for biological markers that are normally for biological markers that are normally distributed, the observations fall within the 2.5% of the lower distributed, the observations fall within the 2.5% of the lower and upper tails of a large set of observations obtained from a and upper tails of a large set of observations obtained from a group of presumably healthy personsgroup of presumably healthy persons

Calculated by adding and subtracting two standard deviations Calculated by adding and subtracting two standard deviations from the meanfrom the mean

Normal distribution is also called Gaussian distributionNormal distribution is also called Gaussian distributionWhy the normal distribution is popularWhy the normal distribution is popular

Possess certain mathematical properties that make it attractive Possess certain mathematical properties that make it attractive and and easy to manipulateeasy to manipulateThe assumption of much statistical theory and methodologyThe assumption of much statistical theory and methodologyBasic for the use of inferential statisticsBasic for the use of inferential statistics

Properties of the Normal DistributionA symmetrical bell-shaped curveThe area between any two designated points is always the same

The amount of area under the normal curve is directly proportional to the percentage of raw scores

A symmetrical bellA symmetrical bell--shaped curveshaped curveThe area between any two designated points is always the sameThe area between any two designated points is always the same

The amount of area under the normal curve is directly proportionThe amount of area under the normal curve is directly proportional to the al to the percentage of raw scorespercentage of raw scores

Figure 6.1 Important divisions of the Normal Distribution of SAT scores

The normal distribution is a theoretical distribution defined bytwo parameters: the mean and the standard deviationThe normal distribution is a theoretical distribution defined byThe normal distribution is a theoretical distribution defined bytwo parameters: the mean and the standard deviationtwo parameters: the mean and the standard deviation

Properties of the Normal Distribution

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−=2

21exp

21

σµ

πσxy

Areas under the Normal CurveStandard normal distribution – 0 mean and 1 varianceZ score, Z value, or standard score

Standard normal distribution Standard normal distribution –– 0 mean and 1 variance0 mean and 1 varianceZ score, Z value, or standard scoreZ score, Z value, or standard score

σµ−

=xZ

Areas under the Normal CurveStandardizationStandardizationStandardization

Areas under the Normal CurveLet us assume that the SAT scores of a given population are normally distributed with mean of 500 and standard deviation of 100.What is the proportion of persons having SAT math scores between 500 and 650?What proportion of persons has SAT scores greater than 650?

Let us assume that the SAT scores of a given population are normLet us assume that the SAT scores of a given population are normally ally distributed with mean of 500 and standard deviation of 100.distributed with mean of 500 and standard deviation of 100.What is the proportion of persons having SAT math scores betweenWhat is the proportion of persons having SAT math scores between 500 500 and 650?and 650?What proportion of persons has SAT scores greater than 650?What proportion of persons has SAT scores greater than 650?

Areas under the Normal CurveWhat is the proportion of persons with SAT scores between 380 and 620?What is the proportion of persons with SAT scores between What is the proportion of persons with SAT scores between 380 and 620?380 and 620?

Areas under the Normal CurveWhat is the proportion of persons with SAT scores between 450 and 670?What is the proportion of persons with SAT scores between What is the proportion of persons with SAT scores between 450 and 670?450 and 670?

Areas under the Normal CurveWhat is the Z value of the normal curve that marks the upper 10% (or 90th percentile) of the area?What is the 90th percentile of SAT scores?

What is the Z value of the normal curve that marks the upper What is the Z value of the normal curve that marks the upper 10% (or 90th percentile) of the area?10% (or 90th percentile) of the area?What is the 90th percentile of SAT scores?What is the 90th percentile of SAT scores?

Areas under the Normal CurveIn an Australian national study of risk factor prevalence, two of the populations investigated are men whose blood pressures are within a normal or accepted range and who are not taking any corrective medication, and men who have had high blood pressure but who are at present undergoing antihypertensive drug therapyFor the population of men who are not taking corrective medication, diastolic blood pressure is appropriately normally distributed with mean of 80.7 mmHg and standard deviation of 9.2 mmHg. For the men who are using antihypertensive drugs, diastolic blood pressure is also approximately normally distributed with mean of 94.9 mmHg and standard deviation of 11.5 mm Hg

In an Australian national study of risk factor prevalence, two oIn an Australian national study of risk factor prevalence, two of the populations investigated f the populations investigated are men whose blood pressures are within a normal or accepted raare men whose blood pressures are within a normal or accepted range and who are not taking nge and who are not taking any corrective medication, and men who have had high blood pressany corrective medication, and men who have had high blood pressure but who are at present ure but who are at present undergoing antihypertensive drug therapyundergoing antihypertensive drug therapyFor the population of men who are not taking corrective medicatiFor the population of men who are not taking corrective medication, diastolic blood pressure on, diastolic blood pressure is appropriately normally distributed with mean of 80.7 mmHg andis appropriately normally distributed with mean of 80.7 mmHg and standard deviation of 9.2 standard deviation of 9.2 mmHg. For the men who are using antihypertensive drugs, diastolimmHg. For the men who are using antihypertensive drugs, diastolic blood pressure is also c blood pressure is also approximately normally distributed with mean of 94.9 mmHg and stapproximately normally distributed with mean of 94.9 mmHg and standard deviation of 11.5 andard deviation of 11.5 mm Hgmm Hg

Areas under the Normal Curve

Our goal is to be able to determine whether a man has normal blood pressure or whether he is taking antihypertensive medication solely on the basis of his diastolic blood pressure reading. If our goal is to identify 90% of the individuals who are currently taking medication, what value of diastolic blood pressure should be designated as the lower cutoff point?What proportion of the men with normal blood pressures will be incorrectly labeled as antihypertensive drug users?What if the cutoff were 90 mm Hg?

Our goal is to be able to determine whether a man has normal Our goal is to be able to determine whether a man has normal blood pressure or whether he is taking antihypertensive blood pressure or whether he is taking antihypertensive medication solely on the basis of his diastolic blood pressure medication solely on the basis of his diastolic blood pressure reading. If our goal is to identify 90% of the individuals who reading. If our goal is to identify 90% of the individuals who are currently taking medication, what value of diastolic blood are currently taking medication, what value of diastolic blood pressure should be designated as the lower cutoff point?pressure should be designated as the lower cutoff point?What proportion of the men with normal blood pressures will What proportion of the men with normal blood pressures will be incorrectly labeled as antihypertensive drug users?be incorrectly labeled as antihypertensive drug users?What if the cutoff were 90 mm Hg?What if the cutoff were 90 mm Hg?