The Normal CurveThe Normal Curve

IntroductionIntroduction

The normal curveThe normal curveWill need to understand it to Will need to understand it to understand inferential statisticsunderstand inferential statisticsIt is a theoretical modelIt is a theoretical model

Most actual distributions donMost actual distributions don’’t look like t look like this, but may be closethis, but may be closeIt is a frequency polygon that is perfectly It is a frequency polygon that is perfectly symmetrical and smoothsymmetrical and smoothIt is bell-shaped and unimodalIt is bell-shaped and unimodalIts tails extend infinitely in both Its tails extend infinitely in both directions and never intersect with the directions and never intersect with the horizontal axishorizontal axis

Distances on The Distances on The Normal CurveNormal Curve

Distances along the horizontal axis are Distances along the horizontal axis are divided into standard deviations and divided into standard deviations and will always include the same will always include the same proportion of the total areaproportion of the total area

This is true for a flat curve or a tall, This is true for a flat curve or a tall, narrow curvenarrow curveIt still divides into the same It still divides into the same percentagespercentagesSo, as the standard deviation of a So, as the standard deviation of a normal distribution increases, the normal distribution increases, the percentage of the area between plus percentage of the area between plus and minus one standard deviation will and minus one standard deviation will stay the samestay the same

Distances from Distances from the Meanthe Mean

Between plus and minus 1 standard Between plus and minus 1 standard deviations lies 68.26% of the areadeviations lies 68.26% of the area

Between plus and minus 2 standard Between plus and minus 2 standard deviations lies 95.44 % of the areadeviations lies 95.44 % of the area

Between plus and minus 3 standard Between plus and minus 3 standard deviations lies 99.72 % of the areadeviations lies 99.72 % of the area

On all normal curves, the area between the On all normal curves, the area between the mean and plus one standard deviation will be mean and plus one standard deviation will be 34.13%34.13%

Normally Normally Distributed Distributed VariablesVariables

The normal curve will tell you what percentage The normal curve will tell you what percentage of people are in any area of the curveof people are in any area of the curve

A normal distribution of 1,000 cases will have A normal distribution of 1,000 cases will have 683 people between plus and minus 1 683 people between plus and minus 1 standard deviation, about 954 people between standard deviation, about 954 people between plus and minus 2 standard deviations, and plus and minus 2 standard deviations, and nearly all people (997) between plus or minus nearly all people (997) between plus or minus 3 standard deviations3 standard deviations

Only 3 people will be outside 3 standard Only 3 people will be outside 3 standard deviations from the mean, if the sample size is deviations from the mean, if the sample size is 1,0001,000

Computing Z Computing Z ScoresScoresIf your score on an exam was exactly 1 If your score on an exam was exactly 1

standard deviation above the mean, you standard deviation above the mean, you would know that you did better than would know that you did better than 84.13 percent of the students (the 50% 84.13 percent of the students (the 50% below half, added to the 34.13% below half, added to the 34.13% between half and the first standard between half and the first standard deviation)deviation)

However, itHowever, it’’s not likely that your score s not likely that your score will be the same as the mean plus will be the same as the mean plus exactly one standard deviationexactly one standard deviation

So, Z scores are used to find the So, Z scores are used to find the percentage of scores below yours from percentage of scores below yours from any place on the horizontal axisany place on the horizontal axis

The standardized normal distribution (or The standardized normal distribution (or Z distribution) has a mean of 0 and a Z distribution) has a mean of 0 and a standard deviation of 1standard deviation of 1

The curve becomes generic, or universal, and The curve becomes generic, or universal, and you can plug in any mean and standard you can plug in any mean and standard deviation into itdeviation into it

Formula for Converting Raw Formula for Converting Raw Scores Into Z ScoresScores Into Z Scores

You plug your score into the X sub i You plug your score into the X sub i positionposition

You will be given the mean and the You will be given the mean and the standard deviation of the samplestandard deviation of the sample

s

XXZ i

Calculating Z Calculating Z ScoresScores

The Z score table gives the area The Z score table gives the area between a Z score and the meanbetween a Z score and the mean

For a Z score of -1.00, that area (in For a Z score of -1.00, that area (in percentages) is 34.13%percentages) is 34.13%

If a Z score is 0, what would that tell If a Z score is 0, what would that tell you?you?

The value of the corresponding raw score The value of the corresponding raw score would be the same as the mean of the would be the same as the mean of the empirical distributionempirical distribution

Using the Normal Curve to Using the Normal Curve to Estimate ProbabilitiesEstimate Probabilities

Can also think about the normal curve Can also think about the normal curve as a distribution of probabilitiesas a distribution of probabilities

Can estimate the probability that a Can estimate the probability that a case randomly picked from a normal case randomly picked from a normal distribution will fall in a particular distribution will fall in a particular areaarea

To find a probability, a fraction needs To find a probability, a fraction needs to be usedto be used

The numerator will equal the number of The numerator will equal the number of events that would constitute a successevents that would constitute a success

The denominator equals the total number The denominator equals the total number of possible events where a success could of possible events where a success could occuroccur

ExampleExampleThe example in your book of your The example in your book of your chances of drawing a king of hearts chances of drawing a king of hearts from a well-shuffled deck of cardsfrom a well-shuffled deck of cards

The fraction is 1/52The fraction is 1/52

Or can express the fraction as a Or can express the fraction as a proportion by dividing the numerator by proportion by dividing the numerator by the denominatorthe denominator

So 1/52 = .0192308 = .0192So 1/52 = .0192308 = .0192

In the social sciences, probabilities are In the social sciences, probabilities are usually expressed as proportionsusually expressed as proportions

Or 1.92 percent of the timeOr 1.92 percent of the time

ProbabilitiesProbabilitiesTherefore, the areas in the normal Therefore, the areas in the normal curve table can also be thought of curve table can also be thought of as probabilities that a randomly as probabilities that a randomly selected case will have a score in selected case will have a score in that areathat area

So, the probability is very high that So, the probability is very high that any case randomly selected from a any case randomly selected from a normal distribution will have a score normal distribution will have a score close in value to that of the meanclose in value to that of the mean

The normal curve shows that most The normal curve shows that most cases are clustered around the mean, cases are clustered around the mean, and they decline in frequency as you and they decline in frequency as you move farther away from the mean valuemove farther away from the mean value

ProbabilitiesProbabilitiesCan also say that the probability that Can also say that the probability that a randomly selected case will have a a randomly selected case will have a score within plus or minus 1 standard score within plus or minus 1 standard deviations of the mean is 0.6826deviations of the mean is 0.6826

If we randomly select a number of If we randomly select a number of cases from a normal distribution, we cases from a normal distribution, we will most often select cases that have will most often select cases that have scores close to the mean—but rarely scores close to the mean—but rarely select cases that have scores far select cases that have scores far above or below the meanabove or below the mean