psy b07 chapter 3slide 1 the normal distribution

Chapter 3Chapter 3 Slide Slide 11

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THE NORMAL THE NORMAL DISTRIBUTIONDISTRIBUTION


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A quick look backA quick look back The normal distributionThe normal distribution Relationship between bars and Relationship between bars and

lineslines Area under the curveArea under the curve Standard Normal DistributionStandard Normal Distribution z-scoresz-scores

OutlineOutline


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A quick look backA quick look back

In Chapter 2, we spent a lot of time In Chapter 2, we spent a lot of time plotting distributions and plotting distributions and calculating numbers to represent calculating numbers to represent the distributions.the distributions.

This raises the obvious question:This raises the obvious question:

WHY BOTHER?WHY BOTHER?


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A quick look backA quick look back

Answer: because once we know (or Answer: because once we know (or assume) the shape of the assume) the shape of the distribution and have calculated distribution and have calculated the relevant statistics, we are then the relevant statistics, we are then able to make certain inferences able to make certain inferences about values of the variable.about values of the variable.

In the current chapter, this will be In the current chapter, this will be show how this works using the show how this works using the Normal DistributionNormal Distribution


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The Normal DistributionThe Normal Distribution

As shown by Galton (19th century As shown by Galton (19th century guy), just about anything you guy), just about anything you measure turns out to be normally measure turns out to be normally distributed, at least approximately so.distributed, at least approximately so.

That is, usually most of the That is, usually most of the observations cluster around the observations cluster around the mean, with progressively fewer mean, with progressively fewer observations out towards the observations out towards the extremesextremes


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Example:Example:

Thus, if we don’t know how some variable Thus, if we don’t know how some variable is distributed, our best guess is normalityis distributed, our best guess is normality

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A note of cautionA note of caution Although most variables are normally Although most variables are normally

distributed, it is not the case that all distributed, it is not the case that all variables are normally distributed.variables are normally distributed. Values of a dice roll.Values of a dice roll. Flipping a coin.Flipping a coin.

We will encounter some of these We will encounter some of these critters (i.e. distributions) later in the critters (i.e. distributions) later in the coursecourse


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Relationship between Relationship between bars and linesbars and lines

Any Histogram:Any Histogram:

Can be shownCan be shownas a line graph:as a line graph:

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Relationship betweenRelationship betweenbars and linesbars and lines

Example: Pop Quiz #1 Example: Pop Quiz #1

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Relationship betweenRelationship betweenbars and linesbars and lines

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Area under the curveArea under the curve

Line graphs make it easier to talk of Line graphs make it easier to talk of the “area under the curve” between the “area under the curve” between two points where:two points where:

area=proportion (or area=proportion (or percent)=probabilitypercent)=probability

That is, we could ask what That is, we could ask what proportion of our class scored proportion of our class scored between 7 & 9 on the quizbetween 7 & 9 on the quiz


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If we assume that the total area If we assume that the total area under the curve equals one. . . .under the curve equals one. . . .

then the then the areaarea between 7 & 9 equals between 7 & 9 equals the the proportionproportion of our class that of our class that scored between 7 & 9 and also scored between 7 & 9 and also indicates our best guess concerning indicates our best guess concerning the the probabilityprobability that some new data that some new data point would fall between 7 & 9.point would fall between 7 & 9.


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The problem is that in order to The problem is that in order to calculate the area under a curve, calculate the area under a curve, you must either:you must either:

1) use calculus1) use calculus

2) use a table that specifies 2) use a table that specifies the area the area associated with given associated with given values of you values of you variable.variable.


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The good news is that a table does exist, The good news is that a table does exist, thereby allowing you to avoid calculus. thereby allowing you to avoid calculus. The bad news is that in order to use it you The bad news is that in order to use it you must:must:

1) assume that your variable is normally 1) assume that your variable is normally distributeddistributed

2) use your mean and standard deviation 2) use your mean and standard deviation to convert your data into to convert your data into z-scoresz-scores such such that the new distribution has a mean of that the new distribution has a mean of 0 and a standard deviation of 1 - 0 and a standard deviation of 1 - standard normal distributionstandard normal distribution or N(0,1). or N(0,1).


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Standard Normal Standard Normal DistributionDistribution

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Mean to Larger Smallerz z Portion Portion

..... ........ ........ .........98 .3365 .8365 .1635.99 .3389 .8389 .1611

1.00 .3413 .8413 .15871.01 .3438 .8438 .1562..... ........ ........ ........


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z-scoresz-scores

It would be too much work to provide It would be too much work to provide a table of area values for every a table of area values for every possible mean and standard possible mean and standard deviation.deviation.

Instead, a table was created for the Instead, a table was created for the standard normal distribution, and standard normal distribution, and the data set of interest is converted the data set of interest is converted to a standard normal before using to a standard normal before using the table.the table.


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z-scoresz-scores

How do we get our mean equal to zero? How do we get our mean equal to zero? Simple, subtract the mean from each Simple, subtract the mean from each data point.data point.

What about the standard deviation? What about the standard deviation? Well, if we divide all values by a Well, if we divide all values by a constant, we divide the standard constant, we divide the standard deviation by a constant. Thus, to make deviation by a constant. Thus, to make the standard deviation 1, we just divide the standard deviation 1, we just divide each new value by the standard each new value by the standard deviation.deviation.


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z-scoresz-scores

In computational form then,In computational form then,

where z is the z-score for the value of X where z is the z-score for the value of X we enter into the above equationwe enter into the above equation

Xz


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z-scoresz-scores

Once we have calculated a z-score, Once we have calculated a z-score, we can then look at the z table in we can then look at the z table in Appendix Z to find the area we are Appendix Z to find the area we are interested in relevant to that value.interested in relevant to that value.

As we’ll see, the z table actually As we’ll see, the z table actually provides a number of areas provides a number of areas relevant to any specific z-score.relevant to any specific z-score.


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z-scoresz-scores

What percent of students scored What percent of students scored better than 9.2 out of 10 on the better than 9.2 out of 10 on the quiz, given that the mean was 7.6 quiz, given that the mean was 7.6 and the standard deviation was and the standard deviation was 1.6?1.6?


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z-scoresz-scores

I have found the following online applet which you can use to see this process a little more directly.

It allows you to find the area between two points on the “standard normal” distribution.

Try it by clicking here – does this help your understanding?

http://www-stat.stanford.edu/~naras/jsm/FindProbability.html

psy b07 chapter 3slide 1 the normal distribution

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