Psy B07 Chapter 3Slide 1 THE NORMAL DISTRIBUTION

Psy B07 Chapter 3Slide 2  A quick look back  The normal distribution  Relationship between bars and lines  Area under the curve  Standard Normal Distribution  z-scores Outline

Psy B07 Chapter 3Slide 3 A quick look back  In Chapter 2, we spent a lot of time plotting distributions and calculating numbers to represent the distributions.  This raises the obvious question: WHY BOTHER? WHY BOTHER?

Psy B07 Chapter 3Slide 4 A quick look back  Answer: because once we know (or assume) the shape of the distribution and have calculated the relevant statistics, we are then able to make certain inferences about values of the variable.  In the current chapter, this will be show how this works using the Normal Distribution

Psy B07 Chapter 3Slide 5 The Normal Distribution  As shown by Galton (19th century guy), just about anything you measure turns out to be normally distributed, at least approximately so.  That is, usually most of the observations cluster around the mean, with progressively fewer observations out towards the extremes

Psy B07 Chapter 3Slide 6 The Normal Distribution  Example:  Thus, if we don’t know how some variable is distributed, our best guess is normality

Psy B07 Chapter 3Slide 7 The Normal Distribution  A note of caution  Although most variables are normally distributed, it is not the case that all variables are normally distributed.  Values of a dice roll.  Flipping a coin.  We will encounter some of these critters (i.e. distributions) later in the course

Psy B07 Chapter 3Slide 8 Relationship between bars and lines  Any Histogram:  Can be shown as a line graph:

Psy B07 Chapter 3Slide 9 Relationship between bars and lines  Example: Pop Quiz #1

Psy B07 Chapter 3Slide 10 Relationship between bars and lines

Psy B07 Chapter 3Slide 11 Area under the curve  Line graphs make it easier to talk of the “area under the curve” between two points where: area=proportion (or percent)=probability area=proportion (or percent)=probability  That is, we could ask what proportion of our class scored between 7 & 9 on the quiz

Psy B07 Chapter 3Slide 12 Area under the curve  If we assume that the total area under the curve equals one....  then the area between 7 & 9 equals the proportion of our class that scored between 7 & 9 and also indicates our best guess concerning the probability that some new data point would fall between 7 & 9.

Psy B07 Chapter 3Slide 13 Area under the curve  The problem is that in order to calculate the area under a curve, you must either: 1) use calculus 2) use a table that specifies the area associated with given values of you variable.

Psy B07 Chapter 3Slide 14 Area under the curve  The good news is that a table does exist, thereby allowing you to avoid calculus. The bad news is that in order to use it you must: 1) assume that your variable is normally distributed 2) use your mean and standard deviation to convert your data into z-scores such that the new distribution has a mean of 0 and a standard deviation of 1 - standard normal distribution or N(0,1).

Psy B07 Chapter 3Slide 15 Standard Normal Distribution

Psy B07 Chapter 3Slide 16 z-scores  It would be too much work to provide a table of area values for every possible mean and standard deviation.  Instead, a table was created for the standard normal distribution, and the data set of interest is converted to a standard normal before using the table.

Psy B07 Chapter 3Slide 17 z-scores  How do we get our mean equal to zero? Simple, subtract the mean from each data point.  What about the standard deviation? Well, if we divide all values by a constant, we divide the standard deviation by a constant. Thus, to make the standard deviation 1, we just divide each new value by the standard deviation.

Psy B07 Chapter 3Slide 18 z-scores  In computational form then,  where z is the z-score for the value of X we enter into the above equation

Psy B07 Chapter 3Slide 19 z-scores  Once we have calculated a z-score, we can then look at the z table in Appendix Z to find the area we are interested in relevant to that value.  As we’ll see, the z table actually provides a number of areas relevant to any specific z-score.

Psy B07 Chapter 3Slide 20 z-scores  What percent of students scored better than 9.2 out of 10 on the quiz, given that the mean was 7.6 and the standard deviation was 1.6?

Psy B07 Chapter 3Slide 21 z-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?clicking here