Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Psy B07 THE NORMAL DISTRIBUTION Chapter 3 Slide 1 Psy B07 Outline A quick look back The normal distribution Relationship between bars and lines Area under the curve Standard Normal Distribution z-scores Chapter 3 Slide 2 Psy B07 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? Chapter 3 Slide 3 Psy B07 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 Chapter 3 Slide 4 Psy B07 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 Chapter 3 Slide 5 Psy B07 The Normal Distribution Example: 12 10 8 6 4 2 0 Thus, if we don’t know how some variable is distributed, our best guess is normality Chapter 3 Slide 6 Psy B07 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 Chapter 3 Slide 7 Psy B07 Relationship between bars and lines Any Histogram: 14 12 10 8 6 4 2 0 16 Can be shown as a line graph: 14 12 10 8 6 4 2 0 Chapter 3 Slide 8 Psy B07 Relationship between bars and lines Example: Pop Quiz #1 25 22 22 20 17 15 13 10 7 5 2 0 Chapter 3 17 2 2 3 7 8 13 8 3 0 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Slide 9 Psy B07 Relationship between bars and lines 25 20 15 10 5 0 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Chapter 3 Slide 10 Psy B07 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 That is, we could ask what proportion of our class scored between 7 & 9 on the quiz Chapter 3 Slide 11 Psy B07 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. Chapter 3 Slide 12 Psy B07 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. Chapter 3 Slide 13 Psy B07 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). Chapter 3 Slide 14 Psy B07 Standard Normal Distribution 0.5 0.4 0.3 0.2 0.1 0 -3 Chapter 3 -2 -1 0 Z-SCORE 1 2 3 z Mean to z Larger Portion Smaller Portion ..... .98 .99 1.00 1.01 ..... ........ .3365 .3389 .3413 .3438 ........ ........ .8365 .8389 .8413 .8438 ........ ........ .1635 .1611 .1587 .1562 ........ Slide 15 Psy B07 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. Chapter 3 Slide 16 Psy B07 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. Chapter 3 Slide 17 Psy B07 z-scores In computational form then, z X where z is the z-score for the value of X we enter into the above equation Chapter 3 Slide 18 Psy B07 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. Chapter 3 Slide 19 Psy B07 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? Chapter 3 Slide 20 Psy B07 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? Chapter 3 Slide 21