What s a good way to show how the results came out? The relationship between two variables can be represented visually by a SCATTER DIAGRAM.

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1 COUNTING DOTS students were asked to count the dots in the square below without making an marks on their sheet, and then to count them again (There are 87 dots) What s a good wa to show how the results came out? The relationship between two variables can be represented visuall b a SCATTER DIAGRAM Scatter diagram for count the dots How do ou think the results turned out? Would the first or second count tend to be more accurate? Wh? Would there be an association between the first and second counts? Wh? How man students got the correct count the first time? The second time? Both times? One point in the scatter diagram seems to be quite unusual Which one? What are its coordinates? Is there an eplanation? 8

2 How can we find the distribution of the first count? Was the first or second count more accurate? Scatter diagram for count the dots, with one outlier deleted Scatter diagram for count the dots, with one outlier deleted Average SD First count Second count How unusual is a first count of 7? 8 8

3 Is there an association between the first and second counts? Scatter diagram for count the dots, with one outlier deleted Points per quadrant Left Right Upper Lower How can we measure the strength of the association between the first and second counts? SUMMARIZING A SCATTER DIAGRAM A scatter diagram can be summarized b means of five statistics: The average and SD of the -values The average and SD of the -values The correlation coefficient r The averages and the SDs specif the location and spread of the cloud of points, horizontall and verticall The correlation coefficient measures the amount of linear association, ie, clustering about a line Ave+SD Ave Ave SD Ave SD Ave Ave+SD }{{} 9% of the points Strong association Tight clustering Knowing helps in predicting Correlation near Ave SD Ave Ave+SD }{{} 9% of the points Weak association Loose clustering Knowing isn t much help in predicting Correlation near Ave+SD Ave Ave SD 9% of the points 8 8

4 EXAMPLES OF POSITIVE CORRELATIONS In each diagram there are points, with an average of and and SD of, both horizontall and verticall The correlation coefficient r measures the amount of clustering about a line 8 9 r alwas lies between and What values of r are common in the social sciences? The biological sciences? The phsical sciences? 8 7 EXAMPLES, CONTINUED Here are the same scatter diagrams, with ovals showing the main portion of the clouds of points 8 9 Positive association (as increases, tends to increase also) is indicated b a sign in the correlation coefficient r As r gets closer to, the points cluster more tightl around a line 8 8

5 EXAMPLES OF NEGATIVE CORRELATIONS Here are some scatter plots with negative correlation coefficients Again the average is and the SD is, both horizontall and verticall Negative association (as increases, tends to decrease) is indicated b a sign in the correlation coefficient r As r gets closer to, the points cluster more tightl around a line 8 9 EXERCISES In scrambled order, the correlation coefficients in the following diagrams are 9, 7,, 7,, and Match the diagrams with the correlation coefficients The diagrams on pages 8 and 9 can be used to guess r b ee, in cases where the spread of the scatter diagram from side-to-side is about the same as the spread up-and-down 8

6 The following scatter diagram shows the heights of,78 fathers and their full-grown sons There is one dot for each father-son pair The SD of height is 7 inches, both for the fathers and for the sons SON S HEIGHT (INCHES) PEARSON S FATHER-SON DATA FATHER S HEIGHT (INCHES) Is the correlation between heights of fathers and heights of sons around,,,, or 7? If ou took onl the father-son pairs where the father was taller than si feet, would the correlation between the heights be around,,,, or 7? 8 Is the correlation between the heights of husbands and wives in the US around 7,,,, or 7? Eplain Count-the-dots, revisited The horizontal and vertical SDs are about the same correlation coefficient about,, or 7? Eplain 8 Is the

7 THE SD LINE The points in a scatter diagram cluster (tightl or loosel) about the so-called SD line, which goes through all the points which are an equal number of SDs awa from the average, for both variables Ave+SD Ave Ave SD Ave SD Ave Ave+SD Positive correlation The SD line Ave SD Ave Ave+SD Negative correlation Ave+SD Ave Ave SD The SD line goes through the point of averages (marked b the in the diagrams); this is the point with coordinates (, ) When the correlation coefficient is positive, the line rises one vertical SD for each run of one horizontal SD When the correlation coefficient is negative, the line falls one vertical SD for each run of one horizontal SD One stud on male college students found their average height to be 9 inches, with an SD of inches Their average weight was pounds, with an SD of pounds And the correlation was If one of these men is 7 inches tall, how heav would he have to be to fall on the SD line? COMPUTING THE CORRELATION COEFFICIENT To compute a correlation coefficient, ou convert each variable to standard units and take the average of the products: average of ( in standard units) ( in standard units) A TA gives a quiz with questions to students results are: Number right Number wrong 8 7 What is the correlation coefficient? The deviations std units deviations std units product 8 / / / 7 / / / / / 9/ Ave SD Ave SD average product = ( ) + 9 = = Is there an easier wa to reach this conclusion? 8 8

8 Wh does r work as a measure of correlation? Consider the computation of r for the count-the-dots data: + Ave of + Ave of In the upper-right and lower-left quadrants, the products ( in standard units) ( in standard units) are, while in the remaining two quadrants the products are The average all the products is the correlation coefficient r is positive in this case because the points in the two positive quadrants predominate 8