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Part 1: Measures of Center Instructions: Analyze the Simulated Data. You will analyze one quantitative variable (2008). Report your findings in a well-written narrative. Distribution of 2008 Classroom Averages 25% 20% 15% 10% 5% 0% Classroom Averages The Center of the Distribution What is the mean? Interpret the mean in context. Mean = 692.5 The mean is the average of all scores. There are more scores below the mean than there are above the mean. What is the median? Interpret the median in the context. Median = 693 The median is the middle value, determined in this list by looking at index numbers 50 and 51. What is the mode? Interpret the mode in context. Mode = 648 The mode is the number that is repeated more often than any other. Compare the mean, median, and mode. The mean and median being almost the same is not surprising. Because the mode of 648 was repeated only four times, it is not significant in this analysis. However, it is good to look at in case one score was repeated, say 10 or more times, as this may indicate invalid scores. School Average Adams 709 Harrison 666 Jackson 707 Jefferson 660 Madison 703 Monroe 712 Van Buren 688 Washington 693 Score Average by School 850 MEAN 800 750 700 650 600 550 School How does the shape of the histogram relate to the measures of center? The mean (average is the most significant measure of center and by inserting it as a Trendline, we can see how each school performed against mean. What does a side-by-side comparison reveal about the distribution of 2008 averages across the schools? There are four above average -performing schools, two average-performing ones, and two well below average. It also shows that there are more low scores (below mean) and the high scores are relatively higher than the low ones are low. Is there anything interesting, surprising, or noteworthy about the distribution? It is interesting that there are more above-mean performing schools than there are average-performing ones. It is noteworthy that the low-performing schools are well below mean/average. Part 2: Measures of Spread and Outliers School 2008 2009 2008 Monroe 690 716 range 214 Jackson 667 712 stdev 46.77963 Monroe 648 632 IQR 61.25 Monroe 742 714 Variance 2188.333 Harrison 723 727 Q1 658.5 Adams 697 708 Q3 719.75 Van Buren 688 745 1.5 IQR 91.875 Madison 759 753 mean 692.5 Madison 738 745 median 693 Madison 781 789 mode 648 Washington 692 754 min 589 Van Buren 770 810 max 803 Van Buren 648 674 Jackson 724 711 Jefferson 646 675 Jackson 681 680 Harrison 600 637 Adams 684 690 Adams 728 745 Washington 624 672 Washington 678 720 Washington 731 774 Madison 657 702 Jackson 694 706 Van Buren 700 712 jackson 659 671 Harrison 703 695 Washington 589 643 Monroe 704 729 Harrison 677 677 Jefferson 681 709 Madison 703 715 Jackson 681 711 Adams 707 713 Jefferson 620 596 Jefferson 649 700 Van Buren 698 680 Monroe 788 804 Jefferson 674 700 Jackson 773 738 Van Buren 670 673 Washington 710 738 Jefferson 677 680 Washington 803 867 Jefferson 607 638 Van Buren 609 622 Harrison 635 660 Monroe 748 770 Van Buren 722 710 Harrison 621 631 Van Buren 678 653 Washington 711 790 Monroe 742 731 Monroe 699 757 Washington 717 738 Jefferson 714 742 Van Buren 672 709 Monroe 732 765 Jefferson 652 673 Madison 648 670 Washington 718 786 Harrison 619 650 Adams 778 782 Washington 689 701 Harrison 625 651 jackson 714 743 Madison 639 618 Washington 648 690 Monroe 642 642 Van Buren 704 744 Jackson 755 761 Jackson 754 760 Monroe 613 631 Washington 740 771 Monroe 716 733 Adams 670 661 Harrison 747 810 mean e Jefferson 651 635 median Adams 690 685 mode Van Buren 682 730 Jefferson 697 719 Jackson 706 712 Jackson 801 820 Harrison 689 669 Madison 709 694 Jackson 663 679 Washington 651 710 Madison 691 695 Van Buren 735 727 Van Buren 660 693 Adams 719 756 Harrison 713 690 Harrison 643 687 Jefferson 665 694 Adams 706 755 Jackson 631 690 Jefferson 705 727 Monroe 702 738 Monroe 753 771 Monroe 754 780 Analyze the data Simulated Data in Excel. Report your findings in a well-written narrative. The Measures of Spread Range Range = 214. It is the Difference between the smallest average and largest average. IQR IQR = 61.25. I got this answer from your friend, get help with this explanation: IQR tells how spread out the middle values are; it can also be used to tell when some of the other values are too far from the central value. Too far away points are called "outliers," because they "lie outside" the range in which we expect them. An outlier is any value that lies more than one and a half times the length of the box from either end of the box. Variance Variance = 2188.333 I got this answer from your friend, get help with this explanation Standard Deviation What do the measures of spread relative to one-another reveal about the dispersion among 2008 CLASS AVERAGES? You should also use the robust method (1.5 IQR Rule) to test for outliers. Discuss how outliers do, or would, affect the various statistics: Mean? Median? Mode? Range? IQR? Variance? Standard Deviation? Were the outliers obviously in, or absent from, the histogram (Project 1 and above)? Did, or would, the outliers explain the shape of the distribution (skew)? Do you think that the outliers should be excluded from the calculations? Why, or why not? What did you learn that was interesting, unexpected, or noteworthy? Note: Measurements are relative. Be careful of making statements such as "The standard deviation is 100. So the data is very spread out." These measurements are all relative. 100 may be large in some context and small in another. Try comparing one measurement to another. For example, does the standard deviation seem small/large relative to the overall range? Part 3: Regression Equation Instructions: Analyze the data Simulated Data in Excel. Report your findings in a well-written narrative. Analyze the simulated 2008/2009 classroom averages using our bivariate analysis methods. State the regression equation. Interpret the slope and the y-intercept. Do you think that the regression equation would be useful in predicting responses? Why or why not? Excel Instructions: Method 1: Using r, sy, SX, y-bar, and x-bar. Since you already have r, the Excel formulae that you will need are: =average(array) =stdev(array) Recall that you can calculate the slope and the y-intercept of the regression line using the formulae below. Try to use cell references and mathematical operations to perform these calculations in Excel. Round final values to the hundredth place. b=y-mx Sx Method 2: Using linest(). The Excel function =linest(known y's, known x's, const, stats) will produce the slope and y-intercept of the Least Square Regression Equation. In an empty cell, type the formula. Use the range of Salary cells for "known y's" and the range of Age cells for the "known x's". You can leave the other two arguments empty: "const" and "stats". Initially, you will get the slope only. To get the y- intercept, high-light the formula cell and the adjacent cell. Hit the "F2" key and release. Simultaneously press, Crtl, Shift, and Enter. The adjacent cell will populate with the intercept. Note: if you want to add the regression line over your scatter. An easy way to do this is to highlight your original scatterplot by left-clicking on a point in the scatter and then right-clicking to bring up a formatting menu. Select "Add Trendline". The default option is a Least Squares Regression line. Just click on "OK" to return to your scatter.

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Part 1:
The mean of the distribution of 2008 class averages is 692.5, its median is 693, and mode is 648. They are in fact very similar suggesting that the distribution is a typical bell-shaped with a single peak. If they are different, then we may suspect the skewness to the left or to the right. The following histogram proves my expectation of the shape of the distribution. It appears to be a bell-shaped coming from the normal distribution.

The side-by-side comparison barplots show that class averages among all schools are roughly similar. However, without any formal statistical tests, it is hard to see if any of the averages are different from others. But I would say that they are not terribly different from one another.

It is interesting that the observation that averages across all schools are very similar is translated to the shape of the distribution of whole observations being a bell-shaped, and it intuitively makes sense...

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