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Summarizing Data. Graphical Methods. Histogram. Grouped Freq Table. Stem-Leaf Diagram. Box-whisker Plot. Measure of Central Location. Mean Median. Measure of Variability (Dispersion, Spread). Range Inter-Quartile Range Variance, standard deviation Pseudo-standard deviation.
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Summarizing Data Graphical Methods
Histogram Grouped Freq Table Stem-Leaf Diagram Box-whisker Plot
Measure of Central Location • Mean • Median
Measure of Variability (Dispersion, Spread) • Range • Inter-Quartile Range • Variance, standard deviation • Pseudo-standard deviation
Descriptive techniques for Multivariate data In most research situations data is collected on more than one variable (usually many variables)
Graphical Techniques • The scatter plot • The two dimensional Histogram
The Scatter Plot For two variables X and Y we will have a measurements for each variable on each case: xi, yi xi = the value of X for case i and yi = the value of Y for case i.
To Construct a scatter plot we plot the points: (xi, yi) for each case on the X-Y plane. (xi, yi) yi xi
Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial Reading Acheivement Score, and Final Reading Acheivement Score for 23 students who have recently completed a reading improvement program Initial Final Verbal Math Reading Reading Student IQ IQ Acheivement Acheivement 1 86 94 1.1 1.7 2 104 103 1.5 1.7 3 86 92 1.5 1.9 4 105 100 2.0 2.0 5 118 115 1.9 3.5 6 96 102 1.4 2.4 7 90 87 1.5 1.8 8 95 100 1.4 2.0 9 105 96 1.7 1.7 10 84 80 1.6 1.7 11 94 87 1.6 1.7 12 119 116 1.7 3.1 13 82 91 1.2 1.8 14 80 93 1.0 1.7 15 109 124 1.8 2.5 16 111 119 1.4 3.0 17 89 94 1.6 1.8 18 99 117 1.6 2.6 19 94 93 1.4 1.4 20 99 110 1.4 2.0 21 95 97 1.5 1.3 22 102 104 1.7 3.1 23 102 93 1.6 1.9
Circular • No relationship between X and Y • Unable to predict Y from X
Ellipsoidal • Positive relationship between X and Y • Increases in X correspond to increases in Y (but not always) • Major axis of the ellipse has positive slope
Example Verbal IQ, MathIQ
Ellipsoidal (thinner ellipse) • Stronger positive relationship between X and Y • Increases in X correspond to increases in Y (more freqequently) • Major axis of the ellipse has positive slope • Minor axis of the ellipse much smaller
Increased strength in the positive relationship between X and Y • Increases in X correspond to increases in Y (almost always) • Minor axis of the ellipse extremely small in relationship to the Major axis of the ellipse.
Perfect positive relationship between X and Y • Y perfectly predictable from X • Data falls exactly along a straight line with positive slope
Ellipsoidal • Negative relationship between X and Y • Increases in X correspond to decreases in Y (but not always) • Major axis of the ellipse has negative slope slope
The strength of the relationship can increase until changes in Y can be perfectly predicted from X
In a Linear pattern Y increase with respect to X at a constant rate • In a Non-linear pattern the rate that Y increases with respect to X is variable
Growth patterns frequently follow a sigmoid curve • Growth at the start is slow • It then speeds up • Slows down again as it reaches it limiting size
Measures of strength of a relationship (Correlation) • Pearson’s correlation coefficient (r) • Spearman’s rank correlation coefficient (rho, r)
