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Objective: To look for relationships between two quantitative variables

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Scatterplots may be the most common and most effective display for data. o In a scatterplot, you can see patterns, trends, relationships, and even the occasional extraordinary value sitting apart from the others. Scatterplots are the best way to start observing the relationship and the ideal way to picture associations between two quantitative variables.

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When looking at scatterplots, we will look for direction, form, strength, and unusual features. Direction: o A pattern that runs from the upper left to the lower right is said to have a negative direction (just like the graph of a line with a negative slope). o A trend running the other way has a positive direction (just like the graph of a line with a positive slope).

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Direction (cont.) Can the NOAA predict where a hurricane will go? The figure shows a negative direction and a negative association between the year since 1970 and the and the prediction errors made by NOAA. As the years have passed, the predictions have improved (errors have decreased).

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Form: o If the relationship isn’t straight, but curves gently, while still increasing or decreasing steadily, we can often find ways to make it more nearly straight.

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Strength: o At one extreme, the points appear to follow a single stream (whether straight, curved, or bending all over the place).

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Strength (cont.): o At the other extreme, the points appear as a vague cloud with no discernible trend or pattern: o Note: we will quantify the amount of scatter soon.

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It is important to determine which of the two quantitative variables goes on the x-axis and which on the y-axis. This determination is made based on the roles played by the variables. When the roles are clear, the explanatory or predictor variable goes on the x-axis, and the response variable (variable of interest) goes on the y-axis.

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What do you expect the scatterplot to look like? Remember direction, form, strength, and unusual features. 1. Drug dosage and degree of pain relief 2. Calories consumed and weight loss

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Data collected from students in Statistics classes included their heights (in inches) and weights (in pounds): Here we see positive association and a fairly straight form, there seems to a high outlier. Outlier

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How strong is the association between weight and height of Statistics students? If we had to put a number on the strength, we would not want it to depend on the units we used. A scatterplot of heights (in centimeters) and weights (in kilograms) doesn’t change the shape of the pattern:

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Note that the underlying linear pattern seems steeper in the standardized plot than in the original scatterplot. That’s because we made the scales of the axes the same. Equal scaling gives a neutral way of drawing the scatterplot and a fairer impression of the strength of the association

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The points in the upper right and lower left (those in green) strengthen the impression of a positive association between height and weight. The points in the upper left and lower right where z x and z y have opposite signs (those in red) tend to weaken the positive association. Points with z-scores of zero (those in blue) don’t vote either way, because their product is zero.

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correlation coefficient (r) The correlation coefficient (r) gives us a numerical measurement of the strength of the linear relationship between the explanatory and response variables.

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Calculating this by hand can be time consuming and redundant. Below are the steps to calculating it with the use of a calculator: o Make sure your diagnostics are ON (2 nd Catalog, scroll to Diagnostics ON Enter) o Store your values into L1 and L2 (x and y respectively) o Stat Calc 8: LinReg(a+bx) o Before pressing Enter, define the lists: L1, L2 Enter

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Correlation Correlation measures the strength of the linear association between two quantitative variables. Before you use correlation, you must check several conditions: o Quantitative Variables Condition o Straight Enough Condition o Outlier Condition

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Quantitative Variables Condition: o Correlation applies only to quantitative variables. o Don’t apply correlation to categorical data camouflaged as quantitative (zip codes, ID #s, area codes, etc.). o Check that you know the variables’ units and what they measure.

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Straight Enough Condition: o You can calculate a correlation coefficient for any pair of variables. o But correlation measures the strength only of the linear association, and will be misleading if the relationship is not linear.

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Outlier Condition: o Outliers can distort the correlation dramatically. o An outlier can make an otherwise small correlation look big or hide a large correlation. o It can even give an otherwise positive association a negative correlation coefficient (and vice versa). o When you see an outlier, it’s often a good idea to report the correlations with and without the point.

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The sign of a correlation coefficient gives the direction of the association. Correlation is always between –1 and +1. o Correlation can be exactly equal to –1 or +1, but these values are unusual in real data because they mean that all the data points fall exactly on a single straight line. o A correlation near zero corresponds to a weak linear association.

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Correlation treats x and y symmetrically: o The correlation of x with y is the same as the correlation of y with x. Correlation has no units. Correlation is not affected by changes in the center or scale of either variable. o Correlation depends only on the z-scores, and they are unaffected by changes in center or scale.

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Correlation measures the strength of the linear association between the two variables. o Variables can have a strong association but still have a small correlation if the association isn’t linear. Correlation is sensitive to outliers. A single outlying value can make a small correlation large or make a large one small.

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caused Whenever we have a strong correlation, it is tempting to explain it by imagining that the predictor variable has caused the response to help. never Scatterplots and correlation coefficients never prove causation. lurking variable A hidden variable that stands behind a relationship and determines it by simultaneously affecting the other two variables is called a lurking variable.

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It is common in some fields to compute the correlations between each pair of variables in a collection of variables and arrange these correlations in a table.

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Sketch a scatterplot of the following information. Discuss the direction, form, and strength of the association. If the data meet the appropriate conditions, find the correlation coefficient (r).

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Day 1: Day 1: pp. 164-170 # 1, 2, 5, 6, 8 Day 2: Day 2: pp. 164-170 # 11, 12, 16, 25, 34, 35 – 37

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