{"id":487,"date":"2024-10-18T02:01:07","date_gmt":"2024-10-18T02:01:07","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/mat1260\/?post_type=chapter&p=487"},"modified":"2024-12-11T21:12:05","modified_gmt":"2024-12-11T21:12:05","slug":"3-3-correlation-coefficient","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/mat1260\/chapter\/3-3-correlation-coefficient\/","title":{"raw":"3.3: Correlation Coefficient","rendered":"3.3: Correlation Coefficient"},"content":{"raw":"
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Introduction<\/h2>\r\n<\/div>\r\n
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So far we have visualized relationships between two quantitative variables using scatterplots, and described the overall pattern of a relationship by considering its direction, form, and strength. We noted that assessing the strength of a relationship just by looking at the scatterplot is quite difficult, and therefore we need to supplement the scatterplot with some kind of numerical measure that will help us assess the strength.<\/p>\r\nIn this part, we will restrict our attention to the\u00a0special case of relationships that have a linear form<\/em>, since they are quite common and relatively simple to detect. More importantly, there exists a numerical measure that assesses the strength of the linear relationship between two quantitative variables with which we can supplement the scatterplot. We will introduce this numerical measure here and discuss it in detail.\r\n

Even though from this point on we are going to focus only on linear relationships, it is important to remember that\u00a0not every relationship between two quantitative variables has a linear form.<\/em>\u00a0We have actually seen several examples of relationships that are not linear. The statistical tools that will be introduced here are\u00a0appropriate only for examining linear relationships,<\/em>\u00a0and as we will see, when they are used in nonlinear situations, these tools can lead to errors in reasoning.<\/p>\r\nLet\u2019s start with a motivating example. Consider the following two scatterplots.\r\n\r\n\"Two<\/span><\/span>\r\n

We can see that in both cases, the direction of the relationship is\u00a0positive<\/em>\u00a0and the form of the relationship is\u00a0linear<\/em>. What about the strength? Recall that the strength of a relationship is the extent to which the data follow its form.<\/p>\r\n\r\n

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Did I get this?<\/h3>\r\n<\/header>\r\n
\r\n\r\n[h5p id=\"43\"]\r\n\r\n<\/div>\r\n<\/div>\r\nThe purpose of this example was to illustrate how assessing the strength of the linear relationship from a scatterplot alone is problematic, since our judgment might be affected by the scale on which the values are plotted. This example, therefore, provides a motivation for the\u00a0need\u00a0<\/em>to supplement the scatterplot with a\u00a0numerical measure<\/em>\u00a0that will\u00a0measure the strength<\/em>\u00a0of the linear relationship between two quantitative variables.\r\n
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The Correlation Coefficient\u2014r<\/span><\/h2>\r\n

The numerical measure that assesses the strength of a linear relationship is called the\u00a0correlation coefficient<\/em>\u00a0and is denoted by\u00a0r<\/em>. We will<\/p>\r\n\r\n

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  • Define the correlation r.<\/li>\r\n \t
  • Discuss the calculation of r.<\/li>\r\n \t
  • Explain how to interpret the value of r.<\/li>\r\n \t
  • Talk about some of the properties of r.<\/li>\r\n<\/ul>\r\nDefinition:\u00a0<\/em>The correlation coefficient (r) is a numerical measure that measures the\u00a0strength<\/em>\u00a0and\u00a0direction<\/em>\u00a0of a linear relationship between two quantitative variables.\r\n\r\nCalculation:\u00a0<\/em>r is calculated using the following formula: [latex]\\mathcal{r}=\\frac{1}{\\mathcal{n}1}\\sum_{\\mathcal{i}=1}^{\\mathcal{n}}\\left(\\frac{\\mathcal{x}_\\mathcal{i}-\\bar{\\mathcal{x}}}{\\mathcal{S}_\\mathcal{x}}\\right)\\left(\\frac{\\mathcal{y}_\\mathcal{i}-\\bar{\\mathcal{y}}}{\\mathcal{S}_\\mathcal{x}}\\right)[\/latex]\r\n

    However, the calculation of the correlation (r) is not the focus of this course. We will use a statistics package to calculate r for us, and the emphasis of this course is on the\u00a0interpretation<\/em>\u00a0of its value.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    Interpretation<\/span><\/h2>\r\n

    Once we obtain the value of r, its interpretation with respect to the strength of linear relationships is quite simple, as this walkthrough illustrates:<\/p>\r\nhttps:\/\/youtube.com\/watch?v=Bt-Ey2ebfvs\r\n

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    \r\n\r\nTo get a better sense of how the value of r relates to the strength of the linear relationship, take a look at the activity below.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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