{"id":130,"date":"2023-05-05T12:48:46","date_gmt":"2023-05-05T16:48:46","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/chapter\/measures-of-association\/"},"modified":"2024-01-12T18:34:17","modified_gmt":"2024-01-12T23:34:17","slug":"measures-of-association","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/chapter\/measures-of-association\/","title":{"raw":"Measures of Association","rendered":"Measures of Association"},"content":{"raw":"
\r\n

Some Statistics that you may need:<\/h1>\r\n

Correlation<\/h2>\r\n

A [pb_glossary id=\"239\"]correlation[\/pb_glossary] exists between two variables when one of them is related to the other in some way.<\/p>\r\n

A scatter plot is a graph in which the paired [latex](x,y) [\/latex] sample data are plotted with a horizontal [latex]x[\/latex]-axis and a vertical [latex]y[\/latex]-axis.<\/p>\r\n

Linear Correlation means our plot looks like a line<\/p>\r\nIMAGE\r\n

The Linear <\/strong>C<\/strong>orrelation <\/strong>C<\/strong>oefficient<\/strong> or Pearson <\/strong>P<\/strong>roduct <\/strong>M<\/strong>oment <\/strong>C<\/strong>orrelation <\/strong>C<\/strong>oefficient<\/strong> is a way to look at the variances of our data and come up with [latex]r[\/latex], a number which tells us how strong the correlation is.<\/p>\r\n\\[r = \\frac{1}{n-1}\\Sigma \\left( \\frac{x-\\bar{x}}{s_x}\\right)\\left( \\frac{y-\\bar{y}}{s_y}\\right)\\]\r\n\r\n<\/div>\r\n \r\n\r\n\\[r = \\frac{1}{n-1}\\Sigma \\left( \\frac{x-\\bar{x}}{s_x}\\right)\\left( \\frac{y-\\bar{y}}{s_y}\\right)\\]\r\n

\r\n

where the sum \u2211 is over all ordered pairs (<\/em>x,y<\/em>), <\/em>s<\/em>x<\/em><\/sub> is the standard deviation of the x<\/em> values, s<\/em>y<\/em><\/sub> is the standard deviation of the y values, and<\/p>\r\n\r\n

\\[\\bar{x} = E(x) = \\frac{\\Sigma x}{n}\\]\r\n<\/span><\/div>\r\nand\r\n
\\[\\bar{y} = E(x) = \\frac{\\Sigma y}{n}\\]<\/span><\/div>\r\nare the sample means of x<\/em> and y<\/em> respectively.\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

Strong Positive<\/p>\r\n<\/td>\r\n

\r\n

Weak Positive<\/p>\r\n<\/td>\r\n

\r\n

Weak Negative<\/p>\r\n<\/td>\r\n

\r\n

Strong Negative<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

\r\n[latex]r = 0.9[\/latex]<\/p>\r\n<\/td>\r\n

\r\n

\"image\"\r\n[latex]r =0.3[\/latex]<\/p>\r\n<\/td>\r\n

\r\n

\"image\"\r\n[latex]r = -0.3[\/latex]<\/p>\r\n<\/td>\r\n

\r\n

\"image\"\r\n[latex]r = -0.9[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"\" align=\"alignnone\" width=\"567\"]\"image Some values of r[\/caption]\r\n

Covariance<\/h2>\r\n

Covariance is another way of measuring correlation, but can also look at some non-linear relationships. It is defined by:<\/p>\r\n

\\[cov(X, Y) = \\frac{1}{n}\\Sigma (x_i-\\bar{x})(y_i-\\bar{y})\\]\r\n<\/span><\/p>\r\n

where the sum \u2211 is over all ordered pairs (<\/em>x,y<\/em>),<\/em><\/p>\r\n \r\n

Fun fact! When you take the same set of data twice, you get the following identities:<\/p>\r\n

\\[cov(X, X)=\\sigma^2\\]<\/span><\/p>\r\n

\\[r=corr(X,X)=1\\]<\/span><\/p>\r\n

\"image\"\r\n\"image\"Correlation vs. Causation<\/strong><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"430\"]\"image\" This graph really is begging me to give a citation![\/caption]\r\n

In the following chart, we can see a clear correlation between the number of people who drowned by falling in a swimming pool in the USA and number of films that Nicholas Cage appeared in in that year.<\/p>\r\n

CAUSATION: Do you think Nicholas Cage causes drowning?<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"459\"]\"image This is from XKCD, https:\/\/xkcd.com\/552\/[\/caption]\r\n

Or: Does smoking cause lung cancer? It\u2019s harder than you think to prove.<\/p>\r\n

Remember\u2026.. CORRELATION does not imply CAUSATION<\/p>\r\n

\"image\"\r\n\"image\"<\/p>\r\n\r\n

1<\/a> Measures of Centre, Variation and other measures are adapted from \u201cEssentials of Business Statistics\u201d; Black, Goldlist, Edmunds, Castillo; Wiley Publishing 2018.<\/div>\r\n<\/div>","rendered":"
\n

Some Statistics that you may need:<\/h1>\n

Correlation<\/h2>\n

A correlation<\/a> exists between two variables when one of them is related to the other in some way.<\/p>\n

A scatter plot is a graph in which the paired [latex](x,y)[\/latex] sample data are plotted with a horizontal [latex]x[\/latex]-axis and a vertical [latex]y[\/latex]-axis.<\/p>\n

Linear Correlation means our plot looks like a line<\/p>\n

IMAGE<\/p>\n

The Linear <\/strong>C<\/strong>orrelation <\/strong>C<\/strong>oefficient<\/strong> or Pearson <\/strong>P<\/strong>roduct <\/strong>M<\/strong>oment <\/strong>C<\/strong>orrelation <\/strong>C<\/strong>oefficient<\/strong> is a way to look at the variances of our data and come up with [latex]r[\/latex], a number which tells us how strong the correlation is.<\/p>\n

\\[r = \\frac{1}{n-1}\\Sigma \\left( \\frac{x-\\bar{x}}{s_x}\\right)\\left( \\frac{y-\\bar{y}}{s_y}\\right)\\]<\/p>\n<\/div>\n

 <\/p>\n

\\[r = \\frac{1}{n-1}\\Sigma \\left( \\frac{x-\\bar{x}}{s_x}\\right)\\left( \\frac{y-\\bar{y}}{s_y}\\right)\\]<\/p>\n

\n

where the sum \u2211 is over all ordered pairs (<\/em>x,y<\/em>), <\/em>s<\/em>x<\/em><\/sub> is the standard deviation of the x<\/em> values, s<\/em>y<\/em><\/sub> is the standard deviation of the y values, and<\/p>\n

\\[\\bar{x} = E(x) = \\frac{\\Sigma x}{n}\\]
\n<\/span><\/div>\n

and<\/p>\n

\\[\\bar{y} = E(x) = \\frac{\\Sigma y}{n}\\]<\/span><\/div>\n

are the sample means of x<\/em> and y<\/em> respectively.<\/p>\n\n\n\n\n\n
\n

Strong Positive<\/p>\n<\/td>\n

\n

Weak Positive<\/p>\n<\/td>\n

\n

Weak Negative<\/p>\n<\/td>\n

\n

Strong Negative<\/p>\n<\/td>\n<\/tr>\n

\n

\n[latex]r = 0.9[\/latex]<\/p>\n<\/td>\n

\n

\"image\"
\n[latex]r =0.3[\/latex]<\/p>\n<\/td>\n

\n

\"image\"
\n[latex]r = -0.3[\/latex]<\/p>\n<\/td>\n

\n

\"image\"
\n[latex]r = -0.9[\/latex]<\/p>\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n\n

\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\"image
Some values of r<\/figcaption><\/figure>\n

Covariance<\/h2>\n

Covariance is another way of measuring correlation, but can also look at some non-linear relationships. It is defined by:<\/p>\n

\\[cov(X, Y) = \\frac{1}{n}\\Sigma (x_i-\\bar{x})(y_i-\\bar{y})\\]
\n<\/span><\/p>\n

where the sum \u2211 is over all ordered pairs (<\/em>x,y<\/em>),<\/em><\/p>\n

 <\/p>\n

Fun fact! When you take the same set of data twice, you get the following identities:<\/p>\n

\\[cov(X, X)=\\sigma^2\\]<\/span><\/p>\n

\\[r=corr(X,X)=1\\]<\/span><\/p>\n

\"image\"
\n\"image\"Correlation vs. Causation<\/strong><\/p>\n

\"image\"
This graph really is begging me to give a citation!<\/figcaption><\/figure>\n

In the following chart, we can see a clear correlation between the number of people who drowned by falling in a swimming pool in the USA and number of films that Nicholas Cage appeared in in that year.<\/p>\n

CAUSATION: Do you think Nicholas Cage causes drowning?<\/p>\n

\"image
This is from XKCD, https:\/\/xkcd.com\/552\/<\/figcaption><\/figure>\n

Or: Does smoking cause lung cancer? It\u2019s harder than you think to prove.<\/p>\n

Remember\u2026.. CORRELATION does not imply CAUSATION<\/p>\n

\"image\"
\n\"image\"<\/p>\n