Descriptive - Revision history

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‎Example

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‎Example

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Admin: /* Correlation */

2013-08-05T16:01:52Z

‎Correlation

Admin at 15:59, 5 August 2013

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Admin: /* Example */

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‎Example

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 Khan Academy:
-* Intro to Measures of Dispersion: [https://www.khanacademy.org/math/probability/descriptive-statistics/variance_std_deviation/v/range–variance-and-standard-deviation-as-measures-of-dispersion]
+* Intro to Measures of Dispersion: [https://www.khanacademy.org/math/probability/descriptive-statistics/variance_std_deviation/v/range--variance-and-standard-deviation-as-measures-of-dispersion]
 * An example for the calculation of the population variance: [https://www.khanacademy.org/math/probability/descriptive-statistics/variance_std_deviation/v/variance-of-a-population]
 * Why do we need the sample variance? [https://www.khanacademy.org/math/probability/descriptive-statistics/variance_std_deviation/v/sample-variance]

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 Khan Academy:
-* Intro to Measures of Central Tendency: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/statistics-intro-–mean-–median-and-mode]
+* Intro to Measures of Central Tendency: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/statistics-intro--mean--median-and-mode]
 * An example for mean , median and mode calculation: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/mean-median-and-mode]
-* A short discussion of the relation between sample and population mean: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/statistics–-sample-vs–-population-mean]
+* A short discussion of the relation between sample and population mean: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/statistics--sample-vs--population-mean]
 == Dispersion ==

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 Khan Academy:
-* Intro to Measures of Central Tendency: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/statistics-intro–mean–median-and-mode]
+* Intro to Measures of Central Tendency: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/statistics-intro-–mean-–median-and-mode]
 * An example for mean , median and mode calculation: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/mean-median-and-mode]
-* A short discussion of the relation between sample and population mean: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/statistics–sample-vs–population-mean]
+* A short discussion of the relation between sample and population mean: [https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/statistics–-sample-vs–-population-mean]
 == Dispersion ==

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 <math>r=\frac{\frac{1}{n}\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) \left( y_{i}-\bar{y}\right) }{\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right)^{2}\,\frac{1}{n}\sum_{i=1}^{n}\left( y_{i}-\bar{y}\right) ^{2}}}=\frac{\sigma_{y,x}}{\sqrt{\sigma_y^2 \sigma_x^2}}</math>
-Written in this way you should recognise that the denominator is nothing else but the square root of the product of the (population) variance of <math>y</math>, <math>\sigma_y^2</math>, times the variance of <math>x</math>, <math>\sigma_x^2</math>. The term in the numerator is what is called the ''population covariance'' between <math>y</math> and <math>x</math>, <math>\sigma_{y,x}</math>. The ''covariance'' is actually also a measure of the relationship between these two variables, but it has many of the same shortcomings as the variance (see [[Descriptive|Descriptive Statistics]]). Therefore we want a standardised measure (to ensure that <math>-1\leq r \geq 1</math>. This standardisation uses the square root of the two respective variances.
+Written in this way you should recognise that the denominator is nothing else but the square root of the product of the (population) variance of <math>y</math>, <math>\sigma_y^2</math>, times the variance of <math>x</math>, <math>\sigma_x^2</math>. The term in the numerator is what is called the ''population covariance'' between <math>y</math> and <math>x</math>, <math>\sigma_{y,x}</math>. The ''covariance'' is actually also a measure of the relationship between these two variables, but it has many of the same shortcomings as the variance (see [[Descriptive]]Statistics). Therefore we want a standardised measure (to ensure that <math>-1\leq r \geq 1</math>. This standardisation uses the square root of the two respective variances.
 In what we just did we expanded with the term <math>(1/n)</math> and what we got were the correlation calculated as a function of the ''population'' covariance, <math>\sigma_{y,x}</math>, and the ''population'' variances, <math>\sigma_y^2</math> and <math>\sigma_x^2</math>. If we had expanded with the factor <math>(1/(n-1))</math> we would have obtained an expression that relates the correlation to the ''sample'' covariance, <math>s_{y,x}</math> and the sample variances <math>s_y^2</math> and <math>s_x^2</math> as follows:

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 == Additional resources ==
-Khan Academy:
+* Khan Academy: Do not confuse correlation with causation: [https://www.khanacademy.org/math/probability/regression/regression-correlation/v/correlation-and-causality]
+* Ralf Becker: Calculation Example for covariance and correlation: [http://www.youtube.com/watch?v=cw5eTSi7xpU&feature=share&list=PLW7MJJThJQQs3djo1EL6KCRFeCa6wpfYY]
 = Regression =
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 A more important matter arises if we find that a scatter of the some variable <math>y</math> against another, <math>x,</math> does not appear to reveal a linear relationship. In such cases, linearity may be retrieved if <math>y</math> is plotted against some function of <math>x</math> (e.g., <math>\log (x)</math> or <math>x^{2},</math> say). Indeed, there may be cases when <math>Y</math> also needs to be transformed in some way. That is to say, transformations of the data (via some mathematical function) may render a non-linear relationship more linear.
 =Footnotes=
   <references />