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Correlation

The document explains correlation as the relationship between two random variables, highlighting the calculation of the correlation coefficient and its interpretation through scatter diagrams. It details the formulas for variance, co-variance, and the product moment correlation coefficient, r, which ranges from -1 to +1, indicating the strength and direction of the correlation. Additionally, it introduces Spearman’s rank coefficient for assessing correlation based on rankings of values.

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KIN WEI NG
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6 views18 pages

Correlation

The document explains correlation as the relationship between two random variables, highlighting the calculation of the correlation coefficient and its interpretation through scatter diagrams. It details the formulas for variance, co-variance, and the product moment correlation coefficient, r, which ranges from -1 to +1, indicating the strength and direction of the correlation. Additionally, it introduces Spearman’s rank coefficient for assessing correlation based on rankings of values.

Uploaded by

KIN WEI NG
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Correlation

Topic Outcomes
❑ Explain the term “correlation” in relation to data where both
variables must be random.

❑ Calculate correlation coefficient

❑ Evaluate correlation coefficient results in relation to appearance of


the scatter diagrams and with reference to values close to -1, 0,
+1.
Correlation
• Correlation indicates the direction and strength of a relationship between 2 variables.
• The relationship between two variables can be easily seen by plotting a scatter
diagram.
• Bivariate data is data that comes in pairs and there may or may not be a relationship
between them.
Variability of bivariate data

σ(𝑥−𝑥)2
Variance =
𝑛

In correlation : 𝑆𝑥𝑥 = σ(𝑥 − 𝑥)2 and 𝑆𝑦𝑦 = σ(𝑦 − 𝑦)2

σ(𝑥−𝑥)(𝑦−𝑦)
Co-variance =
𝑛

In correlation: 𝑆𝑥𝑦 = σ(𝑥 − 𝑥)(𝑦 − 𝑦)


Variability of bivariate data

σ 𝑥2 2
Variance = −𝑥
𝑛

2 (σ 𝑥)2 2 (σ 𝑦)2
In correlation : 𝑆𝑥𝑥 = σ 𝑥 − and 𝑆𝑦𝑦 = σ 𝑦 −
𝑛 𝑛

σ𝑥σ𝑦
In correlation: 𝑆𝑥𝑦 = σ 𝑥𝑦 −
𝑛
Correlation coefficient r
• You can get a measure of the amount of correlation between two variables by
using the product moment correlation coefficient r.
• This is defined as;
𝑺𝒙𝒚
𝒓=
𝑺𝒙𝒙 𝑺𝒚𝒚
• The value of r varies between -1 and +1.
Correlation coefficient r
If r = 1 there is a perfect positive linear correlation.

If r = -1 there is a perfect negative linear correlation.

If r is zero, there is no linear correlation.


Example
Spearman’s rank coefficient of correlation
• It is called rank correlation because it is based on rankings of the values of
x and y
• Spearman’s measurement is the most common

𝟐
𝟔 ∙ σ 𝒓𝒙 − 𝒓𝒚
𝒓′ =𝟏−
𝒏 ∙ 𝒏𝟐 − 𝟏

• The rankings are usually in ascending order


• But can be descending
• As long as there is consistency in the method of ranking for both x and y, it doesn’t matter
Computation Steps
Example
• The data in the table below shows annual rents and rate bills.
Calculate Spearman’s Rank Correlation Coefficient to assess whether
there is any correlation between rent and rate bills.

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