Understanding Lines of Regression and Coefficient of Correlation

Understanding Lines of Regression and Coefficient of Correlation

Regression and correlation are two fundamental statistical concepts used in data analysis.

• Regression helps us find a relationship between dependent and independent variables. It provides an equation to predict values.
• Correlation measures the strength and direction of the relationship between two variables.

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1. Line of Regression

A regression line is the best-fit line that represents the relationship between two variables, usually denoted as:

$$y = a + bx$$

Where:

• $y$
  is the dependent variable (predicted value).
• $x$
  is the independent variable.
• $a$
  is the intercept (value of 
  $y$
  when 
  $x = 0$
  ).
• $b$
  is the slope (rate of change of 
  $y$
  with respect to 
  $x$
  ).

The slope (

$b$

) is given by:

$$b = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$$

The intercept (

$a$

) is given by:

$$a = \frac{\sum y - b \sum x}{n}$$

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2. Coefficient of Correlation ($r$)

The Pearson correlation coefficient (

$r$

) measures how strong the relationship is between two variables.

$$r = \frac{n \sum xy - \sum x \sum y}{\sqrt{[n \sum x^2 - (\sum x)^2] [n \sum y^2 - (\sum y)^2]}}$$

• $r$
  varies between -1 and 1:
  • $r = 1$
    : Perfect positive correlation.
  • $r = -1$
    : Perfect negative correlation.
  • $r = 0$
    : No correlation.

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3. Sample Data & Computation

Dataset 1

$x$	$y$
1	2
2	3
3	5
4	4
5	6

Step 1: Compute Needed Values

$x$	$y$	$x^2$	$y^2$	$xy$
1	2	1	4	2
2	3	4	9	6
3	5	9	25	15
4	4	16	16	16
5	6	25	36	30

$$\sum x = 15, \quad \sum y = 20, \quad \sum x^2 = 55, \quad \sum y^2 = 90, \quad \sum xy = 69$$

Step 2: Compute Regression Line

Using formulas:

$$b = \frac{5(69) - (15)(20)}{5(55) - (15)^2} = \frac{345 - 300}{275 - 225} = \frac{45}{50} = 0.9$$ $$a = \frac{20 - 0.9(15)}{5} = \frac{20 - 13.5}{5} = \frac{6.5}{5} = 1.3$$

So, the regression equation is:

$$y = 1.3 + 0.9x$$

Step 3: Compute Correlation Coefficient

$$r = \frac{5(69) - (15)(20)}{\sqrt{[5(55) - (15)^2] [5(90) - (20)^2]}}$$ $$= \frac{345 - 300}{\sqrt{(275 - 225)(450 - 400)}}$$ $$= \frac{45}{\sqrt{50 \times 50}} = \frac{45}{50} = 0.9$$

Since 

$r = 0.9$

, there is a strong positive correlation.

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4. Second Sample Dataset

$x$	$y$
10	40
20	30
30	20
40	10
50	5

Step 1: Compute Needed Values

$x$	$y$	$x^2$	$y^2$	$xy$
10	40	100	1600	400
20	30	400	900	600
30	20	900	400	600
40	10	1600	100	400
50	5	2500	25	250

$$\sum x = 150, \quad \sum y = 105, \quad \sum x^2 = 5500, \quad \sum y^2 = 3025, \quad \sum xy = 2250$$

Step 2: Compute Regression Line

$$b = \frac{5(2250) - (150)(105)}{5(5500) - (150)^2} = \frac{11250 - 15750}{27500 - 22500} = \frac{-4500}{5000} = -0.9$$ $$a = \frac{105 - (-0.9)(150)}{5} = \frac{105 + 135}{5} = \frac{240}{5} = 48$$

So, the regression equation is:

$$y = 48 - 0.9x$$

Step 3: Compute Correlation Coefficient

$$r = \frac{5(2250) - (150)(105)}{\sqrt{[5(5500) - (150)^2] [5(3025) - (105)^2]}}$$ $$= \frac{11250 - 15750}{\sqrt{(27500 - 22500)(15125 - 11025)}}$$ $$= \frac{-4500}{\sqrt{5000 \times 4100}}$$ $$= \frac{-4500}{\sqrt{20500000}} = \frac{-4500}{4527} \approx -0.995$$

Since 

$r \approx -0.995$

, this indicates a strong negative correlation.

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5. Summary of Results

Dataset	Regression Equation	Correlation Coefficient ($r$)
1	$$y=1.3+0.9xy = 1.3 + 0.9x	$0.9$ (Strong positive)
2	$$y=48−0.9xy = 48 - 0.9x	$-0.995$ (Strong negative)

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6. Conclusion

• A positive correlation (
  $r > 0$
  ) means 
  $y$
  increases as 
  $x$
  increases.
• A negative correlation (
  $r < 0$
  ) means 
  $y$
  decreases as 
  $x$
  increases.
• The regression equation helps predict values based on given input data.

This explanation provides a clear foundation for writing a program to compute regression and correlation. You can implement this in Python, Java, or any language by following the formulas step-by-step. 🚀

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