Correlation in statistics
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The document discusses different types of correlation that can exist between two variables: positive correlation, negative correlation, and zero correlation. It provides examples like income and expenditure having positive correlation, while price and demand have negative correlation. The document also describes methods for measuring correlation, including scatter diagrams and the coefficient of correlation (ranging from -1 to 1), with formulas provided. Examples are given to demonstrate calculating the coefficient and interpreting the strength and type of correlation.
![Example-3:
An economist gives the following estimates:
Price 1 2 3 4 5
Demand 9 7 6 3 1
Calculate Karl Pearson’s coefficient of correlation and make
Comments about the type of correlation exist.
Solution:
X
Price
(Rs)
Y
Demand
(Kg)
XY 𝑿 𝟐
𝒀 𝟐
1 9 9 1 81
2 7 14 4 49
3 6 18 9 36
4 3 12 16 9
5 1 5 25 1
∑𝑿
= 𝟏𝟓
∑𝒀
= 𝟐𝟔
∑𝑿𝒚
= 𝟓𝟖
∑𝑿 𝟐
=
55
∑𝒀 𝟐
=17
6
Formula of Karl Pearson coefficient of correlation:
r =
𝑛∑𝑥𝑦−∑𝑥∑𝑦
√( 𝑛∑𝑥2−(∑𝑥)2)( 𝑛∑𝑦2 −(∑𝑦)2)
r =
5(58)−(15)(26)
√[5(55)−(15)2][5(176)−(26)2]
r =
290−390
√(275−225)(880−676)
r =
−100
√(50)(204)](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/correlationinstatistics-200227041515/85/Correlation-in-statistics-7-320.jpg)
Correlation in statistics
- 2. Correlation:- Methods of measuring the degree of relationship existing between two variables have been developed by Galton and Karl Pearson. Correlation is defined as “The Relationship which exists between two variables. In other words, when two variables depend on each other in such a way the movements (Increase OR Decrease) are accompanied by movements (Increase OR Decrease) in the other , the variables are said to be correlated For example ; Increase in height of children is accompanied by increase in weight. Some more examples of correlation are defined as : 1. Income and Expenditure 2. Price and Demand Positive Correlation: If both the variables are moving in same direction (Increase OR Decrease) then correlation is said to be direct or Positive, the for example , Income and Expenditure related with positive direction. Negative Correlation: When the movements of the two variables are in opposite direction then the correlation is said to be negative, for example , Price and Demand, if price increases the demand decreases. Zero Correlation: If there is no association between the two variables the correlation is said to be zero correlation (Both the variables are independent). The correlation may be studied by the following two methods. 1. Scatter Diagram 2. Coefficient of Correlation
- 3. Scatter Diagram: The simplest method of investigating the relationship between two variables is to plot a scatter diagram Let there will be two series ‘x’ (Independent Variable) and ‘y’ (Dependent Variable) to be represented graphically. Take the items in ‘x’ series along the axis of ‘x’ and the corresponding items in ‘y’ series along the y-axis. The diagram so formed will be a dotted one and scattered, showing some relationship, such a diagram is called a scattered. Linear Correlation:- If all the points on the scatter diagram tend to lie near a line, the correlation is said to be linear. Positive Linear Correlation:- If all the points tend to lie near an upward sloping line , the linear correlation is said to be positive correlation.
- 4. Negative Linear Correlation:- If all the points tend to lie on a downward sloping line, the linear correlation is said to be negative correlation. Perfect Positive Correlation:- If all the points tend to lie on a rising line the correlation is perfectly positive correlation. Perfect Negative Correlation:- If all the points tend to lie on a falling line the correlation is perfectly negative correlation.
- 5. Null Correlation:- If the points on a scatter diagram do not show a definite movement then there is no correlation between the variables. Coefficient of Correlation:- Numerical Measure of correlation is called coefficient of correlation. It measured the degree of relationship between the variables. The formula is called Karl Pearson’s coefficient of correlation. There are different formulae for the calculation of Karl Pearson coefficient of correlation. 1. r = ∑( 𝑋−𝑋̅)( 𝑌−𝑌̅) √∑( 𝑋−𝑋̅)2∑( 𝑌−𝑌̅)2 2. r = 𝑛∑𝑥𝑦−∑𝑥∑𝑦 √( 𝑛∑𝑥2−(∑𝑥)2)( 𝑛∑𝑦2 −(∑𝑦)2) 3. r = 𝐶𝑜𝑣.(𝑥,𝑦) √𝑉𝑎𝑟 ( 𝑥),𝑉𝑎𝑟( 𝑦) 4. r = 𝑆 𝑥𝑦 √𝑆𝑥2 𝑆𝑦2 5. r = 𝑠 𝑥𝑦 𝑆 𝑥 𝑆 𝑦
- 6. The Limit of correlation is to be from negative one to positive one −1 ≤ 𝑟 ≤ +1 Interpretation of coefficient of correlation:- If r =1 the correlation is said to be perfect positive correlation. If r = -1 the correlation is said to be perfect negative correlation. Coefficient of Correlation Strength 0.90---------0.99 Very strong 0.78---------0.89 Strong 0.64---------0.77 Moderate 0.46---------0.63 Low 0.10---------0.45 Very Low 0.00---------0.09 No
- 7. Example-3: An economist gives the following estimates: Price 1 2 3 4 5 Demand 9 7 6 3 1 Calculate Karl Pearson’s coefficient of correlation and make Comments about the type of correlation exist. Solution: X Price (Rs) Y Demand (Kg) XY 𝑿 𝟐 𝒀 𝟐 1 9 9 1 81 2 7 14 4 49 3 6 18 9 36 4 3 12 16 9 5 1 5 25 1 ∑𝑿 = 𝟏𝟓 ∑𝒀 = 𝟐𝟔 ∑𝑿𝒚 = 𝟓𝟖 ∑𝑿 𝟐 = 55 ∑𝒀 𝟐 =17 6 Formula of Karl Pearson coefficient of correlation: r = 𝑛∑𝑥𝑦−∑𝑥∑𝑦 √( 𝑛∑𝑥2−(∑𝑥)2)( 𝑛∑𝑦2 −(∑𝑦)2) r = 5(58)−(15)(26) √[5(55)−(15)2][5(176)−(26)2] r = 290−390 √(275−225)(880−676) r = −100 √(50)(204)
- 8. r = −100 √10200 r = −100 100⋅995 r = - 0.99 Comments : The Correlation between two variables is very high strong negative correlation. Example-4: Given: ∑( 𝑋 − 𝑋̅)( 𝑌 − 𝑌̅) = 100 ∑( 𝑋 − 𝑋̅)2 = 132 ∑( 𝒀 − 𝒀̅) 𝟐 = 202 Calculate r = ? Solution: r = ∑( 𝑿−𝑿̅)( 𝒀−𝒀̅) √∑(𝑿−𝑿̅) 𝟐∑( 𝒀−𝒀̅) 𝟐 = 𝟏𝟎𝟎 √( 𝟏𝟑𝟐)( 𝟐𝟎𝟐) = 𝟏𝟎𝟎 √ 𝟐𝟔,𝟔𝟔𝟒 = 𝟏𝟎𝟎 𝟏𝟔𝟑⋅𝟐 r = 0.61
- 9. Comments: Therefore, ‘r’ indicates that the coefficient of correlation between two variables is low positive correlation. Example-5: A sample eight paired values give the following results. Cov(x,y) = 0.5 V(x) = 15.5 V(y) = 3.8 Calculate coefficient of correlation ‘r’ Solution: 𝑆 𝑥𝑦= 0.5 , 𝑆 𝑥 = 3.94 , 𝑆 𝑦 = 1.95 r = 𝑠 𝑥𝑦 𝑆 𝑥 𝑆 𝑦 r = 0⋅5 3⋅94× 1⋅95 r = 0⋅5 7⋅683 r = 0 ⋅ 065 Comments: No Correlation exist in betweeen two variables.