Correlation and Regression Class 12 Mathematics Solutions | Exercise - 10.1
Correlation and Regression Class 12 Mathematics Solutions according to the updated syllabus of 2080. You'll get the complete solution of exercise 10.1 in this article. You can also download the PDF of the solutions if you want to view them offline.
Chapter - 10
Correlation and Regression
In correlation and regression chapter we are going to learn about many topics such as what is regression coefficient, nature of correlation, correlation coefficient by Karl Pearson's method, interpretation of correlation coefficient and many other topics related to correlation.
Many people were demanding the notes of correlation and regression as this topic is new for students class 12 students. So we are here with the complete solutions of correlation and regression.
Correlation
In this chapter, we discuss various methods to determine if there exist any relationship between two variables. As for example; the amount of rainfall and the volume of production of certain commodity, age and the blood pressure, etc.
Two variables are said to have "correlation," when they are so related that the change in the value of one variable is accompanied by the change in the value of the other. For example
- the amount of rainfall to some extent is accompanied by an increase in the volume of production
- the decrease in the price of a commodity is accompanied by the increase in the quantity demanded
- increase in advertisement expenditure is accompanied by increase in sales.
The measure of correlation called the 'correlation coefficient summarizes in one figure, the degree and direction of movement. But the important thing that is to be noted here is that, correlation analysis only helps in determining the extent to which the two variables are correlated but it does not tell us about cause and effect relationship.
Exercise - 10.1
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Nature of Correlation
Correlation may be of the following three types:
- Positive and negative
- Linear and non-linear
- Simple, multiple and partial
Positive and negative correlation
If two variables vary in the same direction ie increase (or decrease) in the value of one variable results increase (or decrease) in the value of other variable, then the two variables are said to have positive correlation.
On the other hand, two variables are said to have negative correlation if two variables move in the opposite direction i.e. if one variable increases (or decreases) the second decreases (or increases).
Linear and non-liner correlation
The correlation between two variables is said to be linear when a unit change in one variable results a constant change in the other variable over the entire range of the values.
If corresponding to a unit change in one variable, there is no constant change in other variable, then the correlation is said to be non linear.
Methods of studying correlation
Of the various methods, the rough idea of relation between two variables can be obtained by graphical method known as scatter diagram.
Scatter diagram
It is a graphical method of studying correlation. The simplest method of ascertaining the correlation between two variables is the scatter diagram. For this let X and Y be two variables. each consisting the same number of values.
Points are plotted with the values of X as the x- coordinates and the corresponding values of Y as y-coordinates. The points are represented by dots. The diagram consisting of set of dots thus formed is said to be the scatter diagram.
On seeing the scatterness of the dots, an idea about the degree and the direction of correlation between two variables can be made. More the closeness of the dots to a straight line, higher will be the correlation between two variables. Greater the scatterness, less will be the correlation.
Karl Pearson's correlation coefficient
One of the widely used mathematical methods of calculating the correlation coefficient between two variables is Karl Pearson's correlation coefficient. It is also known as Pearsonian correlation coefficient.
Interpretation of correlation coefficient (r)
From property no.1 of correlation coefficient, its value lies between -1 and +1. After getting the value of r, care should be taken to interpret, otherwise wrong conclusion may be obtained. However the following general rules are mentioned for interpreting the value of r.
- When r = 1 , there is a positively perfect correlation between the two variables.
- When r = - 1 there is negatively perfect correlation between the two variables.
- When r = 0 the variables are uncorrelated.
- Nearer the value of r to +1, closer will be the relationship between two variables and nearer the value of r to 0, lesser will be the relationship.
Properties of correlation coefficient
Important properties of the correlation coefficient are given below:
- Correlation coefficient between two variables is independent of change of origin and scale.
- The correlation coefficient between two variables lies in between -1 and +1. Symbolically, - 1 ≤ r ≤ 1
- The formula for the correlation coefficient between the two variables x and y is symmetrical i.e. rₓᵧ = rᵧₓ
Merits and limitations of Karl Pearson's Correlation coefficient
Merits:
- Karl Pearson's method of finding correlation coefficient is based on all the observations.
- This method summaries in one figure the degree of relationship as well as direction.
Limitations:
- This method always assumes linear relationship between two variables whether this assumption is true or not.
- It is affected by extreme values.
- In comparison to other methods, it is most time consuming
- Interpretation of the value of r is not an easy attempt.
How do you calculate the coefficient of correlation for a given set of data?
To calculate the coefficient of correlation for a given set of data, you can use the following formula: r = [∑(xy) - (n∑x∑y)] / [√{∑x^2 - (n∑x)^2} * √{∑y^2 - (n∑y)^2}]
What is the significance of the coefficient of determination in regression analysis?
The coefficient of determination, denoted as R squared, is a statistical measure that represents the proportion of the variance in the dependent variable that is explained by the independent variable(s) in a regression model. In other words, it indicates how well the regression line fits the observed data points.
What are the assumptions made in linear regression analysis?
In linear regression analysis, there are several assumptions that need to be met for the results to be reliable. These assumptions include: Linearity: The relationship between the independent variable and dependent variable is linear. Independence: The observations are independent of each other. Homoscedasticity: The variance of the errors is constant across all levels of the independent variable. Normality: The errors are normally distributed.
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