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ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation PDF Download

SUMMARY

The change in one variable is reciprocated by a corresponding change in the other variable either directly or inversely, then the two variables are known to be associated or correlated.
There are two types of correlation.
(i) Positive correlation
(ii) Negative correlation
We consider the following measures of correlation:
(a) Scatter diagram: This is a simple diagrammatic method to establish correlation between a pair of variables.
(b) Karl Pearson’s Product moment correlation coefficient:
ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation
A single formula for computing correlation coefficient is given by
ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation
(i) The Coefficient of Correlation is a unit-free measure.
(ii) The coefficient of correlation remains invariant under a change of origin and/or scale of the variables under consideration depending on the sign of scale factors.
(iii) The coefficient of correlation always lies between –1 and 1, including both the limiting values i.e. –1 ≤ r ≤ + 1
(c) Spearman’s rank correlation co-efficient: Spearman’s rank correlation coefficient is given by
ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation where rR denotes rank correlation coefficient and it lies between – 1 and 1 inclusive of these two values.di = xi – yrepresents the difference in ranks for the i-th individual and n denotes the number of individuals.

In case u individuals receive the same rank, we describe it as a tied rank of length u.

In case of a tied rank,
ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation
In this formula, tj represents the jth tie length and the summation extends over the lengths of all the ties for both the series.
(d) Co-efficient of concurrent deviations: The coefficient of concurrent deviation is given by
ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation
If (2c–m) >0, then we take the positive sign both inside and outside the radical sign and if (2c–m) <0, we are to consider the negative sign both inside and outside the radical sign.

  • In regression analysis, we are concerned with the estimation of one variable for given value of another variable (or for a given set of values of a number of variables) on the basis of an average mathematical relationship between the two variables (or a number of variables).
  • In case of a simple regression model if y depends on x, then the regression line of y on x in given by y = a + b, here a and b are two constants and they are also known as regression parameters. Furthermore, b is also known as the regression coefficient of y on x and is also denoted by byx
  • The method of least squares is solving the equations of regression lines
    The normal equations are

ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation

  • The regression coefficients remain unchanged due to a shift of origin but change due to a shift of scale. This property states that if the original pair of variables is (x, y) and if they are changed to the pair (u, v) where ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation
  • The two lines of regression intersect at the point , where x and y are the variables under consideration.
    According to this property, the point of intersection of the regression line of y on x and the regression line of x on y is  i.e. the solution of the simultaneous equations in x and y.
  • The coefficient of correlation between two variables x and y in the simple geometric mean of the two regression coefficients. The sign of the correlation coefficient would be the common sign of the two regression coefficients.
    ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation
  • Correlation coefficient measuring a linear relationship between the two variables indicates the amount of variation of one variable accounted for by the other variable. A better measure for this purpose is provided by the square of the correlation coefficient, Known as ‘coefficient of determination’. This can be interpreted as the ratio between the explained variance to total variance i.e.
    ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation
  • The ‘coefficient of non-determination’ is given by (1–r2) and can be interpreted as the ratio of unexplained variance to the total variance.
  • The two lines of regression coincide i.e. become identical when r = –1 or 1 or in other words, there is a perfect negative or positive correlation between the two variables under discussion. If r = 0 Regression lines are perpendicular to each other.
The document ICAI Notes: Correlation And Regression- 2 | Quantitative Aptitude for CA Foundation is a part of the CA Foundation Course Quantitative Aptitude for CA Foundation.
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FAQs on ICAI Notes: Correlation And Regression- 2 - Quantitative Aptitude for CA Foundation

1. What is correlation and regression?
Ans. Correlation and regression are statistical techniques used to measure and analyze the relationship between two or more variables. Correlation determines the strength and direction of the relationship, while regression helps in predicting the value of one variable based on the values of other variables.
2. How is correlation coefficient calculated?
Ans. The correlation coefficient is a statistical measure that ranges from -1 to +1 and indicates the strength and direction of the relationship between two variables. It is calculated by dividing the covariance of the variables by the product of their standard deviations.
3. What is the difference between positive and negative correlation?
Ans. In positive correlation, two variables increase or decrease together. This means that as one variable increases, the other variable also tends to increase. In negative correlation, one variable increases while the other variable decreases. This indicates an inverse relationship between the two variables.
4. How is regression used in forecasting?
Ans. Regression analysis is commonly used in forecasting to predict the value of a dependent variable based on the values of one or more independent variables. By analyzing the historical relationship between the variables, regression helps in estimating future values and making predictions.
5. Can correlation imply causation?
Ans. No, correlation does not imply causation. Just because two variables are correlated does not mean that one variable causes the other to change. Correlation only indicates a relationship, and further research is required to establish a cause-and-effect relationship between variables.
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