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Correlation Class 11 Economics

Introduction

Correlation is a method used to establish the relationship between two variables. The correlation coefficient, which is a single number ranging from -1 to +1, summarizes the strength and direction of the relationship between the variables.
A correlation coefficient close to zero, either positive or negative, suggests a weak or no relationship between the variables. On the other hand, a correlation coefficient close to +1 indicates a positive relationship, where increases in one variable correspond to increases in the other variable.
A correlation coefficient close to -1 suggests a negative relationship, where increases in one variable correspond to decreases in the other variable. The correlation coefficient can be calculated for variables measured at the ordinal, interval, or ratio level, but it has little significance for variables measured nominally.

Question for Chapter Notes - Correlation
Try yourself:What is the purpose of a correlation coefficient?
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Question for Chapter Notes - Correlation
Try yourself:What does a correlation coefficient close to 0 indicate?
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Type of Correlation

  • Negative Correlation: Negative correlation occurs when two variables move in opposite directions, meaning that an increase in one variable is accompanied by a decrease in the other variable, and vice versa.
  • Positive Correlation: Positive correlation is observed when two variables move in the same direction, meaning that an increase in one variable is accompanied by an increase in the other variable, and vice versa.

Examples of positive correlation are:

  • Price and supply of a commodity.
  • Increase in Height and Weight.
  • Age of husband and age of wife.
  • The family income and expenditure on luxury items.

Examples of negative correlation are:

  • Sale of woollen garments and day temperature.
  • Price and Demand of a commodity.
  • Yield of crops and price.

Methods of estimating correlation:

  • Scatter diagram
  • Karl person’s coefficient of correlation.
  • Spearman’s rank correlation.

A scatter plot provides a visual representation of the strength and direction of correlation between two variables. It involves plotting the x variable on the X-axis and the y variable on the Y-axis, resulting in a cluster of points called a scatter plot. The proximity and overall direction of the scatter points help us to analyze the relationship between the two variables.

Question for Chapter Notes - Correlation
Try yourself:What do the scatter points on a scatter diagram represent?
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Karl person’s coefficient of correlation is a quantitative method of calculating correlation. It gives a precise numerical value of the degree of linear relationship between two variables.
Karl person’s coefficient of correlation is also known as product moment correlation.
Formula:
Correlation Class 11 Economics

Here,
r = Coefficient of correlation
Correlation Class 11 Economics
σx = Standard deviation of X-series.
σy = Standard deviation of Y-series.
N = Number of observations
Correlation Class 11 EconomicsCorrelation Class 11 Economics

  • Karl Person’s coefficient of correlation is calculated by following methods:
    Actual mean method:
    Correlation Class 11 Economics
    Here,
    r = Coeff. Of correlation
    Correlation Class 11 Economics
  • Assumed Mean method:
    Correlation Class 11 Economics
    Correlation Class 11 Economics
    Here,
    dx = Deviations of x-series from assumed mean = (X – A)
    dy = Deviation of Y-series from assumed mean = (Y – A)
    ∑dxdy = Sum of multiple of dx and dy.
    ∑dx2 = Sum of the square of dx.
    ∑dy2 = Sum of the square of dy
    ∑dx = Sum of the deviation of x-series
    ∑dy = Sum of the deviation of Y-series
    N = Number of pairs of observations
    When value of the variables are large, we use step deviation method to reduce the burden of calculation.
  • Step deviation method:
    Correlation Class 11 Economics
    Here, Correlation Class 11 Economics
    Correlation Class 11 Economicsdx = deviation of X-series from assumed mean = (X-A)
    dy = deviation of Y-series from assumed mean = (Y-A)
    ∑dxdy = Sum of multiple of dx and dy.
    ∑dx2 = Sum of the square of dx.
    ∑dy2 = Sum of the square of dy
    ∑dx = Sum of the deviation of x-series
    ∑dy = Sum of the deviation of Y-series
    N = Number of pairs of observations
    C1 is common factor for series -x
    C2 is common factor for series -y

Spearman’s Rank Correlation 

Spearman's rank correlation coefficient is a useful tool for determining the correlation between variables that may not have a clear or objective measurement. It is particularly helpful when analyzing the correlation of qualitative variables.
Correlation Class 11 Economics
Here,
rs = coefficient of rank correlation
D = Rank differences
N = Number of rank
When ranks are repeated the formula is:
Correlation Class 11 Economics
Where m1, m2, .............. are number of repetitions of ranks.

The document Correlation Class 11 Economics is a part of the Commerce Course Economics Class 11.
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FAQs on Correlation Class 11 Economics

1. What is Spearman's rank correlation?
Ans. Spearman's rank correlation is a statistical measure used to determine the strength and direction of the relationship between two variables. It assesses the monotonic relationship, which means it measures the consistency of the relationship rather than the exact linearity. It is often used when the data does not follow a normal distribution or when the relationship between variables is not linear.
2. How is Spearman's rank correlation calculated?
Ans. Spearman's rank correlation is calculated by assigning ranks to the values of each variable, converting the values into their respective ranks. Then, the difference between the ranks for each pair of observations is calculated. The formula for Spearman's rank correlation coefficient is: ρ = 1 - (6 * Σd²) / (n * (n² - 1)) Where: ρ is the Spearman's rank correlation coefficient, Σd² is the sum of the squared differences between the ranks, and n is the number of observations. The resulting coefficient ranges from -1 to +1, where -1 indicates a perfect negative correlation, +1 indicates a perfect positive correlation, and 0 indicates no correlation.
3. When should Spearman's rank correlation be used instead of Pearson's correlation?
Ans. Spearman's rank correlation should be used instead of Pearson's correlation when the relationship between variables is not linear or when the data does not follow a normal distribution. Pearson's correlation assumes a linear relationship and requires the data to be normally distributed. Spearman's rank correlation, on the other hand, assesses the monotonic relationship, which includes both linear and nonlinear relationships, and is robust to non-normality.
4. What is the interpretation of Spearman's rank correlation coefficient?
Ans. The interpretation of Spearman's rank correlation coefficient is as follows: - A coefficient of +1 indicates a perfect positive monotonic relationship, where the ranks of both variables increase together. - A coefficient of -1 indicates a perfect negative monotonic relationship, where the ranks of one variable increase as the ranks of the other variable decrease. - A coefficient of 0 indicates no monotonic relationship between the variables. The magnitude of the coefficient indicates the strength of the relationship, with values closer to +1 or -1 indicating a stronger relationship.
5. Can Spearman's rank correlation coefficient be used for categorical variables?
Ans. Yes, Spearman's rank correlation coefficient can be used for categorical variables. In such cases, the categorical variables are assigned ranks based on their order or importance. The ranks are then used to calculate the correlation coefficient. However, it is important to note that the interpretation of the coefficient may differ for categorical variables compared to numerical variables.
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