Page 1 As the summer heat rises, hill stations, are crowded with more and more visitors. Ice-cream sales become more brisk. Thus, the temperature is related to number of visitors and sale of ice-creams. Similarly, as the supply of tomatoes increases in your local mandi, its price drops. When the local harvest starts reaching the market, the price of tomatoes drops from a princely Rs 40 per kg to Rs 4 per kg or even less. Thus supply is related to price. Correlation analysis is a means for examining such relationships systematically. It deals with questions such as: â€¢ Is there any relationship between two variables? Correlation 7 1. INTRODUCTION In previous chapters you have learnt how to construct summary measures out of a mass of data and changes among similar variables. Now you will learn how to examine the relationship between two variables. Studying this chapter should enable you to: â€¢ understand the meaning of the term correlation; ? understand the nature of relationship between two variables; ? calculate the different measures of correlation; ? analyse the degree and direction of the relationships. CHAPTER Page 2 As the summer heat rises, hill stations, are crowded with more and more visitors. Ice-cream sales become more brisk. Thus, the temperature is related to number of visitors and sale of ice-creams. Similarly, as the supply of tomatoes increases in your local mandi, its price drops. When the local harvest starts reaching the market, the price of tomatoes drops from a princely Rs 40 per kg to Rs 4 per kg or even less. Thus supply is related to price. Correlation analysis is a means for examining such relationships systematically. It deals with questions such as: â€¢ Is there any relationship between two variables? Correlation 7 1. INTRODUCTION In previous chapters you have learnt how to construct summary measures out of a mass of data and changes among similar variables. Now you will learn how to examine the relationship between two variables. Studying this chapter should enable you to: â€¢ understand the meaning of the term correlation; ? understand the nature of relationship between two variables; ? calculate the different measures of correlation; ? analyse the degree and direction of the relationships. CHAPTER 92 STATISTICS FOR ECONOMICS ? If the value of one variable changes, does the value of the other also change? â€¢ Do both the variables move in the same direction? ? How strong is the relationship? 2. TYPES OF RELATIONSHIP Let us look at various types of relationship. The relation between movements in quantity demanded and the price of a commodity is an integral part of the theory of demand, which you will read in class XII. Low rainfall is related to low agricultural productivity. Such examples of relationship may be given a cause and effect interpretation. Others may be just coincidence. The relation between the arrival of migratory birds in a sanctuary and the birth rates in the locality can not be given any cause and effect interpretation. The relationships are simple coincidence. The relationship between size of the shoes and money in your pocket is another such example. Even if relationship exist, they are difficult to explain it. In another instance a third variableâ€™s impact on two variables may give rise to a relation between the two variables. Brisk sale of ice-creams may be related to higher number of deaths due to drowning. The victims are not drowned due to eating of ice-creams. Rising temperature leads to brisk sale of ice-creams. Moreover, large number of people start going to swimming pools to beat the heat. This might have raised the number of deaths by drowning. Thus temperature is behind the high correlation between the sale of ice-creams and deaths due to drowning. What Does Correlation Measure? Correlation studies and measures the direction and intensity of relationship among variables. Correlation measures covariation, not causation. Correlation should never be Page 3 As the summer heat rises, hill stations, are crowded with more and more visitors. Ice-cream sales become more brisk. Thus, the temperature is related to number of visitors and sale of ice-creams. Similarly, as the supply of tomatoes increases in your local mandi, its price drops. When the local harvest starts reaching the market, the price of tomatoes drops from a princely Rs 40 per kg to Rs 4 per kg or even less. Thus supply is related to price. Correlation analysis is a means for examining such relationships systematically. It deals with questions such as: â€¢ Is there any relationship between two variables? Correlation 7 1. INTRODUCTION In previous chapters you have learnt how to construct summary measures out of a mass of data and changes among similar variables. Now you will learn how to examine the relationship between two variables. Studying this chapter should enable you to: â€¢ understand the meaning of the term correlation; ? understand the nature of relationship between two variables; ? calculate the different measures of correlation; ? analyse the degree and direction of the relationships. CHAPTER 92 STATISTICS FOR ECONOMICS ? If the value of one variable changes, does the value of the other also change? â€¢ Do both the variables move in the same direction? ? How strong is the relationship? 2. TYPES OF RELATIONSHIP Let us look at various types of relationship. The relation between movements in quantity demanded and the price of a commodity is an integral part of the theory of demand, which you will read in class XII. Low rainfall is related to low agricultural productivity. Such examples of relationship may be given a cause and effect interpretation. Others may be just coincidence. The relation between the arrival of migratory birds in a sanctuary and the birth rates in the locality can not be given any cause and effect interpretation. The relationships are simple coincidence. The relationship between size of the shoes and money in your pocket is another such example. Even if relationship exist, they are difficult to explain it. In another instance a third variableâ€™s impact on two variables may give rise to a relation between the two variables. Brisk sale of ice-creams may be related to higher number of deaths due to drowning. The victims are not drowned due to eating of ice-creams. Rising temperature leads to brisk sale of ice-creams. Moreover, large number of people start going to swimming pools to beat the heat. This might have raised the number of deaths by drowning. Thus temperature is behind the high correlation between the sale of ice-creams and deaths due to drowning. What Does Correlation Measure? Correlation studies and measures the direction and intensity of relationship among variables. Correlation measures covariation, not causation. Correlation should never be CORRELATION 93 interpreted as implying cause and effect relation. The presence of correlation between two variables X and Y simply means that when the value of one variable is found to change in one direction, the value of the other variable is found to change either in the same direction (i.e. positive change) or in the opposite direction (i.e. negative change), but in a definite way. For simplicity we assume here that the correlation, if it exists, is linear, i.e. the relative movement of the two variables can be represented by drawing a straight line on graph paper. Types of Correlation Correlation is commonly classified into negative and positive correlation. The correlation is said to be positive when the variables move together in the same direction. When the income rises, consumption also rises. When income falls, consumption also falls. Sale of ice-cream and temperature move in the same direction. The correlation is negative when they move in opposite directions. When the price of apples falls its demand increases. When the prices rise its demand decreases. When you spend more time in studying, chances of your failing decline. When you spend less hours in study, chances of your failing increase. These are instances of negative correlation. The variables move in opposite direction. 3. TECHNIQUES FOR MEASURING CORRELATION Widely used techniques for the study of correlation are scatter diagrams, Karl Pearsonâ€™s coefficient of correlation and Spearmanâ€™s rank correlation. A scatter diagram visually presents the nature of association without giving any specific numerical value. A numerical measure of linear relationship between two variables is given by Karl Pearsonâ€™s coefficient of correlation. A relationship is said to be linear if it can be represented by a straight line. Another measure is Spearmanâ€™s coefficient of correlation, which measures the linear association between ranks assigned to indiviual items according to their attributes. Attributes are those variables which cannot be numerically measured such as intelligence of people, physical appearance, honesty etc. Scatter Diagram A scatter diagram is a useful technique for visually examining the form of relationship, without calculating any numerical value. In this technique, the values of the two variables are plotted as points on a graph paper. The cluster of points, so plotted, is referred to as a scatter diagram. From a scatter diagram, one can get a fairly good idea of the nature of relationship. In a scatter diagram the degree of closeness of the scatter points and their overall direction enable us to examine the relation- Page 4 As the summer heat rises, hill stations, are crowded with more and more visitors. Ice-cream sales become more brisk. Thus, the temperature is related to number of visitors and sale of ice-creams. Similarly, as the supply of tomatoes increases in your local mandi, its price drops. When the local harvest starts reaching the market, the price of tomatoes drops from a princely Rs 40 per kg to Rs 4 per kg or even less. Thus supply is related to price. Correlation analysis is a means for examining such relationships systematically. It deals with questions such as: â€¢ Is there any relationship between two variables? Correlation 7 1. INTRODUCTION In previous chapters you have learnt how to construct summary measures out of a mass of data and changes among similar variables. Now you will learn how to examine the relationship between two variables. Studying this chapter should enable you to: â€¢ understand the meaning of the term correlation; ? understand the nature of relationship between two variables; ? calculate the different measures of correlation; ? analyse the degree and direction of the relationships. CHAPTER 92 STATISTICS FOR ECONOMICS ? If the value of one variable changes, does the value of the other also change? â€¢ Do both the variables move in the same direction? ? How strong is the relationship? 2. TYPES OF RELATIONSHIP Let us look at various types of relationship. The relation between movements in quantity demanded and the price of a commodity is an integral part of the theory of demand, which you will read in class XII. Low rainfall is related to low agricultural productivity. Such examples of relationship may be given a cause and effect interpretation. Others may be just coincidence. The relation between the arrival of migratory birds in a sanctuary and the birth rates in the locality can not be given any cause and effect interpretation. The relationships are simple coincidence. The relationship between size of the shoes and money in your pocket is another such example. Even if relationship exist, they are difficult to explain it. In another instance a third variableâ€™s impact on two variables may give rise to a relation between the two variables. Brisk sale of ice-creams may be related to higher number of deaths due to drowning. The victims are not drowned due to eating of ice-creams. Rising temperature leads to brisk sale of ice-creams. Moreover, large number of people start going to swimming pools to beat the heat. This might have raised the number of deaths by drowning. Thus temperature is behind the high correlation between the sale of ice-creams and deaths due to drowning. What Does Correlation Measure? Correlation studies and measures the direction and intensity of relationship among variables. Correlation measures covariation, not causation. Correlation should never be CORRELATION 93 interpreted as implying cause and effect relation. The presence of correlation between two variables X and Y simply means that when the value of one variable is found to change in one direction, the value of the other variable is found to change either in the same direction (i.e. positive change) or in the opposite direction (i.e. negative change), but in a definite way. For simplicity we assume here that the correlation, if it exists, is linear, i.e. the relative movement of the two variables can be represented by drawing a straight line on graph paper. Types of Correlation Correlation is commonly classified into negative and positive correlation. The correlation is said to be positive when the variables move together in the same direction. When the income rises, consumption also rises. When income falls, consumption also falls. Sale of ice-cream and temperature move in the same direction. The correlation is negative when they move in opposite directions. When the price of apples falls its demand increases. When the prices rise its demand decreases. When you spend more time in studying, chances of your failing decline. When you spend less hours in study, chances of your failing increase. These are instances of negative correlation. The variables move in opposite direction. 3. TECHNIQUES FOR MEASURING CORRELATION Widely used techniques for the study of correlation are scatter diagrams, Karl Pearsonâ€™s coefficient of correlation and Spearmanâ€™s rank correlation. A scatter diagram visually presents the nature of association without giving any specific numerical value. A numerical measure of linear relationship between two variables is given by Karl Pearsonâ€™s coefficient of correlation. A relationship is said to be linear if it can be represented by a straight line. Another measure is Spearmanâ€™s coefficient of correlation, which measures the linear association between ranks assigned to indiviual items according to their attributes. Attributes are those variables which cannot be numerically measured such as intelligence of people, physical appearance, honesty etc. Scatter Diagram A scatter diagram is a useful technique for visually examining the form of relationship, without calculating any numerical value. In this technique, the values of the two variables are plotted as points on a graph paper. The cluster of points, so plotted, is referred to as a scatter diagram. From a scatter diagram, one can get a fairly good idea of the nature of relationship. In a scatter diagram the degree of closeness of the scatter points and their overall direction enable us to examine the relation- 94 STATISTICS FOR ECONOMICS ship. If all the points lie on a line, the correlation is perfect and is said to be unity. If the scatter points are widely dispersed around the line, the correlation is low. The correlation is said to be linear if the scatter points lie near a line or on a line. Scatter diagrams spanning over Fig. 7.1 to Fig. 7.5 give us an idea of the relationship between two variables. Fig. 7.1 shows a scatter around an upward rising line indicating the movement of the variables in the same direction. When X rises Y will also rise. This is positive correlation. In Fig. 7.2 the points are found to be scattered around a downward sloping line. This time the variables move in opposite directions. When X rises Y falls and vice versa. This is negative correlation. In Fig.7.3 there is no upward rising or downward sloping line around which the points are scattered. This is an example of no correlation. In Fig. 7.4 and Fig. 7.5 the points are no longer scattered around an upward rising or downward falling line. The points themselves are on the lines. This is referred to as perfect positive correlation and perfect negative correlation respectively. Activity ? Collect data on height, weight and marks scored by students in your class in any two subjects in class X. Draw the scatter diagram of these variables taking two at a time. What type of relationship do you find? Inspection of the scatter diagram gives an idea of the nature and intensity of the relationship. Karl Pearsonâ€™s Coefficient of Correlation This is also known as product moment correlation and simple correlation coefficient. It gives a precise numerical value of the degree of linear relationship between two variables X and Y. The linear relationship may be given by Y = a + bX This type of relation may be described by a straight line. The intercept that the line makes on the Y-axis is given by a and the slope of the line is given by b. It gives the change in the value of Y for very small change in the value of X. On the other hand, if the relation cannot be represented by a straight line as in Y = X 2 the value of the coefficient will be zero. It clearly shows that zero correlation need not mean absence of any type of relation between the two variables. Let X 1 , X 2 , ..., X N be N values of X and Y 1 , Y 2 ,..., Y N be the corresponding values of Y. In the subsequent presentations the subscripts indicating the unit are dropped for the sake of simplicity. The arithmetic means of X and Y are defined as X X N Y Y N == SS ; and their variances are as follows s 2 22 2 x XX N X N X = - =- SS () Page 5 As the summer heat rises, hill stations, are crowded with more and more visitors. Ice-cream sales become more brisk. Thus, the temperature is related to number of visitors and sale of ice-creams. Similarly, as the supply of tomatoes increases in your local mandi, its price drops. When the local harvest starts reaching the market, the price of tomatoes drops from a princely Rs 40 per kg to Rs 4 per kg or even less. Thus supply is related to price. Correlation analysis is a means for examining such relationships systematically. It deals with questions such as: â€¢ Is there any relationship between two variables? Correlation 7 1. INTRODUCTION In previous chapters you have learnt how to construct summary measures out of a mass of data and changes among similar variables. Now you will learn how to examine the relationship between two variables. Studying this chapter should enable you to: â€¢ understand the meaning of the term correlation; ? understand the nature of relationship between two variables; ? calculate the different measures of correlation; ? analyse the degree and direction of the relationships. CHAPTER 92 STATISTICS FOR ECONOMICS ? If the value of one variable changes, does the value of the other also change? â€¢ Do both the variables move in the same direction? ? How strong is the relationship? 2. TYPES OF RELATIONSHIP Let us look at various types of relationship. The relation between movements in quantity demanded and the price of a commodity is an integral part of the theory of demand, which you will read in class XII. Low rainfall is related to low agricultural productivity. Such examples of relationship may be given a cause and effect interpretation. Others may be just coincidence. The relation between the arrival of migratory birds in a sanctuary and the birth rates in the locality can not be given any cause and effect interpretation. The relationships are simple coincidence. The relationship between size of the shoes and money in your pocket is another such example. Even if relationship exist, they are difficult to explain it. In another instance a third variableâ€™s impact on two variables may give rise to a relation between the two variables. Brisk sale of ice-creams may be related to higher number of deaths due to drowning. The victims are not drowned due to eating of ice-creams. Rising temperature leads to brisk sale of ice-creams. Moreover, large number of people start going to swimming pools to beat the heat. This might have raised the number of deaths by drowning. Thus temperature is behind the high correlation between the sale of ice-creams and deaths due to drowning. What Does Correlation Measure? Correlation studies and measures the direction and intensity of relationship among variables. Correlation measures covariation, not causation. Correlation should never be CORRELATION 93 interpreted as implying cause and effect relation. The presence of correlation between two variables X and Y simply means that when the value of one variable is found to change in one direction, the value of the other variable is found to change either in the same direction (i.e. positive change) or in the opposite direction (i.e. negative change), but in a definite way. For simplicity we assume here that the correlation, if it exists, is linear, i.e. the relative movement of the two variables can be represented by drawing a straight line on graph paper. Types of Correlation Correlation is commonly classified into negative and positive correlation. The correlation is said to be positive when the variables move together in the same direction. When the income rises, consumption also rises. When income falls, consumption also falls. Sale of ice-cream and temperature move in the same direction. The correlation is negative when they move in opposite directions. When the price of apples falls its demand increases. When the prices rise its demand decreases. When you spend more time in studying, chances of your failing decline. When you spend less hours in study, chances of your failing increase. These are instances of negative correlation. The variables move in opposite direction. 3. TECHNIQUES FOR MEASURING CORRELATION Widely used techniques for the study of correlation are scatter diagrams, Karl Pearsonâ€™s coefficient of correlation and Spearmanâ€™s rank correlation. A scatter diagram visually presents the nature of association without giving any specific numerical value. A numerical measure of linear relationship between two variables is given by Karl Pearsonâ€™s coefficient of correlation. A relationship is said to be linear if it can be represented by a straight line. Another measure is Spearmanâ€™s coefficient of correlation, which measures the linear association between ranks assigned to indiviual items according to their attributes. Attributes are those variables which cannot be numerically measured such as intelligence of people, physical appearance, honesty etc. Scatter Diagram A scatter diagram is a useful technique for visually examining the form of relationship, without calculating any numerical value. In this technique, the values of the two variables are plotted as points on a graph paper. The cluster of points, so plotted, is referred to as a scatter diagram. From a scatter diagram, one can get a fairly good idea of the nature of relationship. In a scatter diagram the degree of closeness of the scatter points and their overall direction enable us to examine the relation- 94 STATISTICS FOR ECONOMICS ship. If all the points lie on a line, the correlation is perfect and is said to be unity. If the scatter points are widely dispersed around the line, the correlation is low. The correlation is said to be linear if the scatter points lie near a line or on a line. Scatter diagrams spanning over Fig. 7.1 to Fig. 7.5 give us an idea of the relationship between two variables. Fig. 7.1 shows a scatter around an upward rising line indicating the movement of the variables in the same direction. When X rises Y will also rise. This is positive correlation. In Fig. 7.2 the points are found to be scattered around a downward sloping line. This time the variables move in opposite directions. When X rises Y falls and vice versa. This is negative correlation. In Fig.7.3 there is no upward rising or downward sloping line around which the points are scattered. This is an example of no correlation. In Fig. 7.4 and Fig. 7.5 the points are no longer scattered around an upward rising or downward falling line. The points themselves are on the lines. This is referred to as perfect positive correlation and perfect negative correlation respectively. Activity ? Collect data on height, weight and marks scored by students in your class in any two subjects in class X. Draw the scatter diagram of these variables taking two at a time. What type of relationship do you find? Inspection of the scatter diagram gives an idea of the nature and intensity of the relationship. Karl Pearsonâ€™s Coefficient of Correlation This is also known as product moment correlation and simple correlation coefficient. It gives a precise numerical value of the degree of linear relationship between two variables X and Y. The linear relationship may be given by Y = a + bX This type of relation may be described by a straight line. The intercept that the line makes on the Y-axis is given by a and the slope of the line is given by b. It gives the change in the value of Y for very small change in the value of X. On the other hand, if the relation cannot be represented by a straight line as in Y = X 2 the value of the coefficient will be zero. It clearly shows that zero correlation need not mean absence of any type of relation between the two variables. Let X 1 , X 2 , ..., X N be N values of X and Y 1 , Y 2 ,..., Y N be the corresponding values of Y. In the subsequent presentations the subscripts indicating the unit are dropped for the sake of simplicity. The arithmetic means of X and Y are defined as X X N Y Y N == SS ; and their variances are as follows s 2 22 2 x XX N X N X = - =- SS () CORRELATION 95Read More

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### NCERT Solutions - Correlation

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### Scatter Diagram

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### Spearman's Rank

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- Karl Pearson's Coefficient (Part - 2)
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