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## Two-variable Data Analysis

The Law of Supply The law of supply states that, if all other factors remain equal, the higher the price, the higher the quantity supplied. Producers supply more at a higher price because selling a higher quantity at a higher price increases revenue. A, B and C are points on the supply curve. Each point on the curve reflects a direct correlation between quantity supplied (Q) and price (P). So, at point A, the quantity supplied will be Q1 and the price will be P1, and so on. The supply relationship curve illustrates the positive relationship between price and quantity supplied. The higher the price of a good, the higher the quantity supplied (C) and the lower the price of a good, the lower the quantity supplied (A).
The Law of Demand The law of demand states that, if all other factors remain equal, the higher the price of a good, the less people will demand that good. In other words, the higher the price, the lower the quantity demanded. A, B and C are points on the demand curve. Each point on the curve reflects a direct correlation between quantity demanded (Q) and price (P). So, at point B, the quantity demanded will be Q2 and the price will be P2, and so on. The demand relationship curve illustrates the negative relationship between price and quantity demanded. The higher the price of a good the lower the quantity demanded (A) and the lower the price, the more the good will be in demand (C).
Correlation

Introduction: Correlation means the relationship between two variables where with the changes in the values of one variable, the values of other variable also changes. For example, selling price of a commodity and profit earned are correlated to each other because change in a selling price of a commodity will cause a change in profit earned.

Definition: According to Ya-lun Chou, "correlation analysis attempts to determine the degree of relation between the variables" and according to Simpson and Kafka, "Correlation analysis deals with the association between two or more variables". Thus, correlation is a statistical device, which helps us in the analysis of co-variation of two (or) more variables.

Types of correlation: Classification of correlation can be done in following three ways: On the basis of direction of change of variables, On the basis of number of variables, On the basis of change in proportion.

On the basis of direction of change of variables: Depending upon the direction of change of variables, correlation is divided into two types called positive correlation and negative correlation.

• Positive correlation: When the values of two variables move in the same direction, that is, if an increase in the value of one variable is associated with an increase in the value of other variable (or) decrease in the value of one variable is associated with decrease in the value of other variable then the correlation is called as positive correlation. Eg: Relationship between price and supply.
• Negative correlation: When the values of two variables move in different directions, that is, if an increase in the value of one variable is associated with a decrease in the value of other variable and decrease in the value of one variable is associated with the increase in the value of other variable then the correlation is called as negative correlation. Eg: Relationship between price and demand.

On the basis of number of variables: Depending upon the total number of variables, correlation is divided into three types called simple correlation, partial correlation and multiple correlation.

• Simple Correlation: A correlation, in which only two variables are studied is known as simple correlation. Eg: Relationship between height and weight.
• Partial Correlation: In this correlation, we recognize more than two variables but consider only two variables to be influencing each other, the effect of other influencing variables being kept constant. Eg: Relationship between price and demand by keeping supply as constant.
• Multiple Correlation: A correlation, in which three or more variables are studied simultaneously is known as multiple correlation. Eg: Relationship between price, demand and supply.

On the basis of change in proportion: Depending upon the change in proportion, correlation is divided into two types called linear correlation and nonlinear correlation.

• Linear Correlation: If the amount of change in one variable tends to bear constant ratio to the amount of change in other variable then the relation is called linear correlation.
• Curvilinear (or) Nonlinear correlation: If the amount of change in one variable does not bear a constant ratio of the amount of change in the other then the relation is called curvilinear (or) nonlinear correlation.
Methods of Studying the Correlation

Scatterplot method: Scatterplot is a two dimensional graph of ordered pairs which is used for representing the two variable-data. It shows the relationship between two quantitative variables measured on the same cases diagrammatically by representing one variable on horizontal axis and other variable on vertical axis.

In this method, data are plotted on a graph paper in the form of dots, i.e., for each pair of observations of two variables, we put a dot and thus obtain as many dots as number of paired observations of two variables. The direction of dots shows the scatter (or) concentration of various points. This will show the type of correlation. In a scatterplot, if the dots are closer then there is a correlation and if dots are scattered then there is no correlation.

• Example: The cost of entry (in dollars) to an amusement park during 6 years is given in the adjacent table. Represent the data using scatterplot.

Sol: In drawing scatterplot of the above data, we represent horizontal axis as particular year and the vertical axis as cost of entry (in dollar). Thus, scatterplot of the given data is as follows:

Patterns in Scatterplots:

• If all the plotted dots lie on a straight line falling from lower left hand corner to upper right hand corner, then there is a perfect positive correlation between the two variables.
• If all the plotted dots lie on a straight line falling from upper left hand corner to lower right hand corner, then there is a perfect negative correlation between the two variables.
• If the plotted dots in the plane show an increasing tendency from lower left hand corner to upper right hand corner, then there is a positive correlation between the two variables.
• If the plotted dots in the plane show a declining tendency from upper left-hand corner to lower right-hand corner, then there is a negative correlation between the two variables.
• If the plotted dots in the plane are scattered all over the diagram, then there is no correlation between the two variables.

Graphic method: Under this method, the individual values of the two variables are plotted on a graph paper. There will be two curves; one for X variable and another for Y variable. By looking at the direction of the two curves, we can say whether there is any correlation between the two variables or not.

Karl Pearson's Coefficient of Correlation

The first statistic, we have to find out a linear relationship is the Pearson's correlation coefficient (or) simply the correlation coefficient, denoted by the letter 'r'.

Pearson's correlation coefficient is a measure of the strength of the linear relationship between two variables as well as an indicator of the direction of the linear relationship [whether the variables are positively or negatively associated].

If we have a distribution of size 'n' of paired data (x, y) and assuming that we have computed summary statistics [i.e., mean and standard deviation] for x and y, then correlation coefficient, where xi, yi are ith elements in x, y series respectively, and μx, μy are means of x, y series respectively and σx, σy are standard deviations of x, y series respectively.

Similarly, for a sample of size 'n' of paired data (x, y) we have correlation coefficient, where xi, yi are ith elements in x, y series respectively, and x, y are means of x, y series respectively and sx, sy are standard deviations of x, y series respectively.

Properties of correlation coefficient
• The value of coefficient of correlation always lie between 1 and – 1.
• If r > 0, it indicates that the variables are positively associated and if r < 0, it indicates that the variables are negatively associated.
• If r = 1, it indicates that the variables are perfectly positive associated.
• If r = – 1, it indicates that the variables are perfectly negative associated.
• If r = 0, means there is no relationship between the two variables.
• Karl Pearson's coefficient of correlation may not be useful when problem arises while dealing with qualitative character such as honesty, beauty, character and morality etc. (which cannot be measured quantitatively).
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