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Q: How do you use a table of data to check whether the relationship between two variables is a direct variation?

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Scatter chart

By definition, if you graph the relationship between two variables and the result is a straight line (of whatever slope) that is a linear relationship. If it is a curve, rather than a straight line, then it is not linear.

A scatter plot can be used to see if there is any relationship between two variables. It can also give a general idea of the nature of that relationship (linear, quadratic, logarithmic, inverse square, etc; whether or not the relationship remains constant over the domain, whether or not the variation remains constant (homoscedasticity), and so on.

scatter chart

Use a scatter plot to determine whether the relationship is linear, polynomial, exponential or other.

Yes. It can give insight as to whether there is a relationship between two variables, and if so, whether the relationship is direct or indirect; whether it is linear, polynomial, exponential, logarithmic; whether or not there are asysmptotic values; whether or not there is clustering; etc.

Correlation is a way of testing whether there is a linear relationship between two variables of their transformed versions. If there is no correlation then all that it means is that there is no linear relationship between the two or between their transformed versions. It certainly does not mean that there is no relationship. For example, if y = a*x^2 + b where a and b are constants then, because of symmetry, the correlation between x and y is 0.

It allows a quick look at the data to establish whether or not there is any relationship between the variables and also an indication of the nature of the relationship: linear, quadratic, power etc.

The measurement of any statistical variable will vary from one observation to another. Some of this variation is systematic - due to variations in some other variable that "explains" these variations. There may be several such explanatory variables - acting in isolation or in conjunction with one another. Finally, there will be a residual variation which cannot be explained by any of these "explanatory" variables. The statistical technique called analysis of variance first calculates the total variation in the observations. The next step is to calculate what proportion of that variation can be "explained" by other variables, and finding the residual variation. A comparison of the explained variation with the residual variation is an indicator of whether or not the amount explained is statistically significant. The word "explain" is in quotes because there is not always a causal relationship. The causality may go in the opposite direction. Or the variables may be related to another variable that is not part of the analysis.

Regrettably, no. The most a chi-square statistic can do is to participate in the measurement of the level of association of the variation between two variables.

The answer will depend on whether or not the relationship between the pairs of variables is transitive. In mathematics, not all relationships are transitive.For example, if the relationship is "is coprime with", then2 is coprime with 3, 3 is coprime with 4 but 2 is certainly not coprime with 4.

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