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Unit 10: Correlation: Scatter Diagram Method, Karl Pearson's Coefficient of Correlation

two variables is to use the square of coefficient of correlation, which is called coefficient of Notes
determination. The coefficient of determination thus equals r . The coefficient r expresses the
2
2
proportion of the variance in y determined by x; that is, the ratio of the explained variance to
total variance. Therefore, the coefficient of determination expresses the proportion of the total
variation that has been ‘explained’, or the relative reduction in variance when measured about
the regression equation rather than about the mean of the dependent variable. If the value of r
2
= 0.9, r will be 0.81 and this would mean that 81 per cent of the variation in the dependent
variable has been explained by the independent variable. The maximum value of r is unity
2
because it is possible to explain all of the variation in Y, but it is not possible to explain more
than all of it.
• The coefficient of determination is a highly useful measure. However, it is often misinterpreted.
The term itself may be misleading in that it implies that the variable X stands in a determining
or causal relationship to the variable Y. The statistical evidence itself never establishes the
existence of such causality. All that statistical evidence can do is to define covariation, that term
being used in a perfectly neutral sense. Whether causality is present or not, and which way it
runs if it is present, must be determined on the basis of evidence other than the quantitative
observations.
10.4 Key-Words

1. Scatter Diagram : A scatter diagram is a tool for analyzing relationships between
two variables. One variable is plotted on the horizontal axis and
the other is plotted on the vertical axis. The pattern of their
intersecting points can graphically show relationship patterns.
Most often a scatter diagram is used to prove or disprove cause-
and-effect relationships. While the diagram shows relationships,
it does not by itself prove that one variable causes the other. In
addition to showing possible cause-and-effect relationships, a
scatter diagram can show that two variables are from a common
cause that is unknown or that one variable can be used as a
surrogate for the other.
2. Coefficient determination : In statistics, the coefficient of determination, denoted R2, is used
in the context of statistical models whose main purpose is the
prediction of future outcomes on the basis of other related
information. R2 is most often seen as a number between 0 and 1.0,
used to describe how well a regression line fits a set of data. An
R2 near 1.0 indicates that a regression line fits the data well, while
an R2 closer to 0 indicates a regression line does not fit the data
very well. It is the proportion of variability in a data set that is
accounted for by the statistical model. It provides a measure of
how well future outcomes are likely to be predicted by the model.
10.5 Review Questions

1. What is Scatter diagram? How do you interprat a Scatter diagram?
2. What is a ‘Scatter diagram’? How does it help us in studying the correlation between two variables
in respect of both its nature and extent ?
3. How does a scatter diagram help in ascertaining the degree of correlation between two variables?
4. State the properties of Pearson’s coefficient of correlation. How do you interpret a calculated
value of r ? Explain the term ‘Probable error of r’.
5. State the assumptions of Karl Pearson’s Correlation.

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