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

The relationship between r and r may be noted—as the value of r decreases from its maximum value Notes
2
of 1, the value of r decreases much more rapidly. r will, of course, always be larger than r , unless r 2
2
2
= 0 or 1.
r r 2
0.90 0.81
0.80 0.64
0.70 0.49
0.60 0.36
0.50 0.25
Thus the coefficient of correlation is 0.707 when just half the variance in Y is due to X.
It should be clearly noted that the fact that a correlation between two variables has a value of r = 0.60
and the correlation between two other variables has a value of r = 0.30 does not demonstrate that the
first correlation is twice as strong as the second. The relationship between the two given values of r
2
can better be understood by computing the value of r . When r = 0.6, r = 0.36 and when r = 0.30, r =
2
2
0.09.
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.
Properties of the Coefficient of Correlation

The following are the important properties of the correlation coefficient r:
1. The coefficient of correlation lies between – 1 and + 1. Symbolically, –1 ≤≤ +1 or || 1r ≤ .
r
2. The coefficient of correlation is independent of change of scale and origin of the variables X and
Y.
3. The coefficient of correlation is the geometric mean of two regression coefficients.
Symbolically,
r = b xy ×b yx
Self-Assessment

1. Indicate whether the following statements are True or False:
(i) There are no limits to the value of r.
(ii) If r is negative both the variable are decreasing.
(iii) If the values of X variable are 1, 2, 3, 4, 5 and those of Y 4, 6, 8, 10, 12 the Karl Pearson and the
Rank method would give the same answer.
(iv) Pearsonian coefficient is the best under all situations.
(v) Karl Pearson’s coefficient of correlation always lies between 0 and + 1.
10.3 Summary

• A scatter diagram is used to show the relationship between two kinds of data. It could be the
relationship between a cause and an effect, between one cause and another, or even between
one cause and two others.
• In statistics, the Pearson product-moment correlation coefficient (r) is a common measure of the
correlation between two variables X and Y. When measured in a population the Pearson Product

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