CORRELATION
1.1 Introduction
Correlation is often used as a descriptive tool in non-experimental
research. We say that two measures are correlated if they have something in common.
The intensity of the correlation is expressed
by a number called the coefficient of correlation
which is almost always denoted by the letter
r. Although usually called the Pearson coefficient of correlation, it was first
introduced by Galton (in a famous paper published in 1886) and later formalized
by Karl Pearson (1896) and then by Fisher (1935). In this chapter we explain this
coefficient, its rationale, and computation.
The main idea behind the coefficient of correlation is to compute an
index which reflects how much two series of measurements are related to each other.
For convenience, this coefficient will take values from −1 to +1 (inclusive). Avalue
of 0 indicates that the two series of measurements have nothingin common. A value
of +1 says that the two series of measurements are measuring the same thing (e.g.we
have measured the height of a group of persons with metric and imperial1 units). A value of −1 says
that the two measurements are measuring the samething but one measurement varies
inversely to the other (e.g. one variable measure show rich you are, and the other
one measures how poor you are: so the less rich you are, the poorer you are. Both
scales are measuring the same financial status but with ‘opposite’ points of view).
1.2 Types of Correlation
There are two important types of correlation. They are (1)
Positive and negative correlation and (2) Linear and Non – Linear correlation.
1.2.1 Positive and Negative Correlation
If the values of the two variables deviate in the same direction
i.e. if an increase (or decrease) in the values of one variable results, on an
average, in a corresponding increase (or decrease) in the values of the other
variable the correlation is said to be positive.
Some examples of series of positive correlation
are:
(i) Heights and weights;
(ii) Household income and expenditure;
(iii) Price and supply of commodities;
(iv) Amount of rainfall and yield of crops.
Correlation between two variables is said to be negative or
inverse if the variables deviate in opposite direction. That is, if the
increase in the variables deviate in opposite direction. That is, if increase
(or decrease) in the values of one variable results on an average, in
corresponding decrease (or increase) in the values of other variable.
Some examples of series of negative correlation
are
(i) Volume
and pressure of perfect gas;
(ii) Current and resistance [keeping the
voltage constant]
(iii) Price and demand of goods.
1.2.2 Graphs of
Positive and Negative correlation:
Suppose we are given sets of data relating to heights and weights
of students in a class. They can be plotted on the coordinate plane using x –
axis to represent heights and y – axis to represent weights. The different
graphs shown below illustrate the different types of correlations.
Note:
i.
If the points are very close to each other, a fairly good amount
of correlation can be expected between the two variables. On the other hand if
they are widely scattered a poor correlation can be expected between them.
ii.
If the points are scattered and they reveal no upward or downward
trend as
in the case
of (d) then
we say the
variables are uncorrelated.
iii.
If there is an upward trend rising from the lower left hand corner
and going upward to the upper right hand corner, the correlation obtained from
the graph is said to be positive. Also, if there is a downward trend from the
upper left hand corner the correlation obtained is said to be negative.
iv.
The graphs shown
above are generally
termed as scatter diagrams.
1.3 Correlation
Coefficient
The General Idea
Correlation
coefficients (denoted r) are statistics that quantify the relation
between X and Y in unit-free terms. When all points of a scatter plot fall
directly on a line with an upward incline, r = +1; When all points fall directly
on a downward incline, r = !1.
Such
perfect correlation is seldom encountered. We still need to measure correlational
strength, –defined as the degree to which data point adhere to an
imaginary trend line passing through the “scatter cloud.” Strong correlations
are associated with scatter clouds that adhere closely to the imaginary trend
line. Weak correlations are associated with scatter clouds that adhere
marginally to the trend line.
The closer r
is to +1, the stronger the positive correlation. The closer r is to
!1, the stronger the negative correlation. Examples of strong and weak
correlations are shown below. Note: Correlational strength can not be
quantified visually. It is too subjective and is easily influenced by
axis-scaling. The eye is not a good judge of correlational strength.
1.4 Pearson’s
Correlation Coefficient
To
calculate a correlation coefficient, you normally need three different sums of
squares (SS). The sum of squares for variable X, the sum of square for variable
Y, and the sum of the cross-product of XY.
1.5 Linear
and Non – Linear Correlation
The
correlation between two variables is said to be linear if the change of
one unit in one variable result in the corresponding change in the other
variable over the entire range of values.
In general
two variables x and y are said to be linearly related, if there exists a
relationship of the form
where ‘a’ and ‘b’ are real numbers.
This is nothing but a straight line when plotted on a graph sheet with
different values of x and y and for constant values of a and b. Such relations
generally occur in physical sciences but are rarely encountered in economic and
social sciences.
The
relationship between two variables is said to be non – linear if corresponding
to a unit change in one variable, the other variable does not change at a
constant rate but changes at a fluctuating rate. In such cases, if the data is
plotted on a graph sheet we will not get a straight line curve.
