Topic: Linear Correlation
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INTRODUCTION:
Recall that in research we want to not only demonstrate a relationship but also describe and
summarize the relationship. The one remaining type of descriptive statistic for us to discuss is used to
summarize relationships, and it is called the correlation coefficient. In the following sections, we’ll consider
when these statistics are used and what they tell us. Then we’ll see how to compute the two most common
versions of the correlation coefficient.
Correlation, like the mean, is a concept that you were probably familiar with before taking a statistics
course. If we tell you that music ability is correlated with mathematical ability you may be skeptical, but it is
very likely that you know what we mean. If we don’t say that the two variables are negatively correlated, it is
reasonable to assume that we are implying that the variables are positively correlated; that is, when someone
has a good deal of musical ability he or she tends to be good at math (and vice versa), while low musical
ability tends to go with low math ability. And if we say that depression and college grades are negatively
correlated, you probably know that this means low scores on one variable tend to be paired with high scores
on the other.
LESSON PROPER
Correlation is a statistical technique that is used to measure and describe a relationship between
two variables. Correlation means association - more precisely it is a measure of the extent to which two
variables are related. There are three possible results of a correlational study: a positive correlation, a
negative correlation, and no correlation. A relationship—a correlation—can exist, even though one variable
does not cause or influence the other.
Two requirements must be met to confidently conclude that X causes Y.
1. X must occur before Y
2. X must be the only variable that can influence Y
The Characteristics of a Relationship
1. The Direction of the Relationship: The sign of the correlation that is either positive or negative would
describe the relationship direction.
Positive Relationship. This result when one value increases the other value also increases - meaning that
the two variables are moving in the same direction. This is the direct correlation or the positive correlation.
Negative correlation tends to go in the opposite direction. As the x variable increases the y variable
decreases. This is the inverse relationship or the negative relationship.
Positive Correlation
.
Negative Correlation
No Correlation
Source: Understanding Statistics in the Behavioral Sciences, R.R. Pagano
Example: A drink concessioner
Positive relationship: when the temperature is high, you tend to sell a lot of beer. When the temperature is
low, you sell relatively little beer.
Negative relationship: On cold days, you sell much more coffee than on hot days.
In a nonlinear, or curvilinear, relationship, as the scores change, the scores do not tend to only increase
or only decrease: At some point, the scores change their direction of change.
Nonlinear relationships come in many different shapes, but Figure below shows two common ones. The
scatterplot on the left shows the relationship between a person’s age and the amount of time required to
move from one place to another. Very young children move slowly, but as age increases, movement time
decreases. Beyond a certain age,
however, the time scores change
direction and begin to increase.
(Such a relationship is called Ushaped.) The scatterplot on the right
shows the relationship between the
number of alcoholic drinks consumed
and feeling well. At first, people tend
to feel better as they drink, but
Source: Statistics for the Behavioral Sciences, Gravetter & Wallnau
beyond a certain point, drinking more makes them feel progressively worse.
2. The Form of the Relationship: The relationships tend to have a linear form, that is, the points in the
scatter plot tend to form a straight line. However, a special correlations exist for measuring other forms.
The form is specified by the type of correlation used. The Pearson correlation measures linear form.
3. The Strength or Consistency of the Relationship: The consistency of the relationship is measured by
the numerical value of the correlation. A correlation of 1.00 indicates a perfectly consistent relationship and
0.00 indicates no relationship at all. For the Pearson correlation, 𝑟 = 1.00 (or – 1.00)means that the data
points fit perfectly on a straight line.
A perfect relationship is one in which a positive or negative relationship exists and all of the points fall
on the line. An imperfect relationship is one in which a relationship exists, but all of the points do not fall
on the line.
Type of Correlation Coefficients
Correlation Coefficient
Types of Scales
Pearson Product Moment
Spearman Rank-Order
Phi
Point Biserial
Both scales Interval or ratio
Both scales Ordinal
Both scales are naturally dichotomus
One scale dichotomous one scale interval or ratio
Where and Why Correlations are Used
1. Prediction
3. Reliability
2. Validity
4. Theory Verification
The formula for the Pearson Product Moment Coeficient of Correlation is:
𝑛 ∑ 𝑥𝑦 − ∑ 𝑥 ∑ 𝑦
𝑟=
√[𝑛 ∑ 𝑥 2 − (∑ 𝑥)2 ][𝑛 ∑ 𝑦 2 − (∑ 𝑦)2 ]
Where:
𝑟 = The Pearson Product Moment Coefficient of Correlation
𝑛 = sample size
∑ 𝑥𝑦 = the sum of the product of x and y
∑ 𝑥 ∑ 𝑦 = the product of the sum of ∑ 𝑥 and ∑ 𝑦
2
∑ 𝑥 = sum of squares of x
∑ 𝑥2
= sum of squares of y
Hypothesis Tests with the Pearson Correlation
The basic question for this hypothesis test is whether a correlation exists in the population.
The Hypotheses:
Null hypothess
𝐻0 : 𝜌 = 0
Alternative hypothesis 𝐻1 : 𝜌 ≠ 0
(There is no population correlation)
( There is a real correlation)
When there is specific prediction anout the direction of the correlation, it is possible to do a directional, or
one-tailed test.
𝐻0 : 𝜌 ≤ 0
(The population correlation is not positive)
𝐻1 : 𝜌 > 0
(There population correlation is positive)
The purpose of the hypothesis is to decide between the following two alternatives:
1. There is no correlation in the population 𝜌 = 0, and the sample value is the result of sampling error.
2. The non-zero sample correlation accurately represents a real, nonzer correlation in the population. This is
the alternative stated in 𝐻1 .
Degrees of Freedom for the Correlation Test
The hypothesis test for the Pearson correlation has degrees of freedom defined by 𝑑𝑓 = 𝑛 − 2.
Example 1: Below are the midterm (x) and final (y) grades.
75
80
𝑥
𝑦
70
75
65
65
90
95
85
90
85
85
80
90
70
75
65
70
90
90
Solving Stepwise Method
I. Problem: Is there a significant relationship between the midterm and the final examination of 10 students
in Mathematics?
II. Hypotheses:
H0: There is no significant relationship between the midterm grades and the final examination/
grades of 10 students in Mathematics.
H1: There is no significant relationship between the midterm grades and the final examination/
grades of 10 students in Mathematics.
III. Level of Significance:
𝛼 = 0.05
𝑑𝑓 = 𝑛 − 2
= 10 − 2
=8
𝑟. 05 = 0.632
IV. Decision Rule: If the computed 𝑟 value is greater than the 𝑟 tabular value, reject H0.
V. Statistics:
𝑥
𝑦
𝑥2
𝑦2
𝑥𝑦
75
70
65
90
85
85
80
70
65
90
80
75
65
95
90
85
90
75
70
90
5625
4900
4225
8100
7225
7225
6400
4900
4225
8100
6400
5625
4225
9025
8100
7225
8100
5625
4900
8100
6000
5250
4225
8550
7650
7225
7200
5250
4550
8100
∑ 𝑥 = 775
∑ 𝑦 = 815
∑ 𝑥 2 = 60,925
∑ 𝑦 2 = 67,325
∑ 𝑥𝑦 = 64,000
𝑥̅ = 77.5
𝑦̅ = 77.5
𝑟=
𝑛 ∑ 𝑥𝑦 − ∑ 𝑥 ∑ 𝑦
√[𝑛 ∑ 𝑥 2 − (∑ 𝑥)2 ][𝑛 ∑ 𝑦 2 − (∑ 𝑦)2 ]
=
=
10(64,000)−(775)(815)
√[10(60,925)−(775)2 ][10(67,325)−(815)2 ]
=
8,375
√[8,625][9,025]
=
8,375
√[609,250−600,625][673,250−664,225]
8,375
√7,7840,625
=
8,375
8,822.73
𝑟 = 0.949
VI. Conclusion/Implication:
Since the computed value of 𝑟 which is 0.949 is greater than the tabular value of 0.632 at 0.05
level of significance with 8 degrees of freedom, the null hypothesis is disconfirmed in favor of the research
hypothesis. This means that there is a significant relationship between the midterm grades of students and
the final examination. It implies that the higher the midterm graddes the higher also are the final grades
beacause the value of r is positive. Likewise, the lower the midterm grades the loweralso the final grades.
Coefficient of Determination
The value 𝑟 2 is called the coefficient of determination because it measures the proportion of variability in
one variable that can be determined from the relationship with the other variable.
Using example 1: What is the coefficient of determination when 𝑟 = .949?
𝐶𝐷 = 𝑟 2 ∗ 100%
= (0.949)2 ∗ 100%
= 0.9006 ∗ 100%
= 90.06%
This 90.06% indicates that the final examination grade depends on the midterm grades. Thus, the final grade
is influenced by the midterm grade
Example : A correlation of 𝑟 = 0.80 (or −0.80), means 𝑟 2 = 0.64 (or 64%) of the variability in the 𝑌
scores can be predicted from the relationship with 𝑋
In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between
to variables on a scatter plot. The value of r is always between +1 and -1. To interpret its value, see which of
the following values your correlation r is closest to:
Exactly – 1
-0.70 to - 0.99
-0.50 to - 0.69
-0.30 to - 0.49
0
+0.30 to +0.49
+0.50 to +0.69
+0.70 to +0.99
Exactly +1
= A perfect downhill (negative) linear relationship
= A strong downhill (negative) linear relationship
= A moderate downhill (negative) relationship
= A weak downhill (negative) linear relationship
= No linear relationship
= a weak uphill (positive) linear relationship
= a moderate uphill (positive) relationship
= a strong uphill (positive) linear relationship
= A perfect uphill (positive) linear relationship
Scatter Diagrams are convenient mathematical tools to study the correlation between two random
variables. As the name suggests, they are a form of a sheet of paper upon which the data points
corresponding to the variables of interest, are scattered. Judging by the shape of the pattern that the data
points form on this sheet of paper, we can determine the association between the two variables, and can
further apply the best suitable correlation analysis technique.
A scatterplot is a
graph that shows the
location of each data
point formed by a pair of
X–Y scores. A data point
that is relatively far from
the majority of data
points in the scatterplot
is referred to as an
outlier—it lies out of the
general pattern.
Example. Draw the scatter diagram for the given pair of variables and understand the type of correlation
between them.
No. of Students Marks obtained (out of 100)
12
40-50
10
50-60
8
60-70
7
70-80
5
80-90
2
90-100
Solution:
Here, we take the two variables for consideration as:
M: The marks obtained out of 100
S: Number of students
Since the values of M is in the form of bins, we can use the center point of each class in the scatter diagram
instead. So let us first choose the axes of our diagram.
X-axis – Marks obtained out of 100
Y-axis – Number of Students
Spearman Rank Order of Coefficient Correlation (𝒓𝒔 ).
The Spearman Rho is a non-parametric test of correlation for it does not require normality in distribution.
It is used to look into the significant relationship of two variables when ranking is used.
Steps in Computing Spearman Rho
1. Rank the data in x from highest to lowest
(Create a temporary rank column and arrange the score from highest to lowest. The highest score
gets the rank)
2. In case of similar scores, add the RANKS (not the scores) and divide it by the number of similar
scores. The temporary ranks will guide you as to what rank is next.
3. Rank the data in y from highest to lowest.
4. Compute the difference between the rank of x and y
5. Square the difference of the ranks
6. Find the summation of the squared difference.
7. Determine the n
8. Use the formula given
Point Biserial Correlation Coefficient (rpbi)
The point biserial correlation coefficient (rpbi) is a statistics used to estimate the degree of relationship
between a dichotomous and an interval or ratio scale. If a researcher want to find out the degree of
relationships between gender (male and female) and IQ score (scale) the point biserial correlation coefficient
is appropriate. The interpretation of which is similar to Pearson r - that can range -1 to +1.
To find the point biserial correlation, the dichotomous or binomial variable is converted to numerical
values. We can assign a value of zero (0) to one category and a value of one (1) to the other category. Then
use the regular Pearson Correlation formula to convert the data. It is customary to use zero (0) and one (1)
but using other two different numbers would work equally well and would not affect the value of correlation.
Example: A researcher wants to determine if gender is related to IQ. Since IQ is continuous data and gender
is dichotomous, the point-biserial correlation coefficient is appropriate.
Some Examples of Dichotomous variables::
1. Male vs. female
2. College graduate vs. not a college graduate
3. First-born child vs. later born child
4. success vs. failure on a particular task
5. Older than 30 years versus younger than 30 years.
Phi-coefficient. This statistics is used for variables that are both nominal and dichotomous. In this statistics,
contingency table is used to present the data. It is a two dimensional table containing frequencies by
categories. It is more appropriate for two by two since each variable only take two values.
Two-step procedure in computing phi (𝜑):
1. Convert each of the dichotomous variables to numerical values by assigning a 0 to one category and a 1
to the other category for each of the variables.
2. Use the regular Pearson formula with the converted scores.
Example: A researcher is interested in examining the relationship between the birth-order position and
personality. A random sample of n=8 individuals is obtained, and each individual is classified in terms of birthorder position as first-born or only child versus later-born. Then each individual’s personality is classified
either introvert or extrovert.
The original measurements are then converted to numerical values by the following assignments:
Birth Order
Personality
1st or only child = 0
Later-born child = 1
Introvert = 0
Extrovert = 1
The original data and the converted scores are as follows:
Original Data
Converted Scores
Birth Order X
Personality Y
Birth Order X
Personality Y
1st
3rd
Only
2nd
4th
2nd
Only
3rd
Introvert
Extrovert
Extrovert
Extrovert
Extrovert
Introvert
Introvert
Extrovert
0
1
0
1
1
1
0
1
0
1
1
1
1
0
0
1
Interpretation of the Phi coefficient
-1.0 to -0.7
-0.7 to -0.3
-0.3 to +0.3
+0.3 to +0.7
+0.7 to +1.0
= strong negative association
= weak negative association
= little or no association
= weak positive association
= strong positive association