Section 21-22 Correlation, Scattergrams

Question 1

Correlation

Answer 1

CORRELATION refers to the EXTENT to which TWO SETS of scores are RELATED.

• There are 2 TYPES of correlations:

1. DIRECT or POSITIVE CORRELATION where those who score high on one variable tend to score high on the other, and those who score low on one variable tend to score low on the other.
2. INVERSE of NEGATIVE CORRELATION where those who score high on one variable tend to score low on the other.

Question 2

DIRECT or POSITIVE CORRELATION

Answer 2

Keywords: Positive Correlation is UP, UP or Down, Down. They’re positive about each other, so they want to stick together in the same direction ; )

DIRECT or POSITIVE CORRELATION where those who score high on one variable tend to score high on the other, and those who score low on one variable tend to score low on the other.

Question 3

INVERSE of NEGATIVE CORRELATION

Answer 3

Keywords: Negative Correlations go in the OPPOSITE way because they feel negatively about each other ;)

INVERSE of NEGATIVE CORRELATION where those who score high on one variable tend to score low on the other.

Question 4

Causation

Answer 4

a CAUSAL RELATIONSHIP is one where one variable CAUSES changes in the other – that is, one variable AFFECTS the other.

Question 5

PROFOUND LIFE IMPORTANCE of Correlation vs. Causation

Answer 5

Keywords: CAUSATION has INFLUENCE, CORRELATIONS are just RELATED

CORRELATION refers to the DEGREE of RELATEDNESS of two sets of scores.

• But one set of scores may have NO INFLUENCE over the other set of scores. That means that a move in one of the scores will NOT CAUSE the move in the other set of scores even if the other set of scores moves in conjunction with the first set.
• Ex: A group of individuals are BOTH very good at baseball and also get very high grades. In fact, the better rated they are at baseball, the higher their grades are.
  
  • Some people would MISTAKENLY associate the two things and think, “Wow, if I get better at baseball then my grades will go up.”
  • Of course, that is nonsense. Baseball has nothing to do with grades – so why do they appear related?
  • The answer is often (but not always) due to some OTHER 3rd thing that CAUSES each of these things separately. (More on this after we discuss Causation…)

CAUSATION refers to the fact that ONE VARIABLE CAUSES A CHANGE IN THE OTHER VARIABLE.

• With CAUSATION, THERE IS INFLUENCE from one set of scores to the next.
• With the baseball and grades example, there was NO CAUSATION. Being better at baseball has NO EFFECT on grades.
• So why do baseball skills and grades tend to go up together if there is no CAUSAL RELATIONSHIP And only CORRELATION?
• The ANSWER is that there is some 3rd phenomena that IS CAUSAL TO BOTH BOTH baseball AND grades. What might that be?
• One thing that comes to mind would be HARD WORK!
• HARD WORK is CAUSAL to Baseball Skills. The HARDER you work, THE BETTER you get at baseball. The HARD WORK CAUSES Better baseball skills.
• HARD WORK is also CAUSAL to Higher Grades. The HARDER you work, THE HIGHER your grades will be. The HARD WORK CAUSES higher grades.
• So BOTH baseball skills and grades go up with HARD WORK. Even though the RELATIONSHIP between higher baseball skills and higher grades is only CORRELATED because they have NO INFLUENCE on each other.

This is a PROFOUNDLY IMPORTANT CONCEPT in both math and LIFE because many decisions in life are made based on understanding the DIFFERENCE between CORRELATION and CAUSATION.

• People OFTEN confuse the two and mistakenly see the relationship between two phenomena in life as affecting one another, when really, the two phenomena have NOTHING to do with one another and are merely correlated.
• NOTE: that CAUSAL RELATIONSHIPS will ALSO be CORRELATED
  
  • Ex: The HARDER YOU WORK, the BETTER YOUR BASEBALL SKILLS and GRADES.
  • So HARD WORK AND the things they CAUSE are also CORRELATED.
  • But CORRELATED things are NOT NECESSARILY CAUSAL – as in the case of the correlated baseball skills and grades.

Question 6

Using Correlation to Determine Cause and Effect

Answer 6

Even though CORRELATED sets of data are NOT NECESSARILY CAUSAL, correlation is still USEFUL to find clues about causality, which can later be explored in experiments.

• CLUES TO FIND CAUSALITY – Correlations can be used to measure relationships and determine what third thing might be causing these two unrelated things to correlate
• ASSURING TEST VALIDITY – Correlations are helpful in understanding how well tests work. If a test is used to measure the success in some endeavor, then you would expect to find a POSITIVE CORRELATION between test scores and the ultimate success in that endeavor.
• EVIDENCE TO SUPPORT THEORIES – Correlations can help develop theories. Ex: A theory may predict that X should be correlated with Y. If a correlation is found, it helps to support the theory, and you can further attempt to find out WHY X and Y are correlated (i.e. what is causing their correlation).

Question 7

Scattergram

Answer 7

SCATTERGRAM is a graphic representation between two variables. (Also called a “Scatter Diagram” or “scatterplot”).

1. Refer to figure 1. – Using graph paper, draw two axes of equal length. each axis representing one of the measures.
2. Label each axis with the name of a variable. If one variable was measured before the other (e.g., the SAT was administered before college GP was earned), it is customary to place the one measured first on the x-axis (i.e., the horizontal axis).
3. Place one dot where the two scores for each individual intersect.
4. Label the scattergram as a figure and give it a number and a brief title.

LINEAR RELATIONSHIP the dots form a pattern that follows a single straight line (even though there may be scattered around the line).

• Figure 1. has a DIRECT or POSITIVE LINEAR RELATIONSHIP – as SAT goes up, GPA also goes up.
  
  • Note: POSITIVE CORRELATIONSHIPS ALWAYS GO IN THE SAME DIRECTION.
• Figure 3. has a NEGATIVE LINEAR RELATIONSHIP – as Variable X goes UP, Variable Y goes DOWN.
  
  • Note: NEGATIVE CORRELATIONS ALWAYS GO IN THE OPPOSITE DIRECTION.
• Notice that the scatterplot in Figure 1. is much tighter than in Figure 3. – As a result, you can bet that the data used in Figure 1. has a lower Standard Deviation.

CURVILINEAR RELATIONSHIP form a curve that, starting from the left, goes up for a while (indicating a direct relationship) and then turns downward (indicating an inverse relationship). Thus, the overall relationship is neither direct nor inverse. Instead, it is described as curvilinear.

• in Figure 4. are ANXIETY and performance on a test of MANUAL DEXTERITY. The figure indicates that those who are extremely anxious about their performance and those who have very little anxiety both perform poorly on the test of dexterity. Those who are only moderately anxious perform best.

NO RELATIONSHIP Sometimes, there is no pattern in a scattergram, and the dots are scattered throughout, as in Figure 5. In this case, there is no discernible relationship.

• But the only way you would be certain that there is NO RELATIONSHIP would be to run a REGRESSION ANALYSIS in order to find relationships that are not easily visually apparent. (More on this later).