Quantifying Relationships Through Correlation Coefficients

UPI Study › Courses › Research Methods in Psychology › Free lessons › Lesson 5.4 Free with a free account

Quantifying Relationships Through Correlation Coefficients

Research Methods in Psychology · Chapter 5: Non-Experimental and Correlational ResearchSection 5.4 · lesson 32 of 9410 min · free with a free account

← The Nature and Purpose of Correlational ResearchNext lesson: Interpreting and Drawing Insights from Correlation Data →

Earn 3 college credits for Research Methods in PsychologyRecommended for credit by NCCRS · Psychology 111 · self-paced · one payment, lifetime access
Enroll in Research Methods in Psychology · $250 →Download the syllabus (PDF)

In this lesson

Putting numbers on how strongly variables move together.

  • Define a correlation coefficient and read its sign
  • Interpret magnitude as strength, separately from direction
  • Match Pearson, Spearman, Kendall and phi to your data
  • Compute the coefficient of determination and explain it

Free study notes

Get this lesson's notes by email

The key points and 4 practice questions with answers, in your inbox. One email, free.

One email. Unsubscribe in one click.

Transcript

Everything said in the video, section by section (1,433 words). Read along, or search it.

Non-Experimental and Correlational Research

Quantifying Relationships Through Correlation Coefficients

Section five point three showed you what a relationship looks like on a graph. This section turns that picture into a single number. The correlation coefficient is one of the most quoted statistics in psychology. It is also one of the most misread, which is why the details matter. By the end you will define a correlation coefficient, and read its sign correctly. You will interpret its magnitude as strength, separately from its direction. You will know when Pearson's r is the right tool, and when it is not. You will meet the alternatives, including Spearman's rho and Kendall's tau. And you will square the coefficient, to find how much variance it actually explains. That last step is the one that keeps expectations honest.

Pattern To Number

Why the Scatter-Plot Needs a Number

Start with what a scatter-plot can and cannot do. Looking at one, you can see immediately whether two variables move together. Positively related, negatively related, or unrelated. The eye is good at this. But science requires precision, and the eye does not give you precision. Two researchers looking at the same cloud of points will describe it differently. So we need a single number instead. One that captures the direction of the relationship, and also its strength. And one that can be compared across studies, across populations, and across time periods. The correlation coefficient does both jobs at once. The sign carries the direction. Plus or minus, nothing more. The magnitude, meaning the distance from zero, carries the strength.

The Standardized Scale

From Minus One to Plus One

The coefficient is standardized, which is what makes that comparison possible. It runs from minus one point zero zero, through zero, to plus one point zero zero. Nothing can fall outside that range. A coefficient of minus point eight five reflects a strong negative relationship. A coefficient of plus point two zero reflects a weak positive one. And a value near zero indicates no systematic linear association. Now notice something that trips people up constantly. Minus point eight five and plus point eight five are equally strong. Beginners often read a negative correlation as a weak one. It is not. Strength lives in the distance from zero. Direction lives in the sign. Keep those two readings apart, and the number stops being confusing.

Reading The Bands

Conventional Labels for Strength

Researchers attach conventional labels to bands of the coefficient. A value of plus one point zero zero is a perfect positive linear relationship. Every increase in one variable is matched by a proportional increase in the other. From plus point seven zero to plus point nine nine, the relationship is called strong. From plus point three zero to plus point six nine, moderate. From plus point one zero to plus point two nine, weak. Near zero, there is no linear relationship at all. The negative bands mirror these exactly, and carry the same labels. Now the warning that gets buried in most textbooks. These conventions vary across fields. The labels are common, but not universal. So report the coefficient itself, rather than leaning on the adjective.

The Workhorse

Pearson's Product-Moment Coefficient

The most widely used coefficient is Pearson's product-moment correlation coefficient. Almost nobody says that out loud. It is written simply as r. Pearson's r captures the linear relationship between two continuous variables. In plain terms, it asks how closely the points follow a straight line. Its value always falls between minus one and plus one. Sign for direction, distance from zero for strength, exactly as before. The word linear is doing enormous work in that definition. Pearson's r measures only the linear component of a relationship. Anything that is not a straight-line trend is invisible to it. Almost every limitation you will meet later follows from that one restriction.

Four Assumptions

When Pearson's r Is Appropriate

Pearson's r is appropriate when several conditions are met. First, both variables should be measured at the interval or ratio level. Second, the relationship should be approximately linear. Curved relationships produce misleadingly low values of r. Third, both variables should be approximately normally distributed. Fourth, the sample should be free of extreme outliers. A single extreme value can distort the coefficient substantially. These are not paperwork. They are the conditions under which the number means something. Violate them seriously, and the software still returns a coefficient. It simply no longer describes your data. When the assumptions fail badly, alternative correlation indices are more appropriate.

Why Linearity Matters

The Coefficient That Misses the Curve

Linearity deserves its own warning, because this failure is invisible. Two variables can be perfectly related in a curved pattern. A U-shape, for instance. And still produce a coefficient near zero. The classic example is physiological arousal and task performance. Performance rises, then falls, as arousal increases. The relationship is real and strong, and a straight line cannot see it. A scatter-plot reveals that curve instantly. Pearson's r alone could miss it entirely. So always look at the scatter-plot before trusting the coefficient. The number summarizes the data. The picture tells you whether the summary is honest.

Other Indices

When Pearson's Assumptions Fail

Pearson's r is not the only correlation index available. Several alternatives exist for when its assumptions are not met. Spearman's rho works on ranks. Use it when variables are ordinal, or when normality and linearity are violated. Kendall's tau is also rank-based. It suits ordinal variables, particularly with small samples or many tied ranks. Point-biserial r pairs one continuous variable with one dichotomous variable. Phi handles the case where both variables are dichotomous. Eta is used when the relationship is non-linear, and it runs from zero to one. All the others run from minus one to plus one. They share one logic. Near zero means little association, and larger magnitude means more. They differ only in what kinds of variables they can accommodate.

Shared Variance

The Coefficient of Determination

Now the single most useful thing you can do with a correlation. Square it. Squaring Pearson's r produces the coefficient of determination, written r squared. This statistic has an especially useful interpretation. It represents the proportion of variance in one variable associated with the other. Work an example. A study finds r equals plus point five zero. That is between hours of study and exam scores. Square it, and r squared equals point two five. So twenty-five percent of the variance in exam scores is associated with study hours. Which means seventy-five percent is associated with other factors entirely. Even moderately strong relationships leave most of the variation unexplained.

A Useful Humility

How the Coefficient Translates Into Variance

That translation is worth doing every time you meet a coefficient. A correlation of point one zero explains one percent of the variance. Point three zero explains nine percent. Point five zero explains twenty-five percent. Point seven zero explains forty-nine percent. Point nine zero explains eighty-one percent. Only a perfect correlation explains all of it. Notice how slowly the explained variance grows at the low end. A correlation of point three zero is often described as a moderate effect. It explains nine percent of the variance, and leaves ninety-one percent unexplained. This is one reason single correlational findings should rarely drive major decisions. The world is almost always more complicated than one pair of variables can capture.

Chance Or Signal

Statistical Significance of a Correlation

A correlation computed from a sample is itself only an estimate. The thing you actually care about is the correlation in the population. Statistical tests ask whether your observed correlation is unlikely to have arisen by chance. Specifically, unlikely if the true population correlation were zero. The result is reported as a p-value. A p-value below the conventional threshold, often point zero five, means a chance result is unlikely. Here is where careful readers separate from careless ones. Statistical significance and practical importance are not the same thing. With very large samples, even tiny correlations can reach statistical significance. A correlation of point zero five can come out significant, given enough participants. Yet it explains so little variance that it may have minimal real-world importance. Significance says the effect is probably not zero. It does not say it matters.

Reporting Practice

What a Complete Correlation Report Contains

Finally, what a complete report of a correlation actually contains. Name the two variables. Hours of study and exam performance, for example. Give the coefficient used and its value. Pearson's r equals plus point four two. Give the sample size. In this example, an N of one hundred and eighty-four. Give the statistical significance. Here, p less than point zero zero one. Where it helps, give the proportion of variance explained. Here that is r squared of point one eight, an eighteen percent shared variance. And where it helps, give a ninety-five percent confidence interval. In this case, running from point three zero to point five three. That interval tells the reader how precise the estimate actually is. Modern practice emphasizes both the size of the correlation and its significance. Report the number, its precision, and its meaning. Never just the word significant.

About Research Methods in Psychology at UPI Study

Research Methods in Psychology (Psychology 111) is a self-paced online course of 94 video lessons in 16 chapters, from the scientific method and research ethics to designs, data collection, sampling, measurement, validity, experiments, descriptive statistics and hypothesis testing. It carries a NCCRS credit recommendation for 3 semester credits.

Every lesson transcript is free on pages like this one, and the videos are free with a free UPI Study account. Enrolling adds a practice quiz after each lesson, the graded work and final exam that earn the credit, lifetime access, and an official transcript on completion. UPI Study has helped over 130,000 students with college credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Enroll now → Course details Download the syllabus All Psychology and social science courses Every course with PRO →

Compare: Research Methods vs Statistics

Why take it here

Why students take Research Methods in Psychology with UPI Study

How psychologists know what they knowResearch Methods in Psychology teaches how psychological studies are designed, run and judged: research ethics, surveys and observation, correlational and experimental designs, sampling, reliability and validity, and the statistics used to test hypotheses. It is a core course in psychology and many social-science degrees.
Credit at a fraction of the priceThe same 3 credits can cost $750 or more at a US college. Here it is one payment, self-paced, with lifetime access.
Finish your degree soonerTransfer credits reduce the courses you still owe, so you can graduate earlier or lighten a heavy semester.
Strengthen an applicationA completed college-level course on an official transcript shows admissions and scholarship committees you can do the work.
Study on your own scheduleNo start dates, no deadlines. Watch the lessons, practice, and take the final when you are ready.
130,000+students helped with college creditsACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Where the credit goes

Universities that work with NCCRS

Research Methods in Psychology carries a NCCRS credit recommendation. These universities work with NCCRS; some are UPI Study's signed partners. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Thomas Edison State UniversityThomas Edison State UniversitySigned partnerUniversity of the PeopleUniversity of the PeopleSigned partnerCharter Oak State CollegeCharter Oak State CollegeSigned partnerStrayer UniversityStrayer UniversitySigned partnerColumbia Southern UniversityColumbia Southern UniversitySigned partnerIndiana TechIndiana TechSigned partnerPenn StatePenn StateWorks with NCCRSSouthern New Hampshire UniversitySouthern New Hampshire UniversityWorks with NCCRSSUNYSUNYWorks with NCCRS +many more cooperating universitiesFind my college →

Find my college → Not listed? Ask Athena in the corner of this page, or book a 15-minute call.

FAQ

Questions students ask us

About Quantifying Relationships Through Correlation Coefficients

How do you interpret a correlation coefficient?
The sign shows the direction, and the distance from zero shows the strength. Coefficients run from -1.00 to +1.00, so -.85 and +.85 are equally strong, one negative and one positive. Common labels call values from .70 to .99 strong, .30 to .69 moderate, and .10 to .29 weak, though conventions vary across fields, so report the coefficient itself rather than only the label.
When is it appropriate to use Pearson's r?
Pearson's r is appropriate when two variables are measured at the interval or ratio level and their relationship is roughly linear. Both variables should also be approximately normally distributed, and the data should be free of extreme outliers, since a single extreme value can distort the coefficient substantially. Because r measures only straight-line association, a strong curved relationship can produce a misleadingly low value.
What is the difference between Pearson's r and Spearman's rho?
Pearson's r measures the linear relationship between two continuous variables, while Spearman's rho works on ranks. Use Spearman's rho when variables are ordinal or when normality or linearity assumptions are violated. Kendall's tau is another rank-based option, suited to small samples or many tied ranks. Point-biserial r handles one continuous and one dichotomous variable, and phi handles two dichotomous variables.
What does r squared mean in correlation?
R squared, the coefficient of determination, is the proportion of variance in one variable associated with the other, found by squaring Pearson's r. If study hours and exam scores correlate at r = .50, then r squared = .25, so 25 percent of the variance in scores is associated with study hours and 75 percent with other factors. A correlation of .30 explains only 9 percent.

About the Research Methods in Psychology course

How many college credits is Research Methods in Psychology?
Research Methods in Psychology (Psychology 111) carries a NCCRS credit recommendation for 3 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
How much does Research Methods in Psychology cost and what is included?
$250, paid once, with lifetime access: all 94 video lessons, a practice quiz after each, the graded work and final exam, and an official transcript when you finish. Students who want several courses choose UPI Study PRO instead.
Which universities accept UPI Study credits?
UPI Study has signed partner agreements with universities such as Thomas Edison State University, University of the People and Charter Oak State College, and works with universities such as Penn State, SNHU and SUNY. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities. Use "Find my college" on this page to check yours.
How long does Research Methods in Psychology take?
It is self-paced with no deadlines. The 94 lessons total about 16 hours of video; most students add reading and practice time and finish in a few weeks to a few months, on their own schedule.
How are the exams done?
Quizzes and assignments are online in the course. The final exam is taken online with proctoring; a person on the UPI Study team checks assignments, usually within two days.
How do my credits reach my college?
When the course shows 100% complete, request the official transcript from your dashboard. It is sent to the college you name, and you can send it to more than one. Check your college first with "Find my college" on this page, or ask Athena.

Ready for the full course? Earn 3 credits in Research Methods in Psychology.

94 video lessons, a quiz after each one, and the graded work that earns the credit. Recommended for credit by NCCRS.

Enroll in Research Methods in Psychology →Download the syllabusEvery course with PRO →
Research Methods in Psychology
3 credits · NCCRSEnroll →