CORRELATION 1
Correlation
Hannah K. Smith
School of Behavioral Sciences, Liberty University
EDCO 735: Statistics
Dr. Robin Henson
October 5, 2025
Correlation
Prompt 1
A high school counselor is looking at grades. She wants to know if hours studied each
week, which can be represented by X, is connected to the grades they earn, which can be
represented by Y. She conducts a survey amongst each student, and she finds a correlation of r=
+.72. When looking at r, the plus sign tells you that the relationship is positive which can be put
into words by saying that more study hours likely have a positive impact on receiving higher test
scores. When looking at the number of r, .72, that tells us the relationship is strong. The larger
the number, lets us know that the effect is big enough to confidently say that the amount of
time studied and grades do correlate with one another. Now, had the number she found been
CORRELATION 2
smaller, like .13, it would tell us that the amount of time studied had a small relationship with
grades received. A negative would tell us that more studying linked to lower grades, but that can
be influenced by external reasons such as stress or even poor study habits. Now if r had been 0,
that does not necessarily tell us that X and Y are unrelated (Warner, 2021). If r is close to 0, it
can still have a strong but not linear association between X and Y (Warner, 2021).
She cannot just trust the results she found. She needs to put the data in a chart to see it
visually. If she chose to use a scatterplot, she would graph the hours studied versus the grade
received. After graphing the dots, she would then see if she can draw a straight line through it. If
the dots showed a U shape, then Pearson’s r would not be the right tool. This could be an easy
way to identify a pattern or correlation. She could also use histograms to make sure both the
hours and grades look like a bell-shaped curve.
When more tests are run, there is an increased chance of finding an error that looks real
but is not. Say the school counselor is looking at more than the number of hours the student’s
study, that may include their sleep time, how long the are using a screen, and time they play
sports, and number of times they go to tutoring. That gives us five total correlations, so she
could lower the cutoff for significance.
Prompt 2
While there are two forms of the regression equation, both versions prove to be useful
(Warner, 2021). The raw score regression equation can be used when looking at and predicting
real world outcomes, like test scores from study hours. The raw score gives us predictions that
can be interpreted in the original unit, such as for each hour a student studies, their exam score
increased by 5 points. An example of the raw score equation would be if a teacher predicted a
student’s GPA, represented by Y, from the number of hours that study, represented by X. By
using the raw score equation, it gives us the GPA. Basically, this would say if a student were to
study for 10 hours, it is predicted that they have a 3.2 GPA. The raw score equation is better
CORRELATION 3
suited for that example because it gives raw scores in relation to the GPA. The z score regression
equation is useful when you want standardized variable and make comparisons across differing
categories. This form of the equation is beneficial in research as researchers may use multiple
units in their data, such as if they were to compare one’s income amount to their age to their
happiness. An example of this would be a researcher decides to compare if anxiety or
confidence better predicts one’s performance in a given subject. When doing so, they must
standardize into z scores, and the regression is the correlation, represented by r, that makes the
effect sizes comparable across the multiple predictors. The z score equation is better suited for
this example because it must be standardized to appropriately compare them across the various
predictors. Nonetheless, research has shown that regression equations can show and support
results of in various studies (Nagumo et al., 2025)
In bivariate regression, r, R, and B are related in that each represents an important part
of the regression that helps link the information in it to the real world. In fact, there are models
of bivariate regression that that enhance the performance of traditional regression models
(Jeng et al., 2025). Therefore, it could prove useful to integrate them into various research
models. B is the regression coefficient, or slope coefficient, and it corresponds with the
predicted Y when X is zero (Warner, 2021). It is the slope in the equation and shows us how Y
changes based on X (Warner, 2021). The r coefficient tells us the effect size and strength of the
association between variables (Warner, 2021). B and r are related through standard deviations.
Capital R is also called multiple R (Warner, 2021), and that is the correlation between the
observed Y and predicted Y.
R can account for several different predictors combined.
CORRELATION 4
References
Jeng, J., Chuang, C., & Lin, T. (2025). Interval fuzzy c-bivariate regression models with Box–
Cox transformation clustering approach for the interval-valued data. International
Journal of Fuzzy Systems, https://doi.org/10.1007/s40815-024-01951-5
Nagumo, Y., Onodera, S., Hajikazemi, M., & Okabe, T. (2025). Prediction of laminate stiffness
reduction by regression equations for damage variables obtained from the stress-based
variational model. Journal of Applied Mechanics, 92(4), 1-26.
https://doi.org/10.1115/1.4067568
Warner, R. M. (2021). Applied statistics I: Basic bivariate techniques (3rd ed.). Sage Publications.
ISBN: 978-1-5063-5280-0
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