Appendix Y — SPSS Tutorial: Correlation and Bivariate Regression

Computing Pearson’s r, fitting regression models, and interpreting output in SPSS

Learning Objectives

By the end of this tutorial, you will be able to:

Compute Pearson’s correlation coefficient and create a correlation matrix in SPSS
Produce and interpret a scatterplot with a regression line
Conduct a bivariate linear regression and read the full SPSS output
Interpret the slope, intercept, \(R^2\), and unstandardized/standardized coefficients
Check assumptions: linearity, homoscedasticity, normality of residuals, and outliers
Report correlation and regression results following APA guidelines

Y.1 Overview

Correlation and bivariate regression are foundational tools for examining relationships between two continuous variables in Movement Science. SPSS provides a straightforward interface for computing Pearson’s \(r\), testing its significance, fitting a regression model, and producing diagnostic plots. This tutorial demonstrates:

How to produce scatterplots and correlation coefficients in SPSS
How to set up and run a bivariate (simple) linear regression
How to read and interpret SPSS regression output (Coefficients, Model Summary, ANOVA tables)
How to request and evaluate residual diagnostics
How to report results in APA style

Prerequisites: Familiarity with SPSS data entry and basic descriptive statistics.

Y.2 Dataset for this tutorial

We will use the Core Dataset (core_session.csv) introduced in the Core Dataset Overview. This is the same dataset used throughout the book.

Download it here: core_session.csv

For this tutorial, we examine the relationship between two continuous variables measured at the pre-training time point (time = "pre", N = 60):

vo2_mlkgmin — Aerobic capacity (VO₂max) in mL·kg⁻¹·min⁻¹ — predictor/independent variable (\(X\))
sprint_20m_s — 20-meter sprint time in seconds (s) — outcome/dependent variable (\(Y\))

This pairing has clear theoretical grounding: athletes with higher aerobic capacity tend to have faster sprint times, making it a natural candidate for correlation and regression analysis.

Opening the dataset in SPSS:

File → Open → Data…
Change the file type to CSV (*.csv), browse to core_session.csv, and click Open
Follow the Text Import Wizard: choose Delimited, check Variable names included at top of file, set delimiter to Comma, and click Finish
To restrict analyses to the pre-training time point, use Data → Select Cases → If condition is satisfied and enter: time = 'pre'

Which variables to use

See the Core Dataset Codebook for exact variable names, units, and coding. For this tutorial, use vo2_mlkgmin as the predictor and sprint_20m_s as the outcome.

Y.3 Part 1: Creating a scatterplot

Always visualize your data before computing any statistics. A scatterplot reveals the shape, direction, and strength of the relationship—and whether any outliers or nonlinearity exist.

Y.3.1 Procedure

Graphs → Chart Builder…
In the gallery at the bottom, click Scatter/Dot, then double-click the top-left (simple scatter) icon.
Drag vo2_mlkgmin to the X-Axis zone and sprint_20m_s to the Y-Axis zone.
Click OK.

To add a regression line to the existing chart:

Double-click the chart in the output viewer to open the Chart Editor.
From the menu, choose Elements → Fit Line at Total.
Select Linear in the Properties dialog → Apply → Close.
Close the Chart Editor.

Y.3.2 Interpreting the scatterplot

Examine the scatterplot for:

Direction: Do points move upward (positive) or downward (negative) from left to right?
Linearity: Do points roughly follow a straight-line trend, or is there a curved pattern?
Spread: Is the vertical spread of points roughly constant (homoscedastic), or does it fan out?
Outliers: Are any points far away from the overall pattern?

Always plot first

Never skip the scatterplot. Identical correlation coefficients can arise from completely different data patterns (Anscombe’s Quartet). Visual inspection protects against misleading interpretations.

Y.4 Part 2: Computing Pearson’s correlation

Y.4.1 Procedure

Analyze → Correlate → Bivariate…
Move both vo2_mlkgmin and sprint_20m_s to the Variables box.
Under Correlation Coefficients, ensure Pearson is checked.
Under Test of Significance, select Two-tailed (default).
Leave Flag significant correlations checked.
Click OK.

Y.4.2 Interpreting the output

SPSS produces a Correlations table:

Correlations
                          vo2_mlkgmin   sprint_20m_s
vo2_mlkgmin Pearson Corr.  1              -.643**
            Sig. (2-tailed)               .000
            N               60             60
sprint_20m_s Pearson Corr. -.643**         1
            Sig. (2-tailed) .000
            N               60             60

** Correlation is significant at the 0.01 level (2-tailed).

Key elements:

Pearson Correlation = −.643: Moderate-to-strong negative linear relationship — athletes with higher VO₂max tend to have faster (lower) sprint times.
Sig. (2-tailed) = .000: p < .001 — the correlation is statistically significant (SPSS displays “.000” for very small p-values; report as p < .001).
N = 60: Sample size (pre-training time point).
The table is symmetric: \(r_{XY}\) = \(r_{YX}\).

Coefficient of determination

Square the correlation to get \(r^2\):

\[r^2 = (-.643)^2 = .414\]

This means 41.4% of the variance in sprint time is explained by VO₂max in this sample.

Y.5 Part 3: Bivariate linear regression

Correlation quantifies the relationship; regression models it with an equation that enables prediction.

Y.5.1 Procedure

Analyze → Regression → Linear…
Move sprint_20m_s to the Dependent box.
Move vo2_mlkgmin to the Independent(s) box.
Click Statistics…
- ✓ Estimates (regression coefficients) — checked by default
- ✓ Confidence intervals (at 95%)
- ✓ Model fit
- ✓ Descriptives (optional)
- Continue
Click Plots…
- Move *ZRESID (standardized residuals) to the Y axis.
- Move *ZPRED (standardized predicted values) to the X axis.
- ✓ Check Normal probability plot
- Continue
Click Save…
- ✓ Unstandardized Residuals (optional, useful for residual plots)
- Continue
OK

Y.5.2 Interpreting the output

SPSS produces four main output blocks for bivariate regression:

Y.5.2.1 Table 1: Model Summary

Model Summary
Model   R       R Square   Adjusted R Square   Std. Error of the Estimate
1       .643a   .414       .404                .274

a. Predictors: (Constant), vo2_mlkgmin

R = .643: The multiple correlation coefficient (= |Pearson’s \(r\)| in bivariate regression).
R Square = .414: 41.4% of the variance in sprint time is explained by VO₂max.
Adjusted R Square = .404: R² adjusted for sample size and number of predictors (more relevant in multiple regression).
Std. Error of the Estimate = .274 s: Average distance between observed and predicted sprint times.

Y.5.2.2 Table 2: ANOVA

ANOVAa
Model              Sum of Squares   df   Mean Square   F        Sig.
1  Regression      3.067            1    3.067          40.97    .000b
   Residual        4.341            58   .075
   Total           7.408            59

a. Dependent Variable: sprint_20m_s
b. Predictors: (Constant), vo2_mlkgmin

F(1, 58) = 40.97, p < .001: The regression model is statistically significant — VO₂max significantly predicts 20-meter sprint time.
Sum of Squares Regression: Variance in sprint time explained by the model.
Sum of Squares Residual: Unexplained (residual) variance.

Y.5.2.3 Table 3: Coefficients

Coefficientsa
Model                 Unstandardized Coefficients    Standardized    t        Sig.   95% CI for B
                      B           Std. Error         Coefficients
                                                     Beta                           Lower    Upper
1  (Constant)         5.174       .219                               23.641   .000    4.736    5.612
   vo2_mlkgmin       -.033       .005               -.643            -6.401   .000   -.044    -.023

a. Dependent Variable: sprint_20m_s

Key values:

Element	Value	Meaning
B (Constant)	5.174	Intercept (\(a\)): predicted sprint time when VO₂max = 0 (not meaningful here)
B (vo2_mlkgmin)	−.033	Slope (\(b\)): for every 1 mL·kg⁻¹·min⁻¹ increase in VO₂max, sprint time decreases by 0.033 s
Beta (vo2_mlkgmin)	−.643	Standardized slope (equal to \(r\) in bivariate regression)
t (vo2_mlkgmin)	−6.401	t-statistic for the slope
Sig. (vo2_mlkgmin)	.000	p < .001 — slope is significantly different from zero
95% CI for B	[−.044, −.023]	Plausible range for the true slope

The regression equation:

\[\hat{y} = 5.174 + (-0.033) \times \text{VO}_2\text{max}\]

or equivalently:

\[\hat{y} = 5.174 - 0.033 \times \text{VO}_2\text{max}\]

Interpretation:

Slope (−0.033): For every additional 1 mL·kg⁻¹·min⁻¹ of VO₂max, predicted 20-m sprint time decreases by 0.033 seconds on average. The negative slope reflects the expected inverse relationship — fitter athletes sprint faster (lower time).
Intercept (5.174): The predicted sprint time for an athlete with a VO₂max of 0 is 5.174 s. This value is not meaningful in this context (no athlete has zero aerobic capacity) — do not over-interpret the intercept outside the data range.

Extrapolation

Do not use the regression equation to predict sprint times outside the observed range of VO₂max in this sample (approximately 27–57 mL·kg⁻¹·min⁻¹). Predictions beyond the observed data are unreliable.

Y.6 Part 4: Checking assumptions

Regression requires several assumptions to be met for results to be valid and generalizable.

Y.6.1 Assumption 1: Linearity

Check: Examine the scatterplot (Part 1) and the standardized residuals vs. standardized predicted values plot (ZRESID vs. ZPRED).

What to look for: Points should scatter randomly around zero in the residual plot — no curved pattern.

Y.6.2 Assumption 2: Homoscedasticity

Check: Same ZRESID vs. ZPRED plot.

What to look for: The vertical spread of points should be consistent across all values of ZPRED. A funnel shape indicates heteroscedasticity (variance changes with predicted values).

Y.6.3 Assumption 3: Normality of residuals

Check: The Normal Probability Plot (P-P plot) produced by SPSS.

What to look for: Points should fall approximately on the diagonal line. Systematic departures suggest non-normality.

Y.6.4 Assumption 4: Independence

Check: Study design. When using core_session.csv filtered to pre-training, each participant contributes one row — observations are independent. If participants appear in multiple time points, use appropriate repeated-measures methods.

Y.6.5 Assumption 5: No extreme outliers or influential points

Check: Inspect the scatterplot and the ZRESID vs. ZPRED plot for points far from the general pattern. In SPSS, you can save Cook’s Distance values (Save → Cook’s Distance) and examine them in the data file.

Interpreting the residual plot

A good residual plot shows random scatter around zero with no pattern, no funnel shape, and no extreme outliers. Any systematic pattern suggests an assumption violation.

Y.7 Part 5: Making predictions

Using the regression equation from SPSS:

\[\hat{y} = 5.174 - 0.033 \times \text{VO}_2\text{max}\]

Example: Predict 20-m sprint time for an athlete with a VO₂max of 45 mL·kg⁻¹·min⁻¹:

\[\hat{y} = 5.174 - 0.033 \times 45 = 5.174 - 1.485 = 3.689 \approx 3.69 \text{ s}\]

This prediction falls within the observed range of VO₂max in the dataset (~27–57 mL·kg⁻¹·min⁻¹), so it is a valid application of the model.

In SPSS, you can also save predicted values directly:

Analyze → Regression → Linear → Save…
✓ Check Unstandardized Predicted Values
Continue → OK

SPSS adds a new column (PRE_1) to your data file with the predicted value for each case.

Y.8 Part 6: Reporting results in APA style

Y.8.1 Correlation

Report \(r\), degrees of freedom, \(p\)-value, and \(r^2\):

“Aerobic capacity (VO₂max) was significantly and negatively correlated with 20-meter sprint time, \(r(58) = -.643\), \(p < .001\), \(r^2 = .414\), indicating that VO₂max accounted for 41.4% of the variance in sprint time.”

Note: Degrees of freedom for Pearson’s \(r\) = \(n - 2 = 58\).

Y.8.2 Regression

Report the regression equation, unstandardized slope with confidence interval, \(R^2\), and the model F-test:

“A bivariate linear regression was conducted to examine whether aerobic capacity (VO₂max) predicted 20-meter sprint time. The model was statistically significant, \(F(1, 58) = 40.97\), \(p < .001\), \(R^2 = .414\). VO₂max was a significant predictor of sprint time (\(b = -0.033\), 95% CI \([-0.044, -0.023]\), \(\beta = -.643\), \(p < .001\)), indicating that each additional mL·kg⁻¹·min⁻¹ of aerobic capacity was associated with a decrease of 0.033 seconds in 20-meter sprint time, on average.”

Y.8.3 APA formatting rules

Report \(r\) in lowercase italics: r
Report degrees of freedom in parentheses: r(6)
Use p < .001 when the p-value is very small (SPSS shows .000)
Report unstandardized (\(b\)) and standardized (\(\beta\)) coefficients
Include 95% confidence intervals for the slope
Always include \(R^2\) to convey practical (not just statistical) significance

Y.9 Part 7: Common mistakes and troubleshooting

Y.9.1 Mistake 1: Not examining the scatterplot first

Problem: Computing \(r\) without visualizing the data can miss nonlinear relationships, outliers, or heteroscedasticity.

Solution: Always produce a scatterplot before computing any statistics.

Y.9.2 Mistake 2: Reporting only the p-value

Problem: “The correlation was significant, \(p < .05\)” tells the reader almost nothing about the magnitude or practical importance of the relationship.

Solution: Always report \(r\), \(r^2\), and confidence intervals alongside significance tests.

Y.9.3 Mistake 3: Concluding causation from correlation

Problem: “Higher VO₂max causes faster sprint times because \(r = -.643\).”

Solution: Correlation only establishes association. Use cautious language: “VO₂max was associated with sprint time.” Causation requires experimental manipulation and control.

Y.9.4 Mistake 4: Extrapolating predictions

Problem: Using the regression equation to predict sprint times for participants with VO₂max values far outside the observed range (~27–57 mL·kg⁻¹·min⁻¹) in the dataset.

Solution: Restrict predictions to within the observed range of \(X\) in your sample.

Y.9.5 Mistake 5: Over-interpreting the intercept

Problem: Reporting the intercept as a meaningful finding (“the baseline sprint time is 5.17 s”).

Solution: The intercept is only meaningful when \(X = 0\) is within the observed data range and theoretically sensible. In most Movement Science contexts, it is just a mathematical anchor.

Y.10 Summary

This tutorial demonstrated how to:

Produce scatterplots to visualize bivariate relationships in SPSS
Compute Pearson’s \(r\) and test its significance using Analyze → Correlate → Bivariate
Conduct a bivariate regression using Analyze → Regression → Linear and interpret the Model Summary, ANOVA table, and Coefficients table
Check regression assumptions using residual plots and the Normal P-P plot
Make predictions using the regression equation
Report correlation and regression results following APA guidelines

Key takeaways from this example (VO₂max predicting 20-m sprint time, N = 60, pre-training):

\(r(58) = -.643\), p < .001 — a significant, moderate-to-strong negative relationship
\(R^2 = .414\) — VO₂max explained 41.4% of the variance in sprint time
Slope \(b = -0.033\) — each 1 mL·kg⁻¹·min⁻¹ increase in VO₂max predicts a 0.033-s decrease in sprint time
Always visualize before computing — scatterplots reveal what statistics cannot
\(r\) measures only linear relationships; nonlinearity yields misleading coefficients
Correlation does not imply causation — use cautious language
Check all five assumptions before trusting regression results

Next steps

Practice with your own Movement Science datasets: flexibility and balance scores, heart rate and RPE, or body composition and agility
Explore multiple regression (Chapter 12) to model outcomes from more than one predictor
Compare Spearman’s rank correlation when data are ordinal or assumptions are violated
Review Chapter 11 of the textbook for deeper conceptual coverage

Y.11 Additional resources

SPSS manuals: IBM SPSS Statistics Base documentation
APA Style (7th ed.): Guidelines for reporting statistical tests
Textbook website: Download practice datasets and syntax files

Questions or issues? Refer to the textbook’s online support forum or consult your instructor.