Appendix O — SPSS Tutorial: Confidence Intervals
Computing and interpreting confidence intervals for means, differences, and proportions
O.1 Overview
Confidence intervals (CIs) are essential for quantifying uncertainty and communicating the precision of estimates. While point estimates (e.g., sample means) provide single “best guess” values, confidence intervals acknowledge that sample-based estimates vary due to sampling error. A 95% confidence interval provides a range of plausible values for the population parameter, constructed such that if we repeated the study many times, approximately 95% of the intervals would contain the true population value.
SPSS provides confidence intervals automatically in many statistical procedures, but understanding how to access, interpret, and visualize them is critical for responsible statistical practice. This tutorial demonstrates:
- How to obtain 95% and 99% confidence intervals
- How to interpret CI width as a measure of precision
- How to use CIs for assessing statistical and practical significance
- How to create publication-ready error bar plots
Prerequisites: Familiarity with SPSS data entry, descriptive statistics, and basic data management.
O.2 Dataset for this tutorial
We will use the Core Dataset (core_session.csv), filtered to pre-training (N = 60) unless otherwise noted. Download it here: core_session.csv
Key variables used in this tutorial:
sprint_20m_s— 20-meter sprint time in secondsvo2_mlkgmin— VO₂max (mL·kg⁻¹·min⁻¹)group— training vs. control
O.3 Part 1: Confidence interval for a single mean
Let’s start with the most basic scenario: computing a 95% confidence interval for a population mean based on a single sample.
O.3.1 Example: Mean VO₂max at baseline
We measure aerobic capacity (vo2_mlkgmin) in 60 participants at baseline and want to estimate the population mean with a confidence interval.
O.3.2 Procedure: Using Explore
The simplest way to obtain a confidence interval for a mean is through the Explore procedure.
- Analyze → Descriptive Statistics → Explore…
- Move your variable (e.g.,
vo2_mlkgmin) to Dependent List - Click Statistics…
- ✓ Check Descriptives (if not already selected)
- Confidence Interval for Mean: 95% (default)
- Continue
- Display: Select Statistics
- OK
O.3.3 Interpreting the output
SPSS will produce a table labeled Descriptives that includes:
Descriptives
Statistic Std. Error
vo2_mlkgmin Mean 41.340 0.880
95% Confidence Lower Bound 39.579
Interval for Mean Upper Bound 43.101
Median 41.550
Variance 46.477
Std. Deviation 6.817
Minimum 26.9
Maximum 56.5
Range 29.6
Interquartile Range 9.8
Skewness -0.040 0.309
Kurtosis -0.563 0.608Key information:
- Mean = 41.34 mL·kg⁻¹·min⁻¹: Point estimate of population mean
- 95% CI [39.58, 43.10]: We are 95% confident that the true population mean VO₂max lies between 39.58 and 43.10 mL·kg⁻¹·min⁻¹
- Std. Error = 0.880: Standard error of the mean (SE = SD / √n = 6.817 / √60)
Report: “Mean VO₂max was 41.34 mL·kg⁻¹·min⁻¹, 95% CI [39.58, 43.10], based on 60 participants at pre-training.”
The interval width (3.52 mL·kg⁻¹·min⁻¹) reflects moderate precision given the relatively high variability (SD = 6.82).
O.3.4 Changing the confidence level
To compute a 99% confidence interval (wider, more conservative):
- Analyze → Descriptive Statistics → Explore…
- Statistics button
- Change Confidence Interval for Mean to 99
- Continue → OK
The 99% CI will be wider than the 95% CI because we are demanding greater certainty.
Example output (approximate):
- 95% CI: [39.58, 43.10] (width = 3.52)
- 99% CI: [38.99, 43.69] (width = 4.70)
Notice the interval is wider with 99% confidence, reflecting the trade-off between confidence level and precision.
- 95% CI is standard in most research.
- 99% CI is appropriate when greater certainty is required (e.g., safety-critical applications, exploratory research with high false-positive risk).
- 90% CI may be used in exploratory or preliminary studies where narrower intervals are acceptable.
O.4 Part 2: Confidence intervals for independent samples (two groups)
When comparing two independent groups (e.g., training vs. control), we want a confidence interval for the difference in means.
O.4.1 Example: Comparing sprint time between training and control groups
We compare 20-m sprint time (sprint_20m_s) between the training (n = 30) and control (n = 30) groups at pre-training.
O.4.2 Procedure: Independent-Samples T Test
SPSS produces confidence intervals for mean differences through the Independent-Samples T Test procedure.
- Analyze → Compare Means → Independent-Samples T Test…
- Move
sprint_20m_sto Test Variable(s) - Move
groupto Grouping Variable - Click Define Groups… and enter
controlandtraining - Continue → OK
O.4.3 Interpreting the output
SPSS will produce two tables:
Group Statistics:
Group Statistics
group N Mean Std. Deviation Std. Error Mean
sprint_20m_s control 30 3.811 .340 .062
training 30 3.772 .373 .068Independent Samples Test:
Independent Samples Test
t df Sig. Mean 95% CI of the Difference
Difference Lower Upper
sprint_20m_s Equal variances assumed 0.429 58 .669 .039 -.145 .224Key information:
- Mean difference = 0.039 s: Groups differ by less than 1/25 of a second on average
- 95% CI [−0.145, 0.224]: The CI includes zero, indicating no significant difference at α = 0.05
- p = .669: Consistent with the CI conclusion
The confidence interval tells us:
- Statistical significance: Because zero is in the interval, we conclude the two groups did not differ significantly at baseline.
- Effect magnitude: The true difference is plausibly anywhere from −0.145 s to +0.224 s — both very small.
- Precision: The interval width (0.369 s) is moderate; larger samples would narrow it further.
O.4.4 Reporting
“Sprint time at baseline did not differ significantly between control (M = 3.81 s, SD = 0.34) and training groups (M = 3.77 s, SD = 0.37), mean difference = 0.04 s, 95% CI [−0.15, 0.22], t(58) = 0.43, p = .669.”
O.5 Part 3: Confidence intervals for paired samples (within-subjects)
For paired or repeated-measures designs (e.g., pre-test vs. post-test), we use Paired-Samples T Test.
O.5.1 Example: Pre-post sprint time comparison
We compare sprint time at pre-training vs. post-training across all 55 participants with complete data.
O.5.2 Procedure: Paired-Samples T Test
- Analyze → Compare Means → Paired-Samples T Test…
- Select the two variables (e.g.,
sprint_preandsprint_post) and move them to Paired Variables - Click Options… → Continue
- OK
O.5.3 Interpreting the output
Paired Samples Statistics:
Paired Samples Statistics
Mean N Std. Deviation Std. Error Mean
Pair 1 sprint_pre 3.802 55 .365 .049
sprint_post 3.792 55 .402 .054Paired Samples Test:
Paired Samples Test
Mean Std. 95% CI of the Difference t df Sig.
Difference Deviation Lower Upper
Pair 1 sprint_pre - sprint_post .010 .158 -.033 .053 0.469 54 .641Key information:
- Mean difference = 0.010 s: Sprint time barely changed from pre to post
- 95% CI [−0.033, +0.053]: CI includes zero; change is non-significant
- The CI excludes practically meaningful change, confirming minimal intervention effect on sprint time across the full sample
“Sprint time did not change significantly from pre-training (M = 3.80 s, SD = 0.37) to post-training (M = 3.79 s, SD = 0.40), mean change = 0.01 s, 95% CI [−0.03, 0.05], t(54) = 0.47, p = .641.”
The confidence interval shows the true population change is plausibly between −0.033 and +0.053 seconds — practically negligible in either direction.
O.6 Part 4: Visualizing confidence intervals with error bars
Error bar plots are effective for displaying means with confidence intervals across groups or conditions.
O.6.1 Procedure: Creating error bar plots
- Graphs → Legacy Dialogs → Error Bar…
- Select Simple and Define
- Bars Represent: Select Confidence interval for mean
- Level (%): 95 (or desired confidence level)
- Move your continuous variable (e.g.,
JumpHeight) to Variable - Move your grouping variable (e.g.,
TrainingGroup) to Category Axis - OK
O.6.2 Example output
SPSS will produce a plot with:
- Group means displayed as points or bars
- Error bars extending ±95% CI from each mean
- Y-axis labeled with the variable name and units
- Non-overlapping error bars suggest a statistically significant difference (though formal testing is required).
- Wide error bars indicate high uncertainty/low precision.
- Narrow error bars indicate high precision (large sample, low variability).
Caution: Overlapping CIs do not necessarily mean “no difference.” Always examine the confidence interval for the difference, not just overlap of individual group CIs.
O.6.3 Enhancing plots for publication
To improve error bar plots:
- Double-click the chart to enter Chart Editor
- Elements → Show Data Labels (optional, for displaying exact means)
- Edit → Properties to customize colors, fonts, and axis labels
- Export as high-resolution image (File → Export)
O.7 Part 5: Confidence intervals for proportions
While SPSS does not provide confidence intervals for proportions through standard menus, you can compute them manually or use syntax.
O.7.1 Manual calculation example
Suppose 18 out of 50 recreational runners report an injury during a season.
Sample proportion: \(\hat{p} = 18/50 = 0.36\)
95% CI: \(\hat{p} \pm 1.96 \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
\[ 0.36 \pm 1.96 \times \sqrt{\frac{0.36 \times 0.64}{50}} = 0.36 \pm 0.133 = [0.227, 0.493] \]
Interpretation: We are 95% confident that the true injury rate among recreational runners is between 22.7% and 49.3%.
O.7.2 Using SPSS syntax for proportions
COMPUTE p = 18/50.
COMPUTE n = 50.
COMPUTE SE = SQRT((p * (1 - p)) / n).
COMPUTE CI_lower = p - 1.96 * SE.
COMPUTE CI_upper = p + 1.96 * SE.
EXECUTE.Add these commands to a syntax file and run to obtain CI bounds.
O.8 Part 6: Understanding factors affecting confidence interval width
O.8.1 Effect of sample size
To see how sample size affects CI width:
- Split your dataset into subsets of different sizes (e.g., n = 10, n = 30, n = 100)
- Compute 95% CIs for each subset using Explore
- Compare interval widths
Expected result: Larger samples produce narrower CIs.
O.8.2 Effect of confidence level
Compute CIs at different confidence levels (90%, 95%, 99%) for the same dataset:
- 90% CI: Narrowest
- 95% CI: Moderate width
- 99% CI: Widest
Trade-off: Higher confidence requires wider intervals.
O.8.3 Effect of variability
Compare CIs for variables with different standard deviations:
- Higher SD → wider CI (more sampling error)
- Lower SD → narrower CI (less sampling error)
- Sample size has the greatest practical impact on precision. Larger samples always improve precision.
- Confidence level is a choice reflecting desired certainty. 95% is standard but not mandatory.
- Variability is inherent in the population and cannot be controlled, but precise measurement tools can reduce it.
O.9 Part 7: Common mistakes and troubleshooting
O.9.1 Mistake 1: Interpreting CIs as “95% of data fall in this range”
Incorrect: “95% of participants have jump heights between 55 and 61 cm.”
Correct: “We are 95% confident that the population mean jump height is between 55 and 61 cm.”
The CI estimates the population parameter (mean), not the distribution of individual observations.
O.9.2 Mistake 2: Concluding “no difference” from overlapping CIs
Two groups with overlapping 95% CIs may still differ significantly. Always compute the CI for the difference using Independent-Samples T Test.
O.9.3 Mistake 3: Reporting only “significant” without the CI
Report: “Mean difference = 4.3 cm, 95% CI [2.5, 6.1], p < .001”
Not: “p < .001” alone.
O.9.4 Mistake 4: Using z-values instead of t-values for small samples
For small samples (n < 30), use the t-distribution (automatic in SPSS procedures). Using z-values (1.96) underestimates uncertainty.
O.10 Part 8: Exporting and reporting confidence intervals
O.10.1 APA-style reporting template
“[Group 1 description] (M = [mean], SD = [SD], n = [n]) [verb] [Group 2 description] (M = [mean], SD = [SD], n = [n]), mean difference = [diff], 95% CI [lower, upper], t([df]) = [t-value], p = [p-value].”
Example:
“Sprint time at baseline did not differ significantly between control (M = 3.81 s, SD = 0.34, n = 30) and training groups (M = 3.77 s, SD = 0.37, n = 30), mean difference = 0.04 s, 95% CI [−0.15, 0.22], t(58) = 0.43, p = .669.”
O.10.2 Creating a summary table
You can copy-paste SPSS output into Word or Excel, or create a custom table:
| Variable | n | Mean (SD) | 95% CI |
|---|---|---|---|
| VO₂max (mL·kg⁻¹·min⁻¹) | 60 | 41.34 (6.82) | [39.58, 43.10] |
| Sprint Time (s) — Control | 30 | 3.81 (0.34) | [3.68, 3.94] |
| Sprint Time (s) — Training | 30 | 3.77 (0.37) | [3.63, 3.91] |
| Sprint Difference | — | 0.04 | [−0.15, 0.22] |
Note. Pre-training time point.
O.11 Summary
This tutorial covered:
- Computing 95% and 99% CIs for single means using Explore
- Obtaining CIs for independent-samples comparisons using Independent-Samples T Test
- Computing CIs for paired-samples comparisons using Paired-Samples T Test
- Visualizing CIs using Error Bar plots
- Understanding how sample size, confidence level, and variability affect CI width
- Avoiding common misinterpretations of confidence intervals
Confidence intervals are indispensable for responsible statistical reporting. Always report CIs alongside point estimates to communicate uncertainty and precision honestly.
- Practice computing CIs on your own datasets
- Experiment with different confidence levels to see the trade-off between width and certainty
- Use error bar plots in presentations and manuscripts to visualize uncertainty
- Consult Chapter 9 of the textbook for deeper conceptual understanding of confidence intervals
O.12 Additional resources
- SPSS manuals: IBM SPSS Statistics Base documentation
- APA Style (7th ed.): Guidelines for reporting confidence intervals
- Cumming, G. (2014). Understanding the New Statistics: Comprehensive guide to confidence intervals and effect sizes
- Textbook website: Download practice datasets and syntax files
Questions or issues? Refer to the textbook’s online support forum or consult your instructor.