Effect Size and Confidence Intervals Explained for Your Dissertation (2026): Beyond the p-value

Your statistical software prints p = 0.03 and every instinct says: “significant — I can move on.” That instinct has derailed more dissertations than any methodological flaw. A p-value tells you only whether an observed pattern is likely under the null hypothesis; it says nothing about the magnitude of the effect or the precision of your estimate. Effect size and confidence intervals supply precisely that information, and since the American Psychological Association mandated their inclusion in the 7th edition of the Publication Manual, examiners expect to see them reported correctly alongside every inferential test.

This guide explains the logic behind each measure, provides the calculation formulas and worked numeric examples you need to apply them, and shows you the exact APA 7th-edition reporting syntax for your dissertation results chapter. It also addresses the misconceptions that trip up even experienced researchers — including the single most common misreading of a confidence interval.

Why p-values Alone Are Insufficient

The p-value answers a narrow question: “If the null hypothesis were true, how probable is a test statistic at least as extreme as the one I observed?” It does not answer any of the questions a dissertation examiner actually cares about: How large is the effect? Could it occur by chance in a smaller or larger sample? Is it large enough to matter in practice?

The sample-size dependency of p is the most consequential limitation. With a large enough sample, even a trivially small, practically meaningless difference will reach statistical significance. Conversely, a genuinely large effect may fail to reach p < .05 when the sample is small, solely because statistical power is low. The effect has not changed; only the test’s ability to detect it has. This means a non-significant result is not evidence of no effect — it may simply be evidence of an underpowered study.

Major journals and the APA have responded by requiring effect sizes and confidence intervals in all submitted manuscripts. The 7th edition of the APA Publication Manual (2020) states explicitly that “for each inferential test reported, provide information about effect sizes” (section 6.24). Examiners marking quantitative dissertations in 2026 apply the same standard. For a thorough treatment of how these requirements fit within a broader quantitative framework, see the guide to research paradigms and quantitative methodology on this site.

Cohen’s d: Standardised Mean Differences

Cohen’s d is the most widely used effect-size index for comparing two group means. It expresses the difference between means in units of standard deviation, making results comparable across studies that used different scales of measurement.

Formula

d = (M₁ − M₂) / SD_pooled

where:
  SD_pooled = √[ ((n₁ − 1)·s₁² + (n₂ − 1)·s₂²) / (n₁ + n₂ − 2) ]

For paired-samples designs, the denominator is the standard deviation of difference scores rather than the pooled within-group SD. The unbiased variant — Hedges’ g — applies a small-sample correction factor and is preferred when total N is below 20.

Cohen’s Conventional Benchmarks — and Their Limits

Jacob Cohen (1988) proposed the following reference points: a d of 0.2 is “small,” 0.5 is “medium,” and 0.8 is “large.” These values are reproduced in virtually every introductory statistics textbook. However, Cohen himself described them as benchmarks of last resort, to be used “only when no better basis for estimating the index is available.” More recent methodological scholarship (Lakens, Improving Your Statistical Inferences, 2022) argues forcefully that applying the benchmarks outside of their original context — Cohen derived them primarily from social science literature of the 1960s — creates circular reasoning and obscures whether an effect is meaningfully large in your specific domain.

For a dissertation, the appropriate standard is the effect size typical in your research literature, not Cohen’s generic benchmarks. Locate the three or four most comparable prior studies and report your d relative to theirs. If no precedent exists, the benchmarks remain defensible as an explicit, acknowledged heuristic.

Pearson r and R²: Correlation-Family Effect Sizes

When your design produces a correlation or a point-biserial correlation (e.g., continuous predictor and continuous outcome, or dichotomous group variable with continuous outcome), Pearson r is the natural effect-size index.

Cohen’s benchmarks for r are: small = 0.10, medium = 0.30, large = 0.50. Note that these benchmarks for r do not correspond to those for d: r = 0.50 and d = 0.80 are both “large” but cannot be derived from each other by a simple linear transformation. The two families of effect size measure different constructs and serve different designs.

R² (the coefficient of determination) expresses the proportion of variance in the outcome explained by the predictor. An R² of 0.09, for instance, means that 9% of the variance in the dependent variable is accounted for by the independent variable. In multiple regression, the equivalent is R² with the Wherry correction (adjusted R²) applied to account for the number of predictors and avoid optimistic inflation.

Eta-Squared, Partial Eta-Squared, and Omega-Squared for ANOVA

When you move beyond a two-group comparison into a one-way or factorial ANOVA, the standardised mean difference no longer summarises the effect in a single number. Variance-explained indices take over.

Eta-Squared (η²)

Eta-squared is calculated as SS_effect / SS_total. It represents the proportion of total variance in the dependent variable that is associated with the independent variable. While straightforward to compute from ANOVA output, η² has a known upward bias: it systematically overestimates the population effect because SS_total includes all sources of variance in the specific sample, including those attributable to other factors in a factorial design.

Partial Eta-Squared (η²_p)

Partial eta-squared adjusts the denominator to exclude variance attributable to other factors: η²_p = SS_effect / (SS_effect + SS_error). This makes it the standard index for factorial designs and the one most commonly output by SPSS. APA 7th edition explicitly recommends η²_p for factorial ANOVAs. Cohen’s benchmarks for partial eta-squared are: small = 0.01, medium = 0.06, large = 0.14.

Omega-Squared (ω²) and Epsilon-Squared (ε²)

Both omega-squared and epsilon-squared apply bias corrections that account for sample size and number of groups. The formula for ω² in a one-way design is:

ω² = (SS_effect − df_effect · MS_error) / (SS_total + MS_error)

Omega-squared is the preferred index when your sample is small (below 30 per cell) or when your results will feed into a meta-analysis, because its bias is substantially lower than η² or η²_p. The R package effectsize (Ben-Shachar et al., 2020) and the MOTE package by Buchanan et al. both compute ω² and its confidence interval directly from ANOVA output.

Odds Ratios and Risk Ratios for Binary Outcomes

When your dependent variable is binary (e.g., pass/fail, present/absent, yes/no), neither d nor η² applies. The odds ratio (OR) and risk ratio (RR) are the appropriate effect-size indices.

The odds ratio expresses the ratio of the odds of the outcome occurring in one group relative to another. An OR of 1.0 means equal odds; OR > 1.0 means the outcome is more likely in the first group; OR < 1.0 means it is less likely. Cohen’s benchmark for “medium” OR is approximately 2.5 (equivalent to d ≈ 0.5 under the logistic transformation), though here too, domain-specific reference values are more useful.

Important: odds ratios are not the same as relative risks, though the two are often conflated in student dissertations. When the baseline outcome is rare (below ~10%), OR and RR are numerically similar. When the outcome is common, OR exaggerates the relative risk, sometimes substantially. Choose the index that matches your research question and report the other in a footnote if transparency demands it.

For logistic regression, Nagelkerke’s R² is the most commonly reported pseudo-R² in social sciences, though it cannot be interpreted as variance explained in the same way as OLS R². Cox and Snell’s R² has a lower upper bound than 1.0, making Nagelkerke’s preferred for communication purposes.

Confidence Intervals: Definition, Correct Interpretation, and Reporting

A confidence interval (CI) provides a range of plausible values for an unknown population parameter given the observed sample data and the precision of the estimate. The 95% CI is by far the most commonly reported, but 90% and 99% intervals are used in specific contexts (see below on eta-squared).

The Correct Frequentist Interpretation

The most pervasive error in student dissertations — and in published research — is stating: “there is a 95% probability that the true parameter lies within this interval.” That statement is incorrect, and examiners in statistics-heavy fields will penalise it.

The correct interpretation is procedural, not probabilistic with respect to a single interval: if the same sampling procedure were repeated many times and a 95% CI were calculated from each sample, approximately 95% of those intervals would contain the true population parameter. The parameter itself is fixed — only the intervals vary across hypothetical replications. For any single computed interval, the true value either is or is not inside it; there is no probability attached to one specific interval.

This distinction was demonstrated empirically by Hoekstra et al. (2014, Psychonomic Bulletin & Review), who found that a majority of researchers — including those with advanced statistical training — endorsed at least one incorrect interpretation of a confidence interval when presented with six statements about it.

Width of the Interval and Precision

Interval width reflects estimation precision. A narrow CI (e.g., [4.1, 4.9]) means your sample has estimated the parameter with high precision; a wide CI (e.g., [−2.0, 11.0]) signals substantial uncertainty. Width is jointly determined by sample size (larger samples yield narrower intervals), variability in the data, and the chosen confidence level (99% intervals are wider than 95% for the same data).

For effect sizes, narrow CIs are particularly informative when the lower and upper bounds both fall on the same side of zero or the null value: such a result provides both statistical significance and practical precision. An effect-size CI that straddles zero remains consistent with “no effect” even if p < .05 — a scenario that arises when CI and NHST are computed by slightly different methods.

Reporting CIs for Eta-Squared: The 90% Exception

When reporting CIs for eta-squared and partial eta-squared, the APA Task Force on Reporting Statistics recommends 90% rather than 95% intervals. The technical reason: η²_p is a one-tailed quantity (it cannot be negative), and the 90% CI for a one-tailed test corresponds to the 95% CI for a two-tailed test. Using 95% CIs for η²_p can produce situations in which the CI excludes zero while the p-value exceeds .05 — an apparent contradiction that confuses readers.

For your dissertation, the rule is: report 95% CIs for d, r, R², and regression coefficients; report 90% CIs for η²_p and ω².

APA 7th-Edition Reporting Format with Templates

The APA Publication Manual (7th ed., 2020) specifies that statistics are italicised, followed by their values rounded to two decimal places, and that effect sizes and CIs always accompany their parent inferential test. Below are templates for the most common dissertation scenarios. For guidance on writing up the broader results chapter in which these statistics appear, see the step-by-step thesis results chapter guide, which covers how to organise findings, format tables, and present statistics before reporting individual tests.

Independent-Samples t-Test with Cohen’s d

t(df) = [value], p = [value], d = [value], 95% CI [LL, UL]

Example:
"An independent-samples t-test revealed a significant difference in
post-test scores between the intervention and control groups,
t(58) = 3.42, p = .001, d = 0.87, 95% CI [0.37, 1.36]."

One-Way ANOVA with Partial Eta-Squared

F(df_between, df_within) = [value], p = [value], η²p = [value], 90% CI [LL, UL]

Example:
"A one-way ANOVA indicated a significant main effect of teaching
method on examination scores, F(2, 87) = 8.14, p < .001,
η²p = 0.16, 90% CI [0.05, 0.27]."

Pearson Correlation

r(df) = [value], p = [value], 95% CI [LL, UL]

Example:
"A Pearson correlation indicated a moderate positive association
between study hours and grade point average,
r(98) = .43, p < .001, 95% CI [.26, .57]."

Binary Logistic Regression with Odds Ratio

b = [value], SE = [value], Wald χ²(1) = [value], p = [value],
OR = [value], 95% CI [LL, UL]

Example:
"Prior tutoring significantly predicted dissertation completion,
b = 0.89, SE = 0.31, Wald χ²(1) = 8.27, p = .004,
OR = 2.43, 95% CI [1.32, 4.48]."

Note: APA 7th edition requires the use of a zero before the decimal point for values that can exceed 1.0 (e.g., 0.87, not .87 — this changed from the 6th edition). For values that cannot exceed 1.0, such as p, r, and η², no zero is used (e.g., p = .001, r = .43). Verify this distinction in your institution’s dissertation guidelines, as some departments still follow the 6th-edition convention.

For the broader context of how to handle your citations and references in a dissertation, consult the site’s systematic literature review guide, which addresses how to identify prior effect-size estimates during your literature search.

Five Common Misinterpretations to Avoid

1. Treating a Significant Result as a Large Effect

Statistical significance and practical significance are independent. A two-tailed t-test on a sample of 2,000 can detect a d as small as 0.09 at α = .05 with 80% power. If your study uses a large administrative dataset, expect many tiny effects to be “significant.”

2. Treating a Non-Significant Result as Evidence of No Effect

Absence of evidence is not evidence of absence. A non-significant result may simply reflect insufficient power. Always report the effect size and CI regardless of p: an effect of d = 0.42, 95% CI [−0.03, 0.87], p = .07 tells a very different story from d = 0.04, 95% CI [−0.11, 0.19], p = .63.

3. The 95% Probability Fallacy

As discussed above, a 95% CI does not mean “95% probability the parameter is inside this interval.” The parameter is fixed; the procedure is what carries the 95% property. Restate this in the limitations section of your discussion chapter to pre-empt examiner queries.

4. Applying Cohen’s Benchmarks Uncritically

Cohen’s d = 0.2 might be a clinically meaningful effect in a pharmacological trial (where small drug effects on large populations translate to large health gains) and a negligible effect in a psychology experiment. Always anchor your interpretation to domain-specific benchmarks where they exist.

5. Confusing η² and η²_p When Using SPSS

By default, SPSS labels its factorial ANOVA output “Partial Eta Squared” in the Univariate output window, but the General Linear Model → Options tick-box says “Estimates of effect size.” Some students copy this value and report it as plain η² in their text, which is technically incorrect. Check which quantity your software computed and label it precisely in your dissertation.

Worked Examples: Three Dissertation Scenarios

Scenario 1: Psychology Experiment (Independent Samples)

A student testing whether mindfulness training reduces test anxiety compared with a control group collects scores on the State-Trait Anxiety Inventory (range 20–80) from two groups of 30 participants each.

• Intervention group: M = 38.4, SD = 9.2
• Control group: M = 44.1, SD = 10.3
• Pooled SD = √[((29 × 9.2²) + (29 × 10.3²)) / 58] = √[(2454.76 + 3076.01) / 58] ≈ 9.77
• Cohen’s d = (44.1 − 38.4) / 9.77 = 5.7 / 9.77 ≈ 0.58

Reported as: t(58) = −2.25, p = .028, d = 0.58, 95% CI [0.07, 1.09]. By Cohen’s benchmarks this is a medium effect, consistent with the mindfulness literature’s typical effect sizes on self-reported anxiety (Hofmann et al., 2010, meta-analytic d ≈ 0.55).

Scenario 2: Education Research (One-Way ANOVA)

A student compares three teaching formats (lecture-only, blended, fully online) on end-of-module grades (0–100) in a sample of 90 students (30 per cell).

• F(2, 87) = 6.53, p = .002
• SS_effect = 1820, SS_error = 12110, SS_total = 13930
• η²_p = 1820 / (1820 + 12110) = 1820 / 13930 ≈ 0.13

Reported as: F(2, 87) = 6.53, p = .002, η²_p = 0.13, 90% CI [0.03, 0.23]. Teaching format explained approximately 13% of the variance in grades, approaching Cohen’s “large” threshold for η²_p.

Scenario 3: Health Sciences (Logistic Regression)

A student models the probability of dissertation submission within the scheduled window (yes/no) as a function of weekly supervisor contact (hours per week, continuous predictor).

• b = 0.61, SE = 0.22, Wald χ²(1) = 7.70, p = .006
• OR = e^0.61 ≈ 1.84
• 95% CI for OR: [e^(0.61−1.96×0.22), e^(0.61+1.96×0.22)] = [1.19, 2.84]

Reported as: Each additional hour of weekly supervisor contact was associated with 1.84 times the odds of on-time submission, Wald χ²(1) = 7.70, p = .006, OR = 1.84, 95% CI [1.19, 2.84].

Once your results are reported correctly, the next task is interpreting them in your discussion chapter. For guidance on how to frame the significance of your effect sizes relative to prior literature — and where the boundary falls between the results chapter and the discussion — see the guide to discussion vs conclusion chapter differences.

If your dissertation includes a systematic review or meta-analysis strand, effect size reporting takes on additional importance because pooled estimates depend on consistent effect size measurement across studies. The best AI tools for systematic reviews and meta-analysis guide covers how platforms such as JASP and RevMan handle effect size pooling and forest plot generation.

Understanding the philosophical assumptions underlying quantitative designs is also essential context. The research paradigms guide covers how positivist assumptions underpin inferential statistics and why those assumptions shape how you interpret effect sizes.

Finally, if you are working on a qualitative strand alongside your quantitative analysis, the thematic analysis guide explains how qualitative findings can contextualise the effect sizes you report in your quantitative chapter.

Frequently Asked Questions