Writing Statistical Results — Write results with correct scientific/statistical language - Biostatistics

SECTION 1

Historical Context & Motivation

The practice of reporting statistical results in a standardized, precise manner did not arise overnight; it evolved across more than a century of methodological debate, replication crises, and editorial reform. In the late nineteenth century, researchers like Francis Galton and Karl Pearson began formalizing correlational and inferential techniques, yet there was no consensus on how to communicate the outputs of those techniques in published manuscripts. Early statistical reports were often narrative and qualitative, lacking the precision modern readers expect. As quantitative methods proliferated in medicine, psychology, and public health, the gap between what researchers computed and what they communicated widened—spawning confusion, irreproducibility, and outright misinterpretation.

1900 • Pearson's Chi-Square Test

Karl Pearson publishes the chi-square goodness-of-fit test, introducing formal test statistics into scientific literature. Early reports include test values but lack standardized formatting, leading to inconsistent interpretations across journals.

1925 • Fisher Formalizes p-Values

Ronald Fisher's Statistical Methods for Research Workers introduces the p-value as a continuous measure of evidence. Fisher proposes the 0.05 threshold as a convenient benchmark, though he never endorses the rigid binary interpretation that would later dominate.

1974 • APA Publication Manual Standardizes Reporting

The second edition of the American Psychological Association's Publication Manual begins codifying how to report test statistics, degrees of freedom, and p-values. These conventions gradually spread to biomedical and epidemiological journals.

1996 • CONSORT Statement for Clinical Trials

The Consolidated Standards of Reporting Trials (CONSORT) checklist is published, mandating that randomized controlled trial reports include confidence intervals, effect sizes, and transparent descriptions of statistical methods—shifting the emphasis from p-values alone to a richer reporting framework.

2016 • ASA Statement on p-Values

The American Statistical Association releases a formal statement clarifying common misinterpretations of p-values. This landmark document catalyzes a broader movement toward reporting effect sizes, confidence intervals, and Bayesian measures alongside traditional null-hypothesis significance testing.

Against this backdrop, a central question emerges: How should a researcher translate numerical output from statistical software into written prose that is precise, interpretable, and reproducible? Answering that question requires understanding both the conventions that journals enforce and the conceptual meaning behind the numbers. A misplaced parenthesis, an imprecise verbal claim about 'significance,' or a missing confidence interval can undermine an otherwise rigorous study. This lesson equips you with the language, formatting rules, and interpretive caution necessary to write statistical results that meet contemporary standards in biostatistics and clinical research.

SECTION 2

Core Principles of Statistical Reporting

Writing statistical results well is not merely a matter of copying numbers from software output; it demands adherence to a set of principles that ensure clarity, accuracy, and interpretive integrity. These principles reflect decades of editorial consensus across organizations such as the APA, ICMJE (International Committee of Medical Journal Editors), and STROBE (Strengthening the Reporting of Observational Studies in Epidemiology). Whether you are reporting a simple t-test or a complex multivariable regression, the same foundational ideas apply.

Precision in Language

Use exact statistical terminology. Say 'statistically significant' rather than 'significant' when referring to a test result, and avoid causal language unless the study design supports causal inference (e.g., randomized experiments). Always distinguish between 'association' and 'causation.'

Complete Reporting

Every inferential test should include the test statistic with degrees of freedom (if applicable), the exact p-value (not just p < 0.05), the effect size, and a confidence interval. Omitting any of these elements leaves readers unable to fully evaluate the finding.

Effect Sizes Over p-Values

A p-value tells you whether an effect is likely due to chance under the null hypothesis, but it does not tell you how large or clinically meaningful the effect is. Cohen's d, odds ratios, risk ratios, and R² all provide complementary information about magnitude.

Confidence Intervals for Uncertainty

A 95% confidence interval provides a range of plausible values for the population parameter. Reporting CIs communicates the precision of your estimate and allows readers to assess whether the effect is practically meaningful, not just statistically detectable.

Reproducibility and Transparency

State the statistical software used, the alpha level selected a priori, assumptions checked (e.g., normality, homogeneity of variance), and how missing data were handled. Transparency in methods enables other researchers to replicate your analysis.

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KEY TAKEAWAY

Think of writing statistical results like filing an engineering report on a bridge inspection: you would never write 'the bridge passed' without specifying the load tested, the deflection measured, and the safety margin. Similarly, writing p < 0.05 without the test statistic, degrees of freedom, effect size, and confidence interval is like declaring a bridge safe without providing the data. Complete reporting is the foundation of scientific trust.

SECTION 3

Anatomy of a Well-Written Statistical Result

A properly reported statistical result has a consistent internal structure regardless of the specific test used. The diagram below dissects the anatomy of a well-constructed results sentence, illustrating how each component—from the descriptive context to the inferential statistics—fits together to produce a clear and complete report.

A complete results sentence includes six elements: context, descriptive statistics, test statistic with degrees of freedom, exact p-value, effect size, and confidence interval. Omitting any of these components leaves the reader with an incomplete picture of the finding.

Notice that the example sentence in the diagram begins with a plain-English description of the direction and magnitude of the finding—this contextualizes the numbers for readers who may skim past the statistics. The parenthetical descriptives (means and standard deviations) let the reader gauge the raw data before encountering the inferential statistics. The test statistic is italicized by convention (e.g., t, F, r) when written in prose, degrees of freedom appear in parentheses immediately after, and the p-value uses a lowercase italic p without a zero before the decimal (since p cannot exceed 1.0). The effect size and confidence interval round out the sentence by quantifying how large the effect is and how precisely it was estimated.

SECTION 4

Formatting Conventions & Notation

Precise formatting is not pedantic ornamentation; it is the grammar of statistical communication. Just as misspelling a drug name in a medical chart can have serious consequences, misplacing a parenthesis in a statistical report can lead readers to misidentify the test, the degrees of freedom, or the significance level. The conventions below draw primarily from APA Style (7th edition) and common biostatistical journal requirements, though individual journals may have minor variations.

Test Statistic Notation Templates

INDEPENDENT-SAMPLES t-TEST

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

Where t is the test statistic (italicized), df is degrees of freedom, d is Cohen's d effect size, and CI denotes the confidence interval for the mean difference with lower limit (LL) and upper limit (UL).

ONE-WAY ANOVA

F(df₁, df₂) = value, p = .xxx, η² = value

Where F is the F-ratio, df₁ is the between-groups degrees of freedom, df₂ is the within-groups degrees of freedom, and η² (eta-squared) is the proportion of variance explained.

CHI-SQUARE TEST

χ²(df, N = n) = value, p = .xxx, V = value

Where χ² is the chi-square statistic, N is the total sample size (included because χ² is sensitive to sample size), and V is Cramér's V effect size.

LOGISTIC REGRESSION (ODDS RATIO)

OR = value, 95% CI [LL, UL], p = .xxx

Where OR is the odds ratio (exponentiated regression coefficient), the 95% CI is for the OR, and p is the Wald test p-value for that predictor. An OR > 1 indicates increased odds; an OR < 1 indicates decreased odds.

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Common Formatting Rules

Use a lowercase italic p with no zero before the decimal (e.g., p = .017, not p = 0.017). Report exact p-values to two or three decimal places; use p < .001 when p is very small. Round test statistics to two decimal places. Report means and standard deviations to one decimal place more than the measurement precision. Use square brackets for confidence intervals: 95% CI [2.3, 7.8]. Italicize all statistical symbols that represent test statistics or parameters: t, F, r, M, SD, N, n.

SECTION 5

Common Language Pitfalls & How to Avoid Them

Even researchers who format their statistics correctly often stumble on the verbal interpretation that surrounds the numbers. The following diagram and table highlight the most frequent errors in scientific writing and their corrected alternatives. Mastering these distinctions is what separates technically competent reporting from genuinely rigorous scientific communication.

Four common language pitfalls side by side with their corrected versions. Each incorrect phrasing reflects a subtle but consequential error: ambiguous use of 'significant,' inappropriate causal language, equating non-significance with no difference, and misinterpreting the p-value as a posterior probability.

Pitfall	Why It's Wrong	Better Phrasing
Using 'significant' without the qualifier 'statistically'	In everyday English, 'significant' means 'important.' Readers may conflate statistical significance with clinical or practical importance.	'The difference was statistically significant' or 'clinically meaningful' — choose explicitly.
Causal language from observational data	Observational designs cannot rule out confounding. Causal claims require randomized experiments or rigorous causal-inference frameworks.	Use 'associated with,' 'correlated with,' or 'predicted' rather than 'caused,' 'led to,' or 'produced.'
Claiming 'no effect' from a non-significant result	Failure to reject H₀ does not confirm H₀. The study may have been underpowered or the effect may be small but real.	'The difference was not statistically significant, t(df) = …, p = .xx. The confidence interval … suggests the true difference could range from … to ….'
Interpreting p as P(H₀ \| data)	The p-value is P(data \| H₀), not P(H₀ \| data). Confusing these is the prosecutor's fallacy applied to science.	'Under the null hypothesis, the probability of obtaining a result this extreme or more extreme was p = .xxx.'
Reporting only p < 0.05 instead of the exact value	A p of .049 and a p of .001 both satisfy < .05 but represent very different levels of evidence against H₀.	Always report the exact p-value (e.g., p = .034); use p < .001 only when the value is extremely small.

Summary of frequent language errors in statistical reporting

SECTION 6

Worked Example: Writing Results from a Clinical Trial

Suppose you conducted a randomized controlled trial comparing the effect of a new antihypertensive drug versus placebo on systolic blood pressure (SBP). You have 50 participants per group, and after 12 weeks you obtain the following output from your statistical software: treatment group M = 121.4 mmHg, SD = 8.3; placebo group M = 132.7 mmHg, SD = 9.1; t(98) = 6.52, p < .001, mean difference = 11.3 mmHg, 95% CI [7.9, 14.7]. Let's construct the results paragraph step by step.

Drafting a Results Paragraph for an Independent-Samples t-Test

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Step 1 — State the Purpose and Test Used

Begin with a sentence that orients the reader to the research question and the statistical test selected. This establishes the analytical framework before presenting numbers. For example: 'An independent-samples t-test was conducted to compare systolic blood pressure (SBP) between the treatment and placebo groups at 12 weeks.' Note the use of past tense (was conducted) and the explicit naming of both the test and the outcome variable.

Test and purpose clearly stated.
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Step 2 — Report Assumption Checks

Before presenting results, briefly note that assumptions were verified. You might write: 'Levene's test indicated no violation of the equal-variance assumption, F(1, 98) = 0.87, p = .35.' If assumptions were violated, state the corrective action taken (e.g., Welch's t-test).

Assumptions verified and reported.
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Step 3 — Present Descriptive Statistics

Provide the group means and standard deviations so readers can understand the raw data: 'Participants in the treatment group exhibited lower mean SBP (M = 121.4, SD = 8.3) compared with the placebo group (M = 132.7, SD = 9.1).' Note that the direction of the effect is stated in plain English before the numbers appear.

Descriptives: Treatment M = 121.4, SD = 8.3; Placebo M = 132.7, SD = 9.1
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Step 4 — Report the Test Statistic, p-Value, Effect Size, and CI

Now present the inferential statistics in a single clause: 'This difference was statistically significant, t(98) = 6.52, p < .001, d = 1.30, 95% CI [7.9, 14.7].' The effect size d = 1.30 indicates a very large effect by Cohen's conventions (small ≈ 0.2, medium ≈ 0.5, large ≈ 0.8). The 95% CI for the mean difference [7.9, 14.7] shows that the true reduction in SBP is estimated to lie between approximately 8 and 15 mmHg.

t(98) = 6.52, p < .001, d = 1.30, 95% CI [7.9, 14.7]
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Step 5 — Interpret in Context

Close with a sentence that interprets the finding in clinical or practical terms: 'These results suggest that the treatment produced a clinically meaningful reduction in systolic blood pressure, with the mean difference exceeding the 5 mmHg threshold generally considered clinically relevant for cardiovascular risk reduction.' Because this was a randomized experiment, causal language ('produced') is permissible.

Clinical interpretation provided with causal language justified by RCT design.

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Final Assembled Paragraph

An independent-samples t-test was conducted to compare systolic blood pressure (SBP) between the treatment and placebo groups at 12 weeks. Levene's test indicated no violation of the equal-variance assumption, F(1, 98) = 0.87, p = .35. Participants in the treatment group exhibited lower mean SBP (M = 121.4, SD = 8.3) compared with the placebo group (M = 132.7, SD = 9.1). This difference was statistically significant, t(98) = 6.52, p < .001, d = 1.30, 95% CI [7.9, 14.7]. These results suggest that the treatment produced a clinically meaningful reduction in systolic blood pressure.

SECTION 7

Strengths and Limitations of Reporting Conventions

Standardized statistical reporting conventions have dramatically improved the clarity and reproducibility of scientific communication, but they are not without limitations. Understanding both sides allows you to apply these conventions thoughtfully rather than mechanically.

Strengths	Limitations
Standardization enables readers to extract key information quickly, regardless of the specific journal or discipline.	Over-reliance on p-values can reduce nuanced findings to a binary 'significant/not significant' dichotomy, masking the actual evidence gradient.
Including effect sizes and CIs shifts focus from mere statistical significance to practical or clinical importance.	Effect size metrics (d, η², OR) differ across tests, making cross-study comparison complex when different metrics are reported.
Reporting assumption checks increases transparency and helps readers assess the validity of the analysis.	Strict formatting rules can feel formulaic and may discourage nuanced narrative interpretation of results.
Exact p-values allow readers to apply their own evidential thresholds rather than relying on the author's dichotomous conclusion.	Conventions vary across style guides (APA, ICMJE, Vancouver), creating confusion when researchers publish across disciplines.
Checklists (CONSORT, STROBE) reduce selective reporting and publication bias by mandating comprehensive disclosure.	Checklist compliance can become a box-ticking exercise if authors do not understand the rationale behind each item.

Strengths and limitations of current statistical reporting standards

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KEY TAKEAWAY

Reporting conventions are like a shared grammar: they enable communication but do not guarantee eloquence. A beautifully formatted result with a misinterpreted p-value is like a grammatically perfect sentence that says something factually wrong. The conventions give you the structure; understanding the underlying statistics gives you the meaning.

SECTION 8

Connection to Advanced Reporting Frameworks

The reporting conventions covered so far represent the standard frequentist framework. However, contemporary biostatistics increasingly embraces additional approaches that demand their own reporting vocabularies. Understanding these extensions positions you for advanced coursework in Bayesian analysis, meta-analysis, and precision-medicine analytics.

Feature	Standard Frequentist Reporting	Advanced/Bayesian Reporting
Evidence metric	p-value: P(data ≥ observed \| H₀)	Bayes factor (BF₁₀): ratio of evidence for H₁ vs. H₀; posterior probability of hypothesis
Interval estimate	95% confidence interval: if repeated many times, 95% of CIs would contain the true value	95% credible interval: there is a 95% probability the parameter lies in this interval, given the data and prior
Effect estimation	Point estimate with CI (e.g., d = 0.80, 95% CI [0.45, 1.15])	Posterior distribution: full probability distribution for the parameter, often summarized by median and 95% HDI
Reporting language	'The difference was statistically significant, t(df) = …, p = …'	'The data provided strong evidence for H₁ over H₀, BF₁₀ = 42.5, posterior median = …, 95% HDI […, …]'
Meta-analysis	Pooled effect with heterogeneity: d = 0.55, 95% CI [0.38, 0.72], I² = 32%	Bayesian meta-analysis: posterior predictive distribution for effect in a new study

Standard vs. advanced reporting frameworks

As the field moves toward estimation-based and Bayesian reporting, the fundamental principles remain unchanged: be precise, be complete, and let readers evaluate the evidence on their own terms. Whether you report a p-value or a Bayes factor, the imperative to provide context, effect magnitude, and uncertainty quantification endures. Many journals now encourage or require registered reports, where the analysis plan (including how results will be written) is peer-reviewed before data collection—further underscoring that how you write results is inseparable from how you design the study.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A researcher writes: 'The treatment significantly improved patient outcomes (p < 0.05).' Identify at least three specific problems with this sentence and explain why each is problematic in the context of rigorous statistical reporting.

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First, 'significantly' is ambiguous—it should read 'statistically significantly' to distinguish from clinical or practical significance. Second, 'p < 0.05' is an inequality rather than the exact p-value; the exact value (e.g., p = .032) should be reported to allow readers to assess the strength of evidence. Third, no test statistic or degrees of freedom are provided (e.g., t(df) = …), leaving the reader unable to verify or interpret the test. Fourth, there is no effect size reported (e.g., Cohen's d, odds ratio), so the magnitude of improvement is unknown. Fifth, there is no confidence interval, so precision of the estimate is hidden. In sum, the sentence fails the 'complete reporting' principle by omitting most of the six essential elements of a well-written statistical result.

Question 2: BASIC CALCULATION

You conduct a chi-square test of independence with 2 rows and 3 columns, N = 200, and obtain χ² = 11.47, p = .003, Cramér's V = 0.24. Write a complete, correctly formatted results sentence for this analysis.

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A properly written result would be: 'A chi-square test of independence indicated a statistically significant association between [Variable A] and [Variable B], χ²(2, N = 200) = 11.47, p = .003, V = 0.24.' Note the key formatting elements: the degrees of freedom for a 2 × 3 table is (2 − 1)(3 − 1) = 2, which appears in parentheses after χ²; the total sample size N is included because χ² is sensitive to sample size; the exact p-value (.003) is reported without a leading zero; and Cramér's V provides the effect size. A V of 0.24 represents a small-to-medium effect by Cohen's benchmarks (small ≈ 0.10, medium ≈ 0.30 for df* = 1).

Question 3: INTERMEDIATE

A one-way ANOVA compares fasting glucose levels across three diet groups (n = 30 per group). The results are: F(2, 87) = 5.64, p = .005, η² = .115. Post-hoc Tukey HSD tests reveal that the low-carb group (M = 88.2, SD = 7.1) differs from the control group (M = 96.4, SD = 8.9, p = .003) but not from the Mediterranean group (M = 92.1, SD = 7.8, p = .18). Write the full results paragraph, including the omnibus test and post-hoc comparisons.

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A complete paragraph would read: 'A one-way ANOVA was conducted to compare fasting glucose levels across three diet groups (low-carb, Mediterranean, and control). There was a statistically significant difference among the groups, F(2, 87) = 5.64, p = .005, η² = .115, indicating that diet group accounted for approximately 11.5% of the variance in fasting glucose. Post-hoc comparisons using Tukey's HSD test revealed that the low-carb group (M = 88.2, SD = 7.1) had significantly lower fasting glucose than the control group (M = 96.4, SD = 8.9), p = .003. The difference between the low-carb group and the Mediterranean group (M = 92.1, SD = 7.8) was not statistically significant, p = .18. These results suggest that a low-carbohydrate diet may be associated with meaningfully lower fasting glucose levels relative to standard dietary patterns.' Note the reporting of η² as a proportion-of-variance effect size, the reporting of each post-hoc comparison with its own p-value, and the use of hedged language ('may be associated with') appropriate for interpreting differences among pre-existing diet groups.

Question 4: APPLIED

You are writing the results section of a manuscript reporting a logistic regression predicting hospital readmission (yes/no) from three predictors: age (continuous), comorbidity index (continuous), and discharge disposition (home vs. skilled nursing facility). The key predictor is discharge disposition: OR = 2.15, 95% CI [1.32, 3.50], p = .002. Age: OR = 1.03 per year, 95% CI [1.01, 1.05], p = .008. Comorbidity index: OR = 1.28 per unit, 95% CI [0.97, 1.69], p = .08. Model: χ²(3) = 24.87, p < .001, Nagelkerke R² = .14. Write the complete results paragraph.

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A well-constructed paragraph would read: 'A multivariable logistic regression was performed to examine the effects of age, comorbidity index, and discharge disposition on the likelihood of 30-day hospital readmission. The overall model was statistically significant, χ²(3) = 24.87, p < .001, Nagelkerke R² = .14, indicating that the predictors collectively explained approximately 14% of the variance in readmission likelihood. Patients discharged to a skilled nursing facility had significantly higher odds of readmission than those discharged home, OR = 2.15, 95% CI [1.32, 3.50], p = .002. Age was also a statistically significant predictor, with each additional year associated with a 3% increase in odds of readmission, OR = 1.03, 95% CI [1.01, 1.05], p = .008. The comorbidity index was not a statistically significant predictor in the adjusted model, OR = 1.28, 95% CI [0.97, 1.69], p = .08, although the confidence interval suggests that a clinically meaningful effect cannot be ruled out.' Note the interpretation of each OR, the reporting of the CI for each predictor, the interpretation of the non-significant predictor with reference to the CI rather than simply dismissing it, and the use of the model-level χ² test to establish overall model fit before discussing individual predictors.

Question 5: CRITICAL THINKING

A colleague shows you a draft results section that reads: 'Our analysis proved that the intervention had no effect on mortality (p = .07). The confidence interval for the hazard ratio was [0.62, 1.04].' Critique this passage on at least four grounds, referencing both statistical and linguistic principles. Then rewrite it correctly, incorporating all six elements of a complete results sentence.

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The passage contains at least four errors. First, 'proved' is far too strong: statistical analysis does not prove anything; it provides evidence. Second, 'no effect' conflates failure to reject the null hypothesis with confirmation of the null. The 95% CI [0.62, 1.04] spans values consistent with a meaningful protective effect (HR = 0.62) as well as a negligible harmful effect (HR = 1.04), so the data are ambiguous rather than conclusive. Third, the test statistic and degrees of freedom are missing. Fourth, the hazard ratio point estimate is not stated. A corrected version: 'A Cox proportional-hazards regression indicated that the intervention was not statistically significantly associated with mortality, HR = 0.81, 95% CI [0.62, 1.04], z = −1.82, p = .07. The confidence interval includes the null value of 1.0 but also encompasses hazard ratios consistent with a substantial protective effect, suggesting that the study may have been underpowered to detect a clinically meaningful reduction in mortality.' This revision avoids causal language (appropriate if this were observational), provides the HR, CI, test statistic, and exact p-value, and interprets the non-significant result with appropriate nuance.

SUMMARY

Lesson Summary

Summary

Writing statistical results with correct scientific language requires mastery of both formatting conventions and interpretive precision. Every inferential finding should be reported with six core elements: context and direction, descriptive statistics (means, standard deviations, or medians and interquartile ranges), the test statistic with degrees of freedom, the exact p-value (using a lowercase italic p with no leading zero), an effect size (Cohen's d, η², OR, etc.), and a confidence interval in square brackets.

Equally important is avoiding common language pitfalls: never use causal language from observational designs, always distinguish statistical significance from clinical importance, never interpret a non-significant p-value as evidence of no effect, and never describe the p-value as the probability that the null hypothesis is true. Whether reporting a simple t-test or a multivariable logistic regression, these principles—rooted in decades of methodological consensus from the APA, CONSORT, STROBE, and the ASA—ensure that your results are transparent, reproducible, and scientifically rigorous.